Thesis Submitted in Fulfilment of the Requirements for the Degree of Doctor of Philosophy

A Demazure character formula for the product monomial crystal

Joel Gibson

September 2020

School of Mathematics and Statistics Faculty of Science

The University of Sydney

A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy

Statement of originality

This is to certify that to the best of my knowledge, the content of this thesis is my own work. This thesis has not been submitted for any degree or other purposes.

I certify that the intellectual content of this thesis is the product of my own work and that all the assistance received in preparing this thesis and sources have been acknowledged.

Signed:

Joel Gibson

Abstract

The product monomial crystal was defined by Kamnitzer, Tingley, Webster, Weekes, and Yacobi for any semisimple simply-laced Lie algebra and a multiset of parameters. The crystal is closely related to the representation theory of truncated shifted Yangians, a family of algebras quantising transversal slices to Schubert varieties in the affine Grassmannian. In this thesis we give a systematic study of the product monomial crystal usingthe novel tool of truncations, resulting in a Demazure-type character formula which is valid in any symmetric bipartite Kac-Moody type. We establish results on stability of the crystal, and use these and the character formula to show that in type A the product monomial crystal is the crystal of a generalised Schur module associated to a column-convex diagram, as defined by Magyar, Reiner, and Shimozono.

Acknowledgements

Firstly I would like to express my sincere gratitude to my supervisor Oded Yacobi, for his guidance, encouragement, enthusiasm, and support throughout my candidature. His mentorship and guidance have helped me navigate the academic world, as well as become a better communicator. I consider myself very fortunate to have had his supervision and mentorship over the past four years. Thanks also go to my associate supervisor An- thony Henderson for some great discussions and direction to valuable references, and Geordie Williamson for a fantastic MSRI workshop and his infectious love of representation theory. I thank all the friends who have supported me over the last four years, both in the faculty and outside. Thanks to Kitty Chang and Joachim Worthington for showing me the ropes as a brand-new PhD student, Alex Casella and Dom Tate for the endless effort they put in to the postgraduate maths community at USyd, and Alex Kerschlfor being an excellent conference companion. The members of the representation theory group we’ve had at Sydney over the last few years have been nothing short of amazing, and particular thanks to Anna Romanov, Yusra Naqvi, and Emily Cliff for fostering such a sense of community here. Also thanks to my peers Joe, Josh, Giulian, Gaston, Kane, and Yee for being a great bunch of people. Between mathematical discussions, postgraduate solidatity, and impromptu performances of Gilbert and Sullivan, they’ve been an endless well of support and encouragement. Finally I’d like to thank my family, without whom none of this would have been possible. My partner Anneka has now seen me through both an Honours thesis and a PhD thesis, and has remained enthusiastic and encouraging through all of the ups and downs of a postgraduate degree. Her constant love, support, patience, and energy has been invaluable, and it is in a large part due to her that this thesis (working title ‘The Crystal Matrix’) is finally complete. My parents have been (as always) supporting and encouraging of me, and their advice has helped me slow down and enjoy myself. Thanks for encouraging and reinforcing both my love of science as well as my love of music — taking both in equal parts over the last four years has been a real treat. Finally, my sister Maddie has always believed in me, always reminded me to be proud of myself, and has been a constant irrepressible source of good vibes and ‘dank’ memes.

Contents

1 Introduction 11 1.1 Structure ...... 12

2 Notation 13

3 Lie Theory 15 3.1 Cartan data ...... 16 3.2 The Coxeter graph ...... 17 3.3 The Braid, Weyl, and Cactus groups ...... 18 3.4 Kac-Moody root data ...... 20 3.5 Kac-Moody algebras ...... 24 3.6 Quantum groups ...... 26

4 Crystals 31 4.1 Crystal bases of integrable modules ...... 31 4.2 The category of crystals ...... 34 4.3 Tensor product of crystals ...... 39 4.4 Recognition theorems ...... 40 4.5 Weyl group action ...... 41 4.6 Cactus group action ...... 42 4.7 Addendum ...... 43

5 Demazure modules and crystals 45 5.1 Formal Characters ...... 45 5.2 Demazure modules and the character formula ...... 47 5.3 Demazure crystals ...... 50 5.4 History: The Demazure character formula ...... 53

6 Monomial crystals 55 6.1 Nakajima’s Monomial Crystal ...... 55 6.2 A variation on the monomial crystal ...... 60 6.3 The product monomial crystal ...... 63 6.4 Labelling elements of the crystal ...... 65 6.5 A partial order ...... 66 6.6 Supports of monomials ...... 67

7 Truncations and the character formula 69 7.1 Truncations defined by upward sets ...... 69 7.2 A Demazure character formula ...... 71 7.3 Truncations are Demazure crystals ...... 73

8 Nakajima quiver varieties 77 8.1 Representations of quivers ...... 77 8.2 Moduli spaces of quiver representations ...... 78 8.3 Nakajima quiver varieties ...... 79 8.4 Vector bundles on quiver varieties ...... 81 8.5 Graded quiver varieties ...... 81

9 Contents

9 Generalised Schur modules 85 9.1 Generalised Specht modules ...... 85 9.2 Generalised Schur modules ...... 88 9.3 Flagged Schur modules ...... 91 9.4 Polynomial characters of GL푛 ...... 93 9.5 Type A truncations ...... 95

10 Stability of decomposition 99 10.1 Stability of restriction ...... 99 10.2 Application: decomposing a product ...... 102 10.3 Stability of the product monomial crystal for GL푛 ...... 103

Bibliography 107

10 1 Introduction

Let 퐺 be a simply-laced reductive group over the complex numbers ℂ, with Repℂ 퐺 the category of its finite- dimensional algebraic representations. This category is semisimple with simple objects 퐿(휆) indexed by dominant weights 휆, and an interesting problem is to define ‘natural’ constructions of these representations forany dominant weight. This problem has been very fruitful, with three such constructions realising 퐿(휆) as:

1. The space of sections Γ(퐺/퐵, ℒ휆) of a line bundle ℒ휆 on the flag variety 퐺/퐵, 2. The cohomology of a Nakajima quiver variety associated to the pair (퐺, 휆), generalising a previous construction of Ginzburg using Springer fibres in the case of 퐺 = GL푛, and 3. The intersection homology 퐼퐻(Gr휆) inside the dual affine Grassmannian Gr = 퐺∨((푧))/퐺∨[[푧]] of the spher- ical orbit Gr휆.

The fact that these three realisations all give rise to 퐿(휆) might be surprising, since the underlying geometric spaces are rather different, and consequently there has been some progress made to state relationships between these spaces. Throughout the papers [Kam+14; Kam+19a; Kam+19b] the authors investigate the relationship between the second and third realisations, establishing the fact that transverse slices in the dual affine Grass- mannian are symplectic dual to Nakajima quiver varieties. Throughout their study they investigate a family of 휆 휆 non-commutative deformations 푌휇 (R) of the coordinate ring of the transverse slice Gr휇 depending on an integral 휆 휆 set of parameters R, and define a category 풪(푌휇 (R)) of their representations. The sum 풱 (휆, R) = ⨁휇≤휆 풪(푌휇 (R)) carries a categorical (Lie 퐺)-action (in the sense of Chuang and Rouquier [CR04; Rou08]), making the complexi- fied Grothendieck group 푉 (휆, R) = 퐾ℂ(풱 (휆, R)) a representation of 퐺. This thesis concerns the representation 푉 (휆, R) and its crystal ℳ(휆, R), called the product monomial crystal after its embedding into Nakajima’s crystal of monomials. We give a novel new method of analysing the crystal ℳ(휆, R) by certain global truncations, which we use to give a Demazure-type character formula for the crystal, our first main result. Our second main result is specific totype A, where we show that the product monomial crystal is in fact the crystal of a previously-studied family of modules called the generalised Schur modules for column- convex diagrams, and give a bijection between the parameters R defining the crystal and the column-convex diagrams defining the modules. The column-convex diagrams include skew shapes as a special case, and so a corollary of our result is that for any skew Schur module there is a weight 휆 and a parameter multiset R such that 풱 (휆, R) is a categorification of the skew Schur module. This is the first such categorification of a skew-Schur module known to theauthor. Although the original setting started with a simply-laced reductive group 퐺, the product monomial crystal makes sense for any symmetric bipartite Cartan type, and we prove all results in this generality. The only obstruction to our results being valid in arbitrary symmetrisable bipartite types is Theorem 6.3.5, which is proved via the theory of Nakajima quiver varieties. If there were some alternative proof of this theorem valid in arbitrary symmetrisable bipartite type (and various computer experiments suggest it is true), the results within would automatically hold in that generality.

11 1 Introduction

1.1 Structure

This thesis is divided into three parts. The first part (Chapters 2 to 5) is concerned with setting up notation and reviewing the theory of Cartan and root data, Kac-Moody algebras, quantum groups, crystals, characters, Demazure modules, and Demazure crystals. The second part (Chapters 6 to 8) examines the product monomial crystal in-depth, developing the novel concept of truncations of the crystal and proving our first main result: the character formula. The third part (Chapters 9 and 10) relates the product monomial crystal in Type A to the generalised Schur modules, proves our second main result: the crystal of a column-convex generalised Schur module is given by the product monomial crystal. A more detailed reader’s guide is given below. The first part begins with Chapter 2, a very brief overview of the notation used throughout to which the reader can refer. In Chapter 3 we review the notion of a Cartan datum (퐼, ⋅) and a root datum Φ so as to put the representation theory of semisimple groups and reductive groups on equal footing with that of Kac-Moody algebras and their quantum analogues. We review the theory of crystals in Chapter 4, first reminding the reader of the original definition of a crystal base in terms of integrable modules over the quantum group, and then connect- ing it with that of an abstract crystal, which is much more common in the combinatorial-oriented literature. In Chapter 5 we define both Demazure modules and Demazure crystals, and state the main theorems concerning their characters. In Chapter 6 we introduce Nakajima’s crystal of monomials, and define the product monomial crystal as apar- ticular subset of the monomial crystal. After establishing some basic notation and results about this crystalwe move on to Chapter 7 where we introduce our truncations and give a character formula for each (Theorem 7.2.3), show that each truncation is a Demazure crystal (Theorem 7.3.7), and deduce a complete character formula for the product monomial crystal in finite type (Corollary 7.3.9). With this main result out of the way, we take a step backwards and introduce Nakajima quiver varieties in Chapter 8 so as to give proof of the fact that the product monomial subset is indeed a crystal (Theorem 6.3.5). In the third part we focus mainly on type A phenomena. Chapter 9 reviews the definition of a Schur functor and Schur module for any arrangement of boxes on a grid (a straightforward generalisation of the well-known definition in the case when the arrangement is a Young diagram), along with the the less well-known notionof a flagged Schur module. We show that the character of the product monomial crystal and the generalised Schur module agree when the 푛 in GL푛 is taken to be ‘large enough’. In Chapter 10 we prove some stability results about the product monomial crystal, showing that when the data R is held fixed and the underlying Cartan type increases in size the decomposition into highest weight crystals stabilises. Using this we can leverage the previous result to show that the product monomial crystal is the crystal of a generalised Schur module for all 푛, and as a consequence that 풱 (휆, R) categorifies a skew Schur module.

12 2 Notation

The blackboard bold letters ℤ, ℚ, ℝ, and ℂ denote (as usual) the integers, rationals, reals, and complex numbers respectively. We further define the natural numbers ℕ = {0, 1, 2, …}, the positive integers ℙ = {1, 2, 3, …}, and for any natural number 푛 ∈ ℕ we set [푛] = {1, 2, … , 푛}. The symmetric group of permutations of the set [푛] is denoted by 픖푛, and composition is written as composition of functions, so we have (using cycle notation) the equation (12)(23) = (123) inside 픖3. A monoid is the same thing as a group which may not have inverses, for instance (ℕ, +) is a monoid, but (ℙ, +) is not as it lacks an identity element. When 푅 is a commutative ring (always assumed to be unital) and 푀 is a monoid, we denote the unital monoid algebra by 푅[푀], also called the group algebra when 푀 is a group. When 푀 is commutative the algebra 푅[푀] will often be written in exponential notation, where 푅[푀] is free as an 푅-module on the basis {푒푚 ∣ 푚 ∈ 푀}, with multiplication on the basis given by 푒푚 ⋅ 푒푛 = 푒푚+푛. For example if 푀 = (ℕ, +) then the monoid algebra 푅[푀] is isomorphic to the polynomial ring 푅[푥] under the map 푒1 ↦ 푥. Just as a subset 푌 ⊆ 푋 may be regarded as an indicator function 푌 ∶ 푋 → {0, 1}, we define a multiset based in 푋 to be a function R∶ 푋 → ℕ. We will always denote multisets by boldface type, like R, S, T, Q. The value of R at the element 푥 ∈ 푋 is called the multiplicity of 푥 in R, written R[푥]. The support of a multiset is the subset Supp R = {푥 ∈ 푋 ∣ R[푥] > 0} ⊆ 푋. A multiset with finite support is called a finite multiset. If 푋 = {푥, 푦, 푧} then the notation R = {푥2, 푦} means that R is a multiset based in 푋, where 푥 has multiplicity 2, 푦 has multiplicity 1, and 푧 has multiplicity 0. If both R and S are multisets based in 푋, then their multiset union is the function R + S. We say S is a sub-multiset of R, denoted S ⊆ R, if S[푥] ≤ R[푥] for all 푥 ∈ 푋, and in this case we define their multiset difference to be the function R−S. When taking sums or products over multisets, they should be taken with multiplicity. For instance, if 푓 ∶ 푋 → 퐺 is a function from 푋 into an abelian group 퐺 (written multiplicatively), and R is a finite multiset R[푥] based in 푋, then the expression ∏푥∈R 푓 (푥) means ∏푥∈푋 푓 (푥) . A partial function 푓 ∶ 푋 99K 푌 between the sets 푋 and 푌 is a function 푓 ∶ 푋 → 푌 ⊔ {⊥} where ⊥ is a special element not belonging to 푌 . Alternatively a partial function may be viewed as a function which is only defined on a subset of its domain. If 푓 (푥) = ⊥ we say that 푓 is undefined at 푥, and the set {푥 ∈ 푋 ∣ 푓 (푥) ∈ 푌 } is called the domain of definition of 푓 . The remainder of the notation here is set up in Chapter 3, but we leave it here as a quick reference guide. The letter 퐼 will always refer to a Cartan datum, which is the data of a symmetrisable generalised Cartan matrix 퐴퐼 = [푎푖푗]푖,푗∈퐼 together with a particular integral choice of symmetrising matrix 퐵퐼 = [푖⋅푗]푖,푗∈퐼 . This data determines the abstract Weyl group (푊퐼 , 푆퐼 ), a Coxeter system with generators (푠푖)푖∈퐼 , and when 푊 = 푊퐼 is finite we denote the longest word by 푤퐼 . The vertices 푖, 푗 are adjacent, written 푖 ∼ 푗, if and only if 푎푖푗 < 0. The letter Φ will always refer to a root datum of type 퐼, which for us means a choice of weight lattice 푋(Φ) and ∨ ∨ coweight lattice 푋 (Φ) in a perfect pairing ⟨−, −⟩, along with simple roots (훼푖)푖∈퐼 and simple coroots (훼푖 )푖∈퐼 . The additive monoid of dominant weights is denoted 푋(Φ)+. We will use sans-serif font to denote common root data, for example GL푛 refers to an algebraic group, while GL푛 refers to a particular root datum of type A푛−1. A root datum determines a Kac-Moody algebra 픤(Φ), its universal enveloping algebra 푈 (Φ), an associated quantum group 푈푞(Φ), and also a combinatorial category of Φ-crystals. To each dominant weight 휆 we associate the highest- weight integrable representation 퐿(휆) of the Kac-Moody algebra, 퐿푞(휆) of the quantum group, and the crystal base int ℬ(휆) of 퐿푞(휆). The modules 퐿(휆) (resp 퐿푞(휆)) are the simple objects of the semisimple category 풪 (Φ) (resp. int 풪푞 (Φ)), and we only ever use the notation 퐿(휆), 퐿푞(휆), and ℬ(휆) when 휆 is dominant.

3 Lie Theory

Working with representations of semisimple groups, reductive groups, Kac-Moody algebras, or quantum groups combinatorially requires the introduction of a lot of notation — the Cartan matrix and Dynkin diagram, Weyl group, roots, coroots, and fundamental weights just to name a few. Since we need to introduce all of this notation anyway, we choose here to do it in a way which treats all of these cases uniformly, by introducing a Cartan datum and a root datum separately, from which we can build out a Kac-Moody algebra or quantum group (or Chevalley group scheme, although we do not cover that here). We briefly recall how the theory develops in the finite type case.

The finite-dimensional semisimple Lie algebras over ℂ (with fixed choice of splitting Cartan subalgebra) are classified up to isomorphism by their associated reduced root systems, in the sense of Bourbaki [BB02]. The classification of split reductive algebraic groups is more complicated, requiring not only the data of theunderlying root system but also a root datum, an embedding of the root system into a pair of dual ℤ-modules. We can further choose to base these data by choosing a Borel subgroup containing the split torus, in which case we need to only remember the dual ℤ-modules along with the simple roots and simple coroots.

The combinatorial data we have chosen to go with is more or less an extension of the notion of a based root datum to arbitrary symmetrisable type, a synthesis of the definitions given in Part I of[Lus10] and Chapter 7 of [Mar18]. Such a datum is made of two parts: the ‘core’ comprises a Cartan datum (playing the role of a root system), a symmetrisable generalised Cartan matrix together with a choice of symmetrisation which makes the square length of each simple root a multiple of 2. The ‘realisation’ part of the datum is analogous to a based root datum in the reductive group sense: a pair of dual ℤ-modules of finite rank, with a choice of simple roots and coroots whose pairing matrix is the generalised Cartan matrix, hence we call it a root datum. The ‘core’ determines the abstract Weyl group as well as the positive and negative parts of the associated Kac-Moody algebra and quantum group, wheras the ‘realisation’ part determines the maximal torus or Cartan subalgebra, as well as the full set of possible weights and coweights.

Many authors only treat Kac-Moody algebras or quantum groups using a specific realisation, rather than allowing it to vary. This does not affect much about the resulting representation theory, which mostly depends onlyon the underlying Cartan datum, but is often more convenient to permit more realisations. For example, while 픰픩푛 is a Kac-Moody algebra in the sense of [Kac90], the often-more-convenient 픤픩푛 is not. Another place we see the utility of allowing more general realisations is in Chapter 10 where we use arguments involving restriction to Levi subcrystals, since being able to have a Levi subgroup share the same set of weights as the larger subalgebra simplifies notation and arguments considerably.

These degree-zero modifications of Kac-Moody algebras are undoubtedly well-known to experts, but itisquite difficult for a non-expert to check what parts of the theory they actually affect. We assure thereaderthatas long as they make the assumption that the simple roots and coroots of a root datum are taken to be linearly independent, virtually all of the well-known theory of Kac-Moody algebras and quantum groups goes through without a hitch (and in finite type this is automatic). Conversely, if either of those sets is dependent the theoryhas to be modified significantly, for example by replacing weight spaces as simultaneous eigenspaces with weight spaces as abstract gradings on a vector space, and forgetting the partial order on weights (which is no longer defined). At points of this chapter we will remark on where these theories diverge, but after this chapterwewill always take the linear independence, or regularity, assumption.

15 3 Lie Theory

3.1 Cartan data

We begin by defining the ‘core’ of our Lie-theoretic data,a Cartan datum. Such a datum is equivalent to a symmetrisable generalised Cartan matrix, together with a choice of symmetrisation making the square length of each simple root a positive multiple of 2 (see Remark 3.1.6). 3.1.1 Definition (Cartan data) A Cartan datum is a pair (퐼, ⋅) of a finite set 퐼 together with a symmetric bilinear form (−⋅−) ∶ ℤ[퐼]×ℤ[퐼] → ℤ, satisfying the two conditions

푖 ⋅ 푖 ∈ {2, 4, 6, …} for all 푖 ∈ 퐼, and 푖 ⋅ 푗 (3.1.2) 2 ∈ {0, −1, −2, …} for all 푖 ≠ 푗. 푖 ⋅ 푖

The 퐼 × 퐼 matrix 퐴 = [푎 ] defined by 푎 = 2 푖⋅푗 is called the Cartan matrix associated to (퐼, ⋅). For each subset 퐼 푖푗 푖푗 푖⋅푖 퐽 ⊆ 퐼, the restricted datum is (퐽, ⋅) with (− ⋅ −) restricted to ℤ[퐽] × ℤ[퐽].

The simplest example of a Cartan datum is the empty datum A0 = ∅, with the next simplest being the A1 datum 퐼 = {푖} with the bilinear form 푖⋅푖 = 2. A familiar example to most readers will be the type A푛 Cartan datum, where 퐼 = {1, … , 푛} and the bilinear form is given by

⎧2 if 푖 = 푗, 푖 ⋅ 푗 = −1 if |푖 − 푗| = 1, (3.1.3) ⎨ ⎩0 otherwise.

Since 푖 ⋅ 푖 = 2 for all 푖 ∈ 퐼, the Cartan matrix 퐴퐼 is identical to the pairing matrix [푖 ⋅ 푗]. For instance, if (퐼, ⋅) is the A4 Cartan datum we have 2 −1 ⎛ ⎞ −1 2 −1 [푖 ⋅ 푗] = [푎푖푗] = ⎜ ⎟ . (3.1.4) ⎜ −1 2 −1⎟ ⎝ −1 2 ⎠ Define a binary relation on 퐼 by declaring 푖 ∼ 푗 if and only if 푖 ⋅ 푗 < 0, in which case we say that 푖 is adjacent to 푗. This defines an undirected simple graph with vertex set 퐼, which we will later upgrade to a Coxeter graph and a Dynkin diagram. In the case of A4, the undirected graph is a path with four vertices:

Using the associated graph to a Cartan datum we can define the following adjectives. 3.1.5 Definition (Properties of Cartan data) Let (퐼, ⋅) be a Cartan datum. We say that the Cartan datum is: 1. Symmetric if 푖 ⋅ 푖 = 2 for all 푖 ∈ 퐼. (Equivalently, 푖 ⋅ 푗 = 푎푖푗 for all 푖, 푗 ∈ 퐼). 2. Simply-laced if it is symmetric, and 푖 ⋅ 푗 ∈ {0, −1} for all 푖 ≠ 푗. 3. Irreducible if its associated graph is connected. 4. Bipartite if its associated graph is bipartite. 5. Finite type if the symmetric bilinear form (− ⋅ −) is positive-definite over ℝ. 6. Infinite type if it is not finite type. 7. Affine type if it is irreducible, not of finite type, and the symmetric bilinear form (− ⋅ −) is positive- semidefinite over ℝ.

It is easy to see that the A푛 Cartan datum satisfies the first four properties above. In addition, some computations with determinants show that the bilinear form of A푛 is positive-definite, and hence A푛 is finite type.

16 3.2 The Coxeter graph

If (Δ ⊆ 푉 ) is a reduced root system in the sense of Bourbaki [BB02], then we can associate to it a Cartan datum (퐼Δ, ⋅Δ) in the following way. If Δ is irreducible then there exists a unique inner product (−, −) on 푉 which is Weyl-invariant, and such that the square length (훼, 훼) of the shortest root is 2. By classification of irreducible root systems there are at most two lengths of roots. In the simply-laced cases A푛, D푛, E6, E7, and E8 every root is short, while in the cases B푛, C푛, and F4 the long roots have square length 4, and for G2 the long roots have square length 6. We may then choose a simple system {훼푖 ∣ 푖 ∈ 퐼Δ} of roots, and define the associated Cartan datum (퐼Δ, ⋅Δ) with 푖 ⋅Δ 푗 = (훼푖, 훼푗). If Δ is reducible, then (퐼Δ, ⋅Δ) is defined by performing this process on each irreducible component. In fact, by the classification of affine Cartan matrices (given in Chapter 4of[Kac90]), we can see that all indecomposable finite type or affine type generalised Cartan matrices have only two square lengths, oneofwhich (2) can be taken to be 2, and the other will be 4, 6, or 8, with 8 occuring only in the case of A2 . 3.1.6 Remark (Symmetrisability) In the usual definition of a Kac-Moody algebra (for example given in[Kac90]), a generalised Cartan matrix is an integral matrix 퐴 ∈ Mat퐼 (ℤ) satisfying the three conditions 1. 푎푖푖 = 2 for all 푖, 2. 푎푖푗 ≤ 0 for all 푖 ≠ 푗, and 3. 푎푖푗 = 0 ⟺ 푎푗푖 = 0 for all 푖 ≠ 푗.

One then says that 퐴 is symmetrisable if there exists an invertible diagonal matrix 퐷 ∈ Mat퐼 (ℚ) and a symmetric matrix 퐵 ∈ Mat퐼 (ℚ) such that 퐴 = 퐷퐵.

When (퐼, ⋅) is a Cartan datum its associated Cartan matrix 퐴퐼 is always symmetrisable, with 퐷퐼 having entries 푑 = 2 , and 퐵 the matrix of the bilinear form (푖 ⋅ 푗). Conversely, when a generalised Cartan matrix 푖푖 푖⋅푖 퐼 퐴 is symmetrisable in the sense of Kac where 퐴 = 퐷퐵, it is possible to simultaneously re-scale 퐷 and 퐵 such that all diagonal entries of 퐵 are nonzero even integers, and by considering indecomposable components of 퐴 we may furthermore take those entries to be positive.

So a Cartan datum is precisely a symmetrisable generalised Cartan matrix 퐴퐼 with a particular choice of integral 퐵퐼 .

To conclude we remark that most definitions in this chapter rely only on the Cartan matrix 퐴퐼 of a Cartan datum (퐼, ⋅), with the only exception being the definition of the quantum enveloping algebra (which requires the integers 푖 ⋅ 푖 rather than just the ratios 푎푖푗).

3.2 The Coxeter graph

Given a Cartan datum (퐼, ⋅) we can label each edge 푖 ∼ 푗 of the associated graph by a number 푚푖푗 ∈ {2, 3, 4, 6, ∞}, producing what is called a Coxeter graph. 3.2.1 Definition (Coxeter matrix and graph)

The Coxeter matrix associated to the Cartan datum (퐼, ⋅) is the symmetric matrix (푚푖푗)푖,푗∈퐼 with diagonal entries 푚푖푖 = 1, and off-diagonal entries 푚푖푗 ∈ {2, 3, 4, 6, ∞} determined by the value of 푎푖푗푎푗푖 as given in the following table: 2 푎 푎 = 4(푖⋅푗) ≥ 4 3 2 1 0 푖푗 푗푖 (푖⋅푖)(푗⋅푗) (3.2.2) 푚푖푗 ∞ 6 4 3 2 The Coxeter graph associated to the Cartan datum (퐼, ⋅) is the same undirected graph defined before with 푖 ∼ 푗 iff 푎푖푗 < 0, where we also label the edge between 푖 and 푗 by 푚푖푗. (Note that for 푖 ≠ 푗 we have 푖 ≁ 푗 ⟺ 푚푖푗 = 2).

2 The table above can be remembered by the ‘equation’ cos2 (− 휋 ) = (푖⋅푗) . 푚푖푗 (푖⋅푖)(푗⋅푗)

17 3 Lie Theory

The irreducible finite type and affine type Cartan data follow the same classification of indecomposable generalised Cartan matrices (see Chapter 4 of [Kac90]), as each such Cartan matrix is symmetrisable, and proportional Cartan data determine identical Cartan matrices. If |퐼| = 0 or |퐼| = 1, then the Cartan datum is necessarily finite type. If |퐼| = 2, then a Cartan datum is of the form

푛 −푘 ( ) where 푛, 푚 ∈ {2, 4, …} , 푘 ∈ ℕ, and both 푛, 푚 divide 2푘. (3.2.3) −푘 푚

Since 푛, 푚 > 0, this matrix is positive-definite if and only if its determinant 푛푚−푘2 is positive, which is equivalent 2 to 4 푘 = 푎 푎 < 4, hence the following result. 푛푚 푖푗 푗푖 3.2.4 Lemma Let (퐼, ⋅) be a Cartan datum with Coxeter matrix (푚푖푗)푖,푗∈퐼 . For each pair 푖 ≠ 푗, the Coxeter entry 푚푖푗 is finite if and only if the restricted rank-two Cartan datum ({푖, 푗} , ⋅) is finite type.

Interestingly, out of all the finite and affine indecomposable Cartan matrices, the only occurrences of 푚푖푗 = ∞ is (1) (2) for the untwisted affinisation A1 and twisted affinisation A1 of A1.

3.2.5 Example (The B4 Cartan datum) Taking the B4 type root system and performing the process outlined in the previous chapter, we get the B4 Cartan datum 퐼 = {1, 2, 3, 4} with the symmetric bilinear form and Cartan matrices

2 −2 2 −2 ⎛ ⎞ ⎛ ⎞ −2 4 −2 −1 2 −1 [푖 ⋅ 푗] = ⎜ ⎟ , [푎푖푗] = ⎜ ⎟ . (3.2.6) ⎜ −2 4 −2⎟ ⎜ −1 2 −1⎟ ⎝ −2 4 ⎠ ⎝ −1 2 ⎠

The associated Coxeter matrix is 2 4 ⎛ ⎞ 4 2 3 [푚푖푗] = ⎜ ⎟ , (3.2.7) ⎜ 3 2 3⎟ ⎝ 3 2⎠ which makes the associated Coxeter graph the labelled path on four vertices: 4

(When an edge has a weight 푚푖푗 = 3, it is typical to leave that edge unlabelled).

3.3 The Braid, Weyl, and Cactus groups

The Weyl group can be presented by generators-and-relations, independent of any realisation as a linear trans- formation on the weight or coweight lattice, hence the name abstract Weyl group. 3.3.1 Definition (Abstract Weyl group)

The abstract Weyl group 푊퐼 corresponding to a Cartan datum (퐼, ⋅) is the group generated by the elements 푆퐼 = {푠푖, 푖 ∈ 퐼} subject to the relations 2 푠푖 = 1 for all 푖 ∈ 퐼, ⏟⏟⏟⏟⏟⏟⏟푠푖푠푗푠푖 ⋯ = 푠⏟⏟⏟⏟⏟⏟⏟푗푠푖푠푗 ⋯ for all 푖 ≠ 푗 with 푚푖푗 < ∞. (3.3.2)

푚푖푗 letters 푚푖푗 letters

The second set of relations are called the braid relations. Note that both sets of relations can be compactly 푚 written as (푠푖푠푗) 푖푗 = 1 for all 푖, 푗 ∈ 퐼 satisfying 푚푖푗 < ∞, presenting (푊퐼 , 푆퐼 ) as a Coxeter system.

18 3.3 The Braid, Weyl, and Cactus groups

We recall some properties of the Coxeter system (푊퐼 , 푆퐼 ), referring the reader to [Hum90] for further details:

1. A reduced expression for 푤 ∈ 푊퐼 is a word (푖1, … , 푖푘) with letters in 퐼 such that 푤 = 푠푖1 ⋯ 푠푖푘 , and 푘 is as small as possible. 2. The length 푙(푤) is the length of any reduced expression for 푤. 3. The Bruhat ordering on 푊퐼 is 푢 ≤ 푤 whenever 푢 ∈ 푊 can be obtained as a subexpression of a reduced expression of 푤. 4. The Bruhat ordering is compatible with the length function: 푢 ≤ 푤 implies 푙(푢) ≤ 푙(푤). 5. The group 푊퐼 is finite if and only if (퐼, ⋅) is finite type. 6. If 퐽 ⊆ 퐼 is of finite type, then 푊퐽 ⊆ 푊퐼 is a finite group and contains a unique element 푤퐽 of maximum length, called the longest word. If 퐼 itself is finite type, we write 푤퐼 (rather than 푤∘ as is sometimes done) for the longest word. 7. If 퐽 ⊆ 퐼 is of finite type, then the automorphism 휔퐽 ∶ 푊퐽 → 푊퐽 defined by 푥 ↦ 푤퐽 푥푤퐽 induces a bijection on the set {푠푗 ∣ 푗 ∈ 퐽} of generators. (This can be seen since 푙(푤퐽 푥) = 푙(푤퐽 ) − 푙(푥) for any 푥 ∈ 푊퐽 ). We also write 휔퐽 ∶ 퐽 → 퐽 for the induced bijection on the set 퐽.

It can be shown that the Weyl group (or rather a finite covering on it) acts on any integrable representation ofa Kac-Moody algebra. This fails in the case of quantum groups, and one has to look to the braid group instead to find an action on integrable representations. 3.3.3 Definition (Braid group)

The Braid group 퐵퐼 corresponding to a Cartan datum (퐼, ⋅) the group generated by the elements 휎푖, 푖 ∈ 퐼 subject to the relations ⏟⏟⏟⏟⏟⏟⏟⏟⏟휎푖휎푗휎푖 ⋯ =⏟⏟⏟⏟⏟⏟⏟⏟⏟ 휎푗휎푖휎푗 ⋯ for all 푖 ≠ 푗 with 푚푖푗 < ∞. (3.3.4)

푚푖푗 letters 푚푖푗 letters

There is a natural surjection 퐵퐼 ↠ 푊퐼 of the braid group onto the Weyl group, defined on generators by 휎푖 ↦ 푠푖.

The map 휔퐽 ∶ 퐽 → 퐽 acts as an automorphism of the induced Coxeter subgraph of 퐽, since conjugation preserves the order of a pair of simple generators. This automorphism can be used to define the cactus group. Here we follow the definition from Section 5Bon16 of[ ]. 3.3.5 Definition (Cactus group)

The cactus group 퐶퐼 corresponding to a Cartan datum (퐼, ⋅) is defined as the group generated by 휏퐽 , where 퐽 ⊆ 퐼 is a connected subgraph of finite type, modulo the relations 2 1. 휏퐽 = 1, 2. 휏퐽 휏퐾 = 휏퐾 휏퐽 if there are no edges between the induced subgraphs 퐽 and 퐾, and

3. 휏퐾 휏퐽 = 휏퐽 휏휔퐽 (퐾), whenever 퐾 ⊆ 퐽.

There is a natural surjection 퐶퐼 ↠ 푊퐼 , defined on the generators by 휏퐽 ↦ 푤퐽 . 2 Of course for each disjoint 퐽, 퐾 of finite type we have 푤퐽 = 1 and 푤퐽 푤퐾 = 푤퐾 푤퐽 , and if 퐾 ⊆ 퐽 then the longest word of 휔퐽 (퐾) is 푤퐽 푤퐾 푤퐽 . Hence 퐶퐼 ↠ 푊퐼 is indeed a well-defined map of groups, surjective since the 휏{푖} are mapped to the simple generators 푠푖.

19 3 Lie Theory

3.4 Kac-Moody root data

We now proceed to the second part of our Lie-theoretic data, the ‘realisation’ of the Cartan datum inside a pair of dual ℤ-modules. 3.4.1 Definition (Kac-Moody root data) A root datum of Cartan type (퐼, ⋅) is a quintuple

∨ ∨ Φ = (푋(Φ), 푋 (Φ), ⟨−, −⟩, (훼푖)푖∈퐼 , (훼푖 )푖∈퐼 ), where (3.4.2)

1. 푋(Φ) and 푋 ∨(Φ) are free abelian groups of finite rank called the weight lattice and coweight lattice, 2. ⟨−, −⟩∶ 푋(Φ) × 푋 ∨(Φ) → ℤ is a perfect pairing, ∨ ∨ 3. The simple roots 훼푖 ∈ 푋(Φ) and simple coroots 훼푖 ∈ 푋 (Φ) are some pairwise distinct elements indexed by 퐼 such that for all 푖, 푗 ∈ 퐼 we have ⟨훼 , 훼∨⟩ = 2 푖⋅푗 = 푎 for all 푖, 푗 ∈ 퐼. 푗 푖 푖⋅푖 푖푗 We will assume in every chapter except this one that a root datum is regular, meaning that both the simple roots and the simple coroots are linearly independent. This is always the case if 퐼 is finite type (Corol- lary 3.4.5).

For each 퐽 ⊆ 퐼 we can form the restricted root datum Φ퐽 of Cartan type (퐽, ⋅), by keeping the same weight ∨ ∨ lattice 푋(Φ퐽 ) = 푋(Φ) and coweight lattice 푋 (Φ퐽 ) = 푋 (Φ), but remembering only those simple roots and coroots indexed by the set 퐽. In the special case 퐽 = {푖} for some 푖 ∈ 퐼, we will write Φ푖 = Φ{푖} for this restriction. ∨ A weight 휆 ∈ 푋(Φ) is a dominant weight if ⟨휆, 훼푖 ⟩ ≥ 0 for all 푖 ∈ 퐼. The set 푋(Φ)+ ⊆ 푋(Φ) of dominant weights form a monoid under addition. Provided the simple roots are linearly independent there is a partial order defined on 푋(Φ) by 휇 ≤ 휆 ⟺ 휆 − 휇 ∈ ∑푖∈퐼 ℕ훼푖. All definitions in this chapter can be made without the regularity assumption on Φ, although the properties of the definitions are sometimes undesirable — for example, ≤ would only be a preorder (lacking the partial order axiom that if 푎 ≤ 푏 and 푏 ≤ 푎, then 푎 = 푏), the action of the Weyl group on the weight and coweight lattices may not be faithful, and the set of dominant weights turns out to be rather useless unless the coroots are linearly independent. The reason we do not rule these out completely from consideration is that they occasionally arise, for example by taking the derived subalgebra of an affine Kac-Moody algebra one arrives at a root datum which is not regular.

We remark that our definition of root datum differs from the term used in the literature on reductive algebraic groups. The two key differences are that our definition is only fixing achoiceof simple roots and coroots (and is more similar to a based root datum for an algebraic group), and the second difference is that our definition permits infinite type root data. Both kinds of root data are used for the same purpose, as some combinatorial data identifying a group scheme, or Kac-Moody algebra, or quantum group. However, ours also builds in a choice of simple/positive system.

We assure the reader that in the familiar cases of semisimple or reductive algebraic groups, they need not worry about regularity. 3.4.3 Lemma If the bilinear form (퐼, ⋅) is nondegenerate, then any root datum Φ of type (퐼, ⋅) is regular. 3.4.4 Proof Since (퐼, ⋅) is invertible as a matrix over ℚ, then so is the Cartan matrix 퐴퐼 . Thinking of the roots and coroots as linear maps 훼∨ ∶ ℤ[퐼] → 푋 ∨(Φ) and 훼 ∶ ℤ[퐼] → 푋(Φ), we see that the Cartan matrix is the matrix of the ∨ bilinear form 퐴(−, −) ∶ ℤ[퐼] × ℤ[퐼] → ℤ defined by 퐴(푖, 푗) = ⟨훼푗, 훼푖 ⟩. Since the Cartan matrix is invertible, the bilinear form 퐴 is nondegenerate, and hence both of 훼∨ and 훼 must be injective. In finite type the bilinear form is positive-definite (by definition) hence the Cartan matrix isinvertible.

20 3.4 Kac-Moody root data

3.4.5 Corollary (Finite-type root data are regular) If Φ is a root datum for the finite-type Cartan datum (퐼, ⋅), then Φ is regular. There is a category associated to a Cartan datum (퐼, ⋅), where the objects are root data of type 퐼 and the morphisms are defined as follows: 3.4.6 Definition (Morphism of root data) Let Φ and Ψ both be root data of Cartan type (퐼, ⋅).A morphism (푓 , 푔)∶ Φ → Ψ is a pair of adjoint maps

푓 푔 푋 ∨(Φ) −→푋 ∨(Ψ), 푋(Φ) ←−푋(Ψ), (3.4.7)

∨ ∨ ∨ such that ⟨휆, 푓 (휇)⟩Ψ = ⟨푔(휆), 휇⟩Φ for all 휇 ∈ 푋 (Φ) and 휆 ∈ 푋(Ψ), and such that 푓 (훼푖 ) = 훼푖 and 푔(훼푖) = 훼푖 for all 푖 ∈ 퐼. As adjoints under the perfect pairings, each of 푓 and 푔 uniquely determines the other. The directions of these maps can be remembered by the fact that a morphism Φ → Ψ should induce morphisms 픤(Φ) → 픤(Ψ) of Kac-Moody algebras, or 푈푞(Φ) → 푈푞(Ψ) of quantum enveloping algebras, therefore it is the map on coweights which should go in the same direction as the map of root data, as the coweights become the Cartan ∨ 0 ∨ subalgebra 픥(Φ) = 푋 (Φ) ⊗ ℂ or the group algebra 푈푞 (Φ) = ℚ(푞)[푋 (Φ)] respectively. The category of root data has initial and final objects, which we call simply-connected and adjoint root data respectively. 3.4.8 Example (Simply-connected and adjoint root data.) Fix a Cartan datum (퐼, ⋅). The simply-connected root datum of type (퐼, ⋅) is the root datum Φ where 푋 ∨(Φ) is the ∨ ∨ free ℤ-module with basis (훼푖 )푖∈퐼 , and 푋(Φ) = Homℤ(푋 (Φ), ℤ) its dual, and the simple roots 훼푖 determined ∨ by the condition ⟨훼푗, 훼푖 ⟩ = 푎푖푗. The simply-connected root datum is cofree, coadjoint, cotorsion-free, and initial in the category of (퐼, ⋅)-root data. The adjoint root datum of type (퐼, ⋅) is defined by interchanging the roles of simple roots and simple coroots in the above definition. The adjoint root datum is free, adjoint, torsion-free, andis final in the category of (퐼, ⋅)-root data.

For example, in the category of A푛−1 root data, the root datum of SL푛 is simply-connected and the root datum of PGL푛 is adjoint. Simply-connected root data are particularly convenient to deal with, since the weight lattice ∨ 푋(Φ) admits a basis of fundamental weights (휛푖)푖∈퐼 dual to the coroots: ⟨휛푖, 훼푗 ⟩ = 훿푖푗. This is useful in finite type, but if (퐼, ⋅) is not finite type then the simply-connected root datum (and the adjoint datum) may not be regular,a fact which we will now demonstrate. (1) ̂ The Cartan datum for the affine type A1 (also called 픰픩2) is 퐼 = {0, 1} with the bilinear form given by the matrix 2 −2 ( ) , (3.4.9) −2 2 hence the simply-connected root datum Φ has 푋(Φ) = ℤ2 = 푋 ∨(Φ) with the linearly independent coroots ∨ ∨ 훼0 = (1, 0) and 훼1 = (0, 1), but the linearly dependent roots 훼0 = (2, −2) and 훼1 = (−2, 2) = −훼0. In order to get a regular root datum, the rank of 푋(Φ) has to be taken to be 3 or more. In general, the rank of a regular root datum must always satisfy rank 푋(Φ) ≥ |퐼| + corank 퐴퐼 .

The last piece of theory we will discuss is the action of the abstract Weyl group 푊퐼 on the weight and coweight lattices. 3.4.10 Lemma (Weyl group action on weights and coweights) ∨ Let Φ be a root datum of type (퐼, ⋅). Then the simple reflections 푟푖 and 푟푖 defined by ∨ ∨ ∨ 푟푖 ∶ 푋(Φ) → 푋(Φ), 푟푖 ∶ 푋 (Φ) → 푋 (Φ), ∨ ∨ ∨ (3.4.11) 푟푖(휆) = 휆 − ⟨휆, 훼푖 ⟩훼푖, 푟푖 (휇) = 휇 − ⟨훼푖, 휇⟩훼푖 ,

2 푚푖푗 satisfy the relations 푟푖 = 1 and (푟푖푟푗) = 1, and hence define an action of the Weyl group 푊퐼 (Φ) on 푋(Φ) ∨ and 푋 (Φ). The pairing is 푊퐼 -invariant under this action, meaning ⟨푤휆, 푤휈⟩ = ⟨휆, 휈⟩ for all 푤 ∈ 푊퐼 .

21 3 Lie Theory

3.4.12 Proof 2 ∨ The property 푟푖 = 1 follows easily from the definition together with the fact that ⟨훼푖, 훼푖 ⟩ = 2. It is also ∨ straightforward to check that ⟨푟푖휆, 푟푖 휈⟩ = ⟨휆, 휈⟩ for each 푖 ∈ 퐼, and so it remains to check that the 푟푖 satisfy 푚 the braid relation (푟푖푟푗) 푖푗 = 1 for 푖 ≠ 푗 and 푚푖푗 ≠ ∞.

Define subspaces 퐾푖푗, 푅푖푗 ⊆ 푋ℝ(Φ) by

∨ ∨ 퐾푖푗 = ker⟨−, 훼푖 ⟩ ∩ ker⟨−, 훼푗 ⟩, 푅푖푗 = ℝ훼푖 + ℝ훼푗, (3.4.13)

so that 푋ℝ(Φ) = 퐾푖푗 ⊕ 푅푖푗. If 훼푖 and 훼푗 are linearly dependent with 푘푖훼푖 = 푘푗훼푗 for some nonzero integers ∨ ∨ 푘푖, 푘푗, then taking the pairing with 훼푖 gives that 2푘푖 = 푘푗푎푖푗, and the pairing with 훼푗 gives that 푘푖푎푗푖 = 2푘푗. Multiplying through shows that 푎푖푗푎푗푖 = 4 and hence 푚푖푗 = ∞, so there is nothing further to check in this case.

Assume now that 훼푖 and 훼푗 are linearly independent. Both 푟푖 and 푟푗 act by the identity on the vector space 퐾푖푗, and by the matrices −1 −푎푖푗 1 0 푟푖|퐾 = ( ) , 푟푗|퐾 = ( ) (3.4.14) 푖푗 0 1 푖푗 −푎푗푖 −1

in the (훼푖, 훼푗) basis. The product 푟푖푟푗 acts on 퐾푖푗 by the matrix

푎푖푗푎푗푖 − 1 푎푖푗 푟푖푟푗|퐾 = ( ) , (3.4.15) 푖푗 −푎푗푖 −1

2 which has characteristic polynomial 휒(푡) = 푡 + (2 − 푎푖푗푎푗푖)푡 + 1. Since 푎푖푗푎푗푖 ≤ 4, the polynomial has two 푎 푎 (perhaps equal) complex conjugate roots 휂, 휂 on the unit circle, whose real part is 푖푗 푗푖 − 1. Therefore: 2

2 1. If 푎푖푗푎푗푖 = 0 then 휂 = 휂 = −1 and 푟푖푟푗|퐾푖푗 is the scalar matrix −1, hence (푟푖푟푗) = 1. 2. If 푎 푎 = 1 then 휂, 휂 are rotations by ± 2휋 , hence (푟 푟 )3 = 1. 푖푗 푗푖 3 푖 푗 3. If 푎 푎 = 2 then 휂, 휂 are rotations by ± 휋 , hence (푟 푟 )4 = 1. 푖푗 푗푖 2 푖 푗 4. If 푎 푎 = 3 then 휂, 휂 are rotations by ± 휋 , hence (푟 푟 )6 = 1. 푖푗 푗푖 3 푖 푗

5. If 푎푖푗푎푗푖 = 4 then 휂 = 휂 = 1, but 푟푖푟푗|퐾푖푗 is not a scalar matrix so 푟푖푟푗 is not diagonalisable and hence has infinite order.

Checking this table with the one appearing in Definition 3.2.1, we that the map 푠푖 ↦ 푟푖 is indeed a group homomorphism 푊퐼 → Autℤ(푋(Φ)). The same argument with roots and coroots swapped works to show ∨ ∨ that 푠푖 ↦ 푟푖 is a group homomorphism 푊퐼 → Autℤ(푋 (Φ)). ∨ The representation 푊퐼 → Autℤ(푋(Φ)) is faithful if and only if 푊퐼 → Autℤ(푋 (Φ)) is faithful, since they are dual representations. If the root datum Φ is regular, then in fact both of these maps are isomorphisms. The proof above shows that if all simple roots are linearly independent, then the relations between 푟푖 and 푟푗 are precisely given by the Coxeter matrix, and in Chapter 1.3 of [Kum02] or 3.13 of [Kac90] it is further shown that when Φ is regular the 푟푖 satisfy the exchange condition, making them the generators of a Coxeter system. For example, (1) when Φ is the simply-connected root datum of affine type A1 , the representation 푊퐼 → Autℤ(푋(Φ)) is faithful despite Φ not being regular, due to the fact that the coroots are linearly independent.

However, without the assumption of regularity the representation 푊퐼 → Autℤ(푋(Φ)) may not be faithful. For (1) ∨ example, there is a type A1 root datum of rank 1, defined by taking any two elements satisfying ⟨훼0, 훼0 ⟩ = 2 ∨ ∨ and defining 훼1 = −훼0 and 훼1 = −훼0 . In that case, both 푟1 and 푟2 act as multiplication by −1 and hence we have 푟1푟2 = 1.

We conclude this section by giving a concrete example of the finite-type root datum SL3.

22 3.4 Kac-Moody root data

3.4.16 Example (The root datum SL3) The SL3 root datum is the simply-connected root datum associated to the Cartan type A2. The standard way 3 to construct this root datum is to take the weight lattice to be the quotient space 푋(SL3) = ℤ /(1, 1, 1), and ∨ 3 the coweight lattice as the subspace 푋 (SL3) = {(푎, 푏, 푐) ∈ ℤ ∣ 푎 + 푏 + 푐 = 0}, with the roots and coroots being given by ∨ 훼1 = (1, −1, 0) , 훼1 = (1, −1, 0) (3.4.17) ∨ 훼2 = (0, 1, −1) , 훼2 = (0, 1, −1).

Writing 휖푖 for the image of the 푖th coordinate vector in 푋(SL3), we can picture the realified weight lattice 푋 (SL ) as follows. ℝ 3 ∨ ℝ휖1 = ker⟨−, 훼2 ⟩

휖 훼1 1 훼2

ℝ휖2 휖2

휖3

∨ ℝ휖3 = ker⟨−, 훼1 ⟩

The heavy black lines are the reflecting hyperplanes ker⟨−, 훼⟩ for some coroot 훼∨, and the thin grey lines ∨ ∨ show the shifted hyperplanes {푥 ∈ 푋ℝ(SL3) ∣ ⟨푥, 훼 ⟩ ∈ ℤ} for some coroot 훼 . The points where the grey lines meet are the integral points 푋(SL3), the ℤ-span of the 휖푖 .

23 3 Lie Theory

3.5 Kac-Moody algebras

A root datum Φ should define three related objects: a group scheme 퐺(Φ), its Lie algebra 픤(Φ), and a quantum group 푈푞(Φ). For 퐼 of finite type, 퐺(Φ) would be the usual split reductive group scheme associated to the based root data Φ, but attempting to define 퐺(Φ) for Φ not of finite type would take us too far afield (in general itis an ind-scheme rather than a scheme), and we instead refer the interested reader to Chapters 7 and 8 of [Mar18]. Since we are working in characteristic zero however, the representation theory of the group 퐺(Φ) is reflected closely by its Lie algebra, which we can define in a straightforward way. These algebras are called Kac-Moody algebras after their independent discoverers Victor Kac and Robert Moody. We don’t intend to develop their theory here, instead we state definitions and results which can be found in [Kac90; Mar18; Kum02]. 3.5.1 Definition (Kac-Moody algebra) Let Φ be a root datum of Cartan type (퐼, ⋅). The Kac-Moody algebra 픤(Φ) is the Lie algebra over ℂ generated ∨ by the Cartan subalgebra 픥 ∶= 푋 (Φ)⊗ℂ and the Chevalley generators 푒푖, 푓푖 for 푖 ∈ 퐼, subject to the relations

[픥, 픥] = 0,

[ℎ, 푒푖] = ⟨훼푖, ℎ⟩푒푖 for ℎ ∈ 픥, [ℎ, 푓푖] = −⟨훼푖, ℎ⟩푓푖 for ℎ ∈ 픥, ∨ (3.5.2) [푒푖, 푓푗] = 훿푖푗훼푖 , 1+|푎 | (ad 푒푖) 푗푖 푒푗 = 0, for 푖 ≠ 푗, 1+|푎 | (ad 푓푖) 푗푖 푓푗 = 0, for 푖 ≠ 푗.

The last two relations are called the Serre relations.

The Chevalley involution is the map of Lie algebras 휔 ∶ 픤(Φ) → 픤(Φ) defined on generators by 휔(푒푖) = −푓푖, + − 휔(푓푖) = −푒푖 and 휔(ℎ) = −ℎ. Let 픫 (Φ) (resp. 픫 (Φ)) be the subalgebra generated by the 푒푖 (resp. 푓푖), then the direct sum of vector spaces 픤(Φ) = 픫−(Φ) ⊕ 픥(Φ) ⊕ 픫+(Φ) is called the triangular decomposition. The universal enveloping algebra is denoted 푈 (Φ), and inherits a triangular decomposition 푈 (Φ) = 푈 −(Φ) ⊗ 푈 0(Φ) ⊗ 푈 +(Φ) by the PBW theorem. Our definition differs slightly from that of Kac[Kac90]. Firstly we permit an arbitrary realisation Φ of the Cartan matrix, rather than taking one of smallest possible dimension |퐼| + corank 퐴퐼 . As we have remarked previously, provided that we always work with regular root data then all the theory remains virtually the same. Secondly our definition of Cartan datum forces 퐴퐼 to be symmetrisable, so we are always working with symmetrisable Kac- Moody algebras. Thirdly we have defined the algebra by the Serre relations, rather than a quotient of the algebra generated by the first four relations by a certain maximal ideal containing the Serre relations. The Gabber-Kac theorem (Theorem 9.11 of [Kac90]) states that when 퐴퐼 is symmetrisable the Serre relations generate this ideal, so we have preferred to go with the explicit relations for our presentation. One should also note that the subalgebras 픫+(Φ) and 픫−(Φ) depend only on the underlying Cartan datum (퐼, ⋅) rather than on Φ. We define representations and their weights as usual, with one exception: since 푋(Φ) is a ℤ-module of finite rank, our definition of weight is what is usually called an integral weight. This is not a problem for us as we wish to restrict to integrable representations, whose weights pair integrally with the coroots automatically anyway. 3.5.3 Definition (Representations of a Kac-Moody algebra)

A representation of 픤(Φ) is a vector space 푉 equipped with a Lie algebra homomorphism 휌푉 ∶ 픤(Φ) → 픤픩(푉 ), or equivalently a left 푈 (Φ)-module. For each weight 휆 ∈ 푋(Φ) the corresponding weight space of the representation 푉 is 푉휆 = {푣 ∈ 푉 ∣ ℎ푣 = ⟨휆, ℎ⟩푣 for all ℎ ∈ 픥(Φ)}.A weight vector is a nonzero vector in a weight space. Weight spaces for distinct weights intersect trivially, and when 푉 = ⨁휆∈푋(Φ) 푉휆 then 푉 is called a weight module. When a weight space 푉휆 is nonzero, we say that 휆 is a weight of 푉 . The category of weight representations is a monoidal category, with the monoidal structure and unit coming

24 3.5 Kac-Moody algebras

from the usual bialgebra structure on 푈 (Φ). The category of weight module of 픤(Φ) is far too large for our purposes, so we will narrow the set of objects we are considering in two separate ways. The first narrowing is to only consider modules in category 풪, meaning weight modules with finite-dimensional weight spaces, whose weights are ‘finitely bounded above’ in aprecise sense written below. The second narrowing is to the integrable modules, which have good symmetry properties with respect to the Weyl group, and can be ‘integrated’ (in a precise sense) to a representation of a Kac-Moody group. When we take the intersection of these we get the category 풪int(Φ), which is a remarkably similar category to the finite-dimensional representations of a semisimple Lie algebra, and is the category we will primarily use for the rest of the thesis. 3.5.4 Definition (Category 풪) Let the category 풪(Φ) be the full subcategory of 픤(Φ) modules 푉 for which: 1. 푉 is a weight module with finite-dimensional weight spaces, and 2. There exist finitely many weights 휆1, … , 휆푠 (depending on 푉 ) such that if 휇 is a weight of 푉 then 휇 ≤ 휆푘 for some 1 ≤ 푘 ≤ 푠. The first condition on Category 풪 means that we can equip it with a reflexive dual: let the graded dual of 푉 be 휔 ∗ 푉 = ⨁휆∈푋(Φ) 푉휆 , with the action (푥 ⋅ 푓 )(푣) = −푓 (휔(푥)푣) where 휔 is the Chevalley involution. This duality is reflexive (meaning (푉 휔)휔 ≅ 푉 ) by the finite-dimensionality of weight spaces, and the fact that 휔 is an involution. Composing with the Chevalley involution means we preserve the second property of 풪: without it, the graded dual would have weights bounded below rather than above. The second condition on Category 풪 means that the category is closed under taking tensor products. The tensor product of two arbitrary modules with finite-dimensional weight spaces may not have finite-dimensional weight spaces, since there could be infinitely many pairs of weights summing to the same weight in the product. The ‘bounded above’ condition on weights in 풪(Φ) implies that each sum must be finite, since for any two weights 휇, 휆 ∈ 푋(Φ) the set {휈 ∣ 휇 ≤ 휈 ≤ 휆} is finite. One could compare this condition to the algebraist’s definition of 푖 formal Laurent series, being a sum of the form ∑푖∈ℤ 푎푖푡 where 푎푖 is eventually zero for 푖 ≪ 0: this boundedness condition means the product of two such series is defined. 3.5.5 Warning When 픤(Φ) is a finite-dimensional semisimple Lie algebra, there is the similarly named “BGG Category 풪”, which we will write as 풪BGG(Φ). It is defined as all finitely generated weight modules which are locally 픫+(Φ)-finite, meaning that 푈 +(Φ)푣 is finite-dimensional for all 푣 ∈ 푉 (Chapter 1.1 of [Hum08]). It is a consequence of these axioms that 풪BGG(Φ) is a full subcategory of 풪(Φ), but in general the containment is strict. One way to see this strict containment is to notice that our category 풪(Φ) is closed under taking tensor products, while the BGG category is not. This can be seen even for 픤(Φ) = 픰픩2 by tensoring two Verma modules (objects of 풪BGG(Φ)) and noticing that the resulting module has weight spaces of unbounded dimension, and hence cannot be finitely generated. One reason for introducing the category 풪(Φ) is to have a category of 픤(Φ)-modules which is not ‘too large’, and still includes the highest-weight modules. 3.5.6 Definition (Highest-weight modules)

Let 푉 be a weight module. A primitive vector is a weight vector 푣휆 ∈ 푉휆 such that 푒푖푣휆 = 0 for all 푖 ∈ 퐼. If 푉 is generated by a primitive vector 푣휆, it is called a highest-weight module with highest-weight 휆. Amongst the highest-weight modules of highest weight 휆, there is a universal such module 푀(휆) such that 푀(휆) surjects onto any other highest-weight module of highest weight 휆. The module 푀(휆) is called a Verma module. One can easily argue using the triangular decomposition of 푈 (Φ) that every highest-weight module is a member of 풪(Φ). Conversely, every module in 풪(Φ) has a (possibly infinite) filtration by submodules such that successive quotients are highest-weight modules. We should remark at this point that the category 풪(Φ) is far from being semisimple.

25 3 Lie Theory

We now briefly discuss our other narrowing of the category of weight modules. 3.5.7 Definition (Integrable module)

A 픤(Φ)-module 푉 is integrable if it is a weight module, and 푒푖 and 푓푖 act locally nilpotently for all 푖 ∈ 퐼, 푁 푁 meaning that for any 푣 ∈ 푉 there exists an 푁 ≥ 0 such that 푒푖 푣 = 0 = 푓푖 푣 for all 푖 ∈ 퐼.

The weight module condition and local nilpotency of the 푒푖 and 푓푖 is equivalent to 푉 being a locally finite 픤(Φ푖) weight module for each 푖 ∈ 퐼, meaning 푉 is a union of finite-dimensional 픤(Φ푖) weight modules. Each subalgebra 픤(Φ푖) ⊆ 픤(Φ) is essentially an 픰픩2 with an enlarged toral subalgebra, and hence we can apply the theory of finite-dimensional 픰픩2 representations to 푉 . For example, the symmetry of weight spaces of finite-dimensional

픰픩2 representations implies that dim 푉휆 = dim 푉푠푖휆 for each 푖 ∈ 퐼. Since this works for all 푖, we get that dim 푉휆 = dim 푉푤휆 for all 푤 ∈ 푊퐼 .

The fact that 푉 is locally finite as a 픤(Φ푖)-module means that we can even define exponentials of the Chevalley generators. For each weight vector 푣 ∈ 푉휆, we have that exp(ℎ) ⋅ 푣 = exp(⟨휆, ℎ⟩)푣 for all ℎ ∈ 픥(Φ), and the local 푛 nilpotency means that exp(푒푖) ⋅ 푣 = ∑푛≥0 푒푖 푣/푛! is a finite sum (and similarly for exp(푓푖)), hence each generator of 픤(Φ) can be exponentiated, leading to an action of the associated Kac-Moody group on 푉 . We won’t have use for Kac-Moody groups here, but this explains the terminology ‘integrable’.

The category of integrable modules is not semisimple, and is distinct from 풪(Φ). For example, the adjoint representation is always an integrable module, but not a member of 풪(Φ) unless (퐼, ⋅) is finite type. On the other hand the Verma module 푀(0) = 푈 −(Φ) is not integrable, but clearly a highest-weight module generated by 1 ∈ 푈 −(Φ). Remarkably, when we intersect category 풪(Φ) with the integrable modules, we get a semisimple category which behaves very much like the category of finite-dimensional representations of a semisimple Lie algebra. 3.5.8 Theorem (Integrable highest weight modules and complete reducibility) The Verma module 푀(휆) admits a unique simple quotient 퐿(휆), which is integrable if and only if 휆 is a dominant1weight. Let 풪int(Φ) be the full subcategory of 풪(Φ) consisting of integrable modules. Then 풪int(Φ) is semisimple, and the set {퐿(휆) ∣ 휆 ∈ 푋(Φ)+} is a complete irredundant set of simple objects. The category 풪int(Φ) looks quite similar to the representations of a finite-dimensional semisimple algebra: it is semisimple, with simple objects indexed by dominant weights. Indeed if (퐼, ⋅) is finite type, then 풪int(Φ) is almost equivalent to the category of finite-dimensional 픤(Φ) representations, the difference being that a module in 풪int(Φ) might be an infinite direct sum of finite-dimensional modules.

3.6 Quantum groups

The ‘quantum group’ is a mythical object 픔(Φ) associated to a root datum Φ, to which one can nevertheless associate a Hopf algebra 푈푞(Φ) over the field ℚ(푞) of rational functions called a quantum enveloping algebra. Although as far as the author knows there is no agreement upon this mythical object, there is a fair amount of agreement on 푈푞(Φ), and the fact that a 푈푞(Φ)-module is called a ‘representation of a quantum group’.

int int There is a similar definition of a category 풪푞 (Φ) of 푈푞(Φ)-modules, which recovers the category 풪 (Φ) in the 푞 → 1 limit. The most remarkable thing we get from representations of quantum groups is the existence of a crystal base for each integrable highest-weight module 퐿푞(휆), an amazing combinatorial basis behaving well under tensor products, a phenomenon which is not possible when just using the Kac-Moody algebra. As before, we do not develop the theory here but merely state definitions and results, referring the reader to[Jos95; Lus10; HK12].

1In the classic treatment one would usually say that 퐿(휆) is integrable if and only if 휆 is a dominant integral weight, but since all weights for us are integral we have omitted this adjective.

26 3.6 Quantum groups

We need to first introduce the quantum analogues of integers, factorials, and binomial coefficients. Forany 푛 ∈ ℤ, define the quantum integer 푞푛 − 푞−푛 [푛] = ∈ ℚ(푞). (3.6.1) 푞 − 푞−1 A convenient way of remembering these numbers is that [0] = 0, [1] = 1, [2] = 푞 + 푞−1, and in general we have that for 푛 ≥ 1 the quantum integer [푛] is the character of the 푛-dimensional 픰픩2 representation. Together with [−푛] = − [푛] this completely defines the quantum integers, for example

[3] = 푞−2 + 1 + 푞2 and [−4] = −(푞3 + 푞 + 푞−1 + 푞−3). (3.6.2)

We define the quantum factorial for 푛 ∈ ℕ as a product of quantum integers:

푛 [푛]! = ∏ [푚] , (3.6.3) 푚=1 and the quantum binomial coefficient for all 푛 ∈ ℤ and 푟 ∈ ℕ:

! 푛 [푛] [ ] = . (3.6.4) 푟 [푟]! [푛 − 푟]!

The quantum integers, factorials, and binomial coefficients are all members of the subring ℤ[푞, 푞−1] ⊆ ℚ(푞) of Laurent polynomials with integer coefficients. When dealing with Cartan data which is not simply-laced (meaning that 푖 ⋅ 푖 > 2 for some 푖 ∈ 퐼), we need to (푖⋅푖)/2 introduce some scaled versions of these quantum numbers. Let 푞푖 = 푞 ∈ ℚ(푞), and let [푛]푖 be the quantum 2 −2 integer [푛] composed with the substitution 푞 ↦ 푞푖. For example, when 푖 ⋅ 푖 = 4, we have [2]푖 = 푞 + 푞 . We can now introduce the quantum enveloping algebra associated to a root datum Φ. 3.6.5 Definition (Quantum enveloping algebra)

The quantum enveloping algebra associated to Φ is the unital associative algebra 푈푞(Φ) over ℚ(푞) generated ∨ by the symbols 퐸푖, 퐹푖 for 푖 ∈ 퐼 and 퐾휇 for 휇 ∈ 푋 (Φ), subject to the following relations:

퐾0 is the unit, 퐾휇퐾훾 = 퐾휇+훾 , ⟨훼 ,휇⟩ 퐾휇퐸푖 = 푞 푖 퐸푖퐾휇, −⟨훼 ,휇⟩ 퐾휇퐹푖 = 푞 푖 퐹푖퐾휇, 퐾̃ − 퐾̃ 퐸 퐹 − 퐹 퐸 = 훿 푖 −푖 , (3.6.6) 푖 푗 푗 푖 푖푗 −1 푞푖 − 푞푖 푎 [푎] [푏] ∑ (−1) 퐸푖 퐸푗퐸푖 = 0 for all 푖 ≠ 푗, 푎+푏=1+|푎푗푖| 푎 [푎] [푏] ∑ (−1) 퐹푖 퐹푗퐹푖 = 0 for all 푖 ≠ 푗. 푎+푏=1+|푎푗푖|

[푛] 푛 The notation 퐸푖 means the quantum divided power 퐸푖 / [푛]푖 for any 푛 ≥ 0. As above, the symbol 푞푖 means (푖⋅푖)/2 푞 , and we set 퐾̃ = 퐾 ∨ . When 퐼 is symmetric, 푞 = 푞 and 퐾̃ = 퐾 ∨ , and each [푛] is just the usual 푖 (푖⋅푖/2)훼푖 푖 푖 훼푖 푖 quantum integer [푛].

Our notation for 푈푞(Φ) follows [Lus10], however our definition is slightly different. Rather than presenting 푈푞(Φ) with the Serre relations, Lusztig instead defines a certain bilinear form on the free ℚ(푞)-algebra generated by the 퐸푖, and takes the quotient by the radical of the bilinear form (and similarly for the 퐹푖). As in the case of Kac-Moody algebras, it is clear that the Serre relations vanish in this quotient, but it requires further careful work to show

27 3 Lie Theory that the Serre relations generate the radical. The statement that these two definitions are equivalent is Corollary 33.1.5 in [Lus10]. ̃ ∨ The scaled terms 퐾푖 are the correct analogue of the coroots 훼푖 ∈ 픥(Φ) in the Kac-Moody case, including the 푖⋅푖/2 quantum scaling 푞푖 = 푞 . They obey the relations

̃ 푎푖푗 ̃ ̃ −푎푖푗 ̃ 퐾푖퐸푗 = 푞푖 퐸푖퐾푖 and 퐾푖퐹푗 = 푞푖 퐹푖퐾푖, (3.6.7)

̃ 2 ̃ ̃ −2 ̃ ̃ and in particular we have the ‘quantum 픰픩2 relations’ 퐾푖퐸푖 = 푞푖 퐸푖퐾푖 and 퐾푖퐹푖 = 푞푖 퐹푖퐾푖. The action of 퐾푖 on ∨ ̃ ⟨휆,훼푖 ⟩ a weight vector 푣휆 ∈ 푉휆 is 퐾푖푣휆 = 푞푖 푣휆, and hence the fourth relation above gives that (퐸푖퐹푖 − 퐹푖퐸푖)푣휆 = ∨ [⟨휆, 훼푖 ⟩]푖 푣휆, which is perhaps how the fourth relation should really be remembered.

In order to define trivial representations, tensor products, and duals, we give a Hopf algebra structure on 푈푞(Φ), following [Lus10].

3.6.8 Definition (Hopf algebra structure on 푈푞(Φ))

The comultiplication on 푈푞(Φ) is the unique extension of

Δ(퐾휇) = 퐾휇 ⊗ 퐾휇,

Δ(퐸푖) = 퐸푖 ⊗ 1 + 퐾̃푖 ⊗ 퐸푖, (3.6.9)

Δ(퐹푖) = 퐹푖 ⊗ 퐾̃−푖 + 1 ⊗ 퐹푖

making Δ∶ 푈푞(Φ) → 푈푞(Φ) ⊗ 푈푞(Φ) a map of algebras, where the tensor product on the right is given the usual algebra structure (푎1 ⊗푏1)⋅(푎2 ⊗푏2) = 푎1푎2 ⊗푏1푏2. Notably, this comultiplication is not commutative.

The counit 휖 ∶ 푈푞(Φ) → ℚ(푞) is the unique map of algebras taking 퐸푖, 퐹푖 to 0 and each 퐾휇 to 1.

The antipode 푆 ∶ 푈푞(Φ) → 푈푞(Φ) is the unique anti-homomorphism of algebras such that

푆(퐸푖) = −퐾̃−푖퐸푖, 푆(퐹푖) = −퐹푖퐾̃푖, 푆(퐾휇) = 퐾−휇. (3.6.10) In direct analogy with the representations of a Kac-Moody algebra, we make the following definitions. 3.6.11 Definition (Representations of a quantum group)

A representation of 푈푞(Φ) is a ℚ(푞)-vector space 푀 equipped with an action making it a left 푈푞(Φ)-module. ⟨휆,휇⟩ ∨ The 휆 weight space of 푀 is 푀휆 = {푣 ∈ 푀 ∣ 퐾휇푣 = 푞 푣 for all 휇 ∈ 푋 (Φ)}, and when 푀 = ⨁휆∈푋(Φ) 푀휆 we say that 푀 is a weight module.

The category 풪푞(Φ) consists of weight modules with finite-dimensional weight spaces, such that the setof weights is bounded above by some finite set 휆1, … , 휆푠 ∈ 푋(Φ) depending on 푀. A weight module is integrable int if 푈푞(Φ푖) acts locally finitely, and the full subcategory of 풪푞(Φ) of integrable objects is denoted by 풪푞 (Φ).

We define primitive vectors, highest-weight modules, quantum Verma modules 푀푞(휆), and their simple quotients int 퐿푞(휆) in direct analogy with the Kac-Moody case. The same complete reducibility theorem applies to 풪푞 (Φ), namely that every object is isomorphic to a possibly infinite direct sum of 퐿푞(휆) for 휆 ∈ 푋(Φ)+. 3.6.12 Remark Sometimes a more general notion of a weight is used, parametrised by a group homomorphism 휎 ∶ 푋 ∨(Φ) → ⟨휆,휇⟩ ∨ {±1} together with a weight 휆: a vector 푣 ∈ 푀 is of weight (휎, 휆) if 퐾휇푣 = 휎(휇)푞 푣 for all 휇 ∈ 푋 (Φ). These weights naturally arise in the study of finite-dimensional representations of 푈푞(Φ). The weight representations of a fixed type 휎 ∈ Hom(푋 ∨(Φ), {±1}) form an abelian subcategory, and all 2rank Φ such subcategories are equivalent. We have chosen the subcategory corresponding to the trivial type 휎(휇) = 1 (sometimes called type (1, … , 1) in the literature), which is also closed under taking tensor products and duals. There is another class of integrable modules which we will come across in passing. The extremal weight modules were first defined by Kashiwara in Section 8of[Kas94], and generalise highest-weight modules. They are used to

28 3.6 Quantum groups study integrable modules for example in [Kas02a], where it is shown that if Φ is affine and 휛푖 ∈ 푋(Φ) is a level-zero fundamental weight, the extremal weight module 푉푞(휛푖) is the affinisation of a finite-dimensional representation 퐿푞(휛푖). 3.6.13 Definition (Extremal weight modules)

A weight vector 푣 ∈ 푉휆 of an integrable 푈푞(Φ)-module 푉 is called 푖-extremal if 푒푖푣 = 0 or 푓푖푣 = 0. In this case, ∨ ∨ [⟨휆,훼푖 ⟩] [−⟨휆,훼푖 ⟩] define 푆푖푣 = 퐹푖 푣 or 푆푖푣 = 퐸푖 푣 respectively (“the” element at the opposite end of the 푖-string). The

weight vector 푣 is further called extremal if 푆푖1 ⋯ 푆푖푟 푣 is 푖-extremal for all 푖 ∈ 퐼, for all words (푖1, … , 푖푟 ).

An integrable 푈푞(Φ)-module 푉 is called extremal of weight 휆 if it contains a vector 푣 ∈ 푉휆 and there exist vectors (푣푤 )푤∈푊 such that 푣 = 푣푒 and

∨ ∨ [⟨푤휆,훼푖 ⟩] if ⟨푤휆, 훼푖 ⟩ ≥ 0, then 퐸푖푣푤 = 0 and 퐹푖 푣푤 = 푣푠푖푤 , ∨ (3.6.14) ∨ [−⟨푤휆,훼푖 ⟩] if ⟨푤휆, 훼푖 ⟩ ≤ 0, then 퐹푖푣푤 = 0 and 퐸푖 푣푤 = 푣푠푖푤 .

If such vectors (푣푤 )푤∈푊 exist then they are unique, and furthermore 푣푤 ∈ 푉푤휆.

For each weight 휆 ∈ 푋(Φ), the extremal weight module 푉푞(휆) is the 푈푞(Φ)-module generated by 푣휆 with the defining relation that 푣휆 is extremal of weight 휆. For each 푤 ∈ 푊 , the map 푣휆 ↦ 푆푤−1 푣푤휆 gives an ∼ isomorphism 푉 (휆) −→푉 (푤휆) of 푈푞(Φ)-modules.

When 휆 is a dominant weight, 푉푞(휆) ≅ 퐿푞(휆) is the irreducible highest-weight module of highest-weight 휆, and similarly if 휆 is antidominant, then 푉푞(휆) is the irreducible lowest-weight module of lowest-weight 휆. When (퐼, ⋅) is finite-type, the extremal-weight modules 푉푞(휆) are finite-dimensional and irreducible. However, if (퐼, ⋅) is not int finite-type, then not all extremal weight modules are members of 풪푞 (Φ).

4 Crystals

int It was first shown by Kashiwara [Kas91] that each integrable highest-weight representation 퐿푞(휆) ∈ 풪푞 (Φ) admits a crystal base ℬ(휆), which can be thought of (almost) as a certain weight basis of 퐿푞(휆) enjoying very special properties. The set ℬ(휆) can be equipped with directed labelled edges coming from the ‘crystallisation’ of the quantum 퐸푖 and 퐹푖 operators (the Kashiwara operators), making ℬ(휆) into a connected graph with a unique highest-weight element 푏휆. Remarkably, there is a purely combinatorial rule (stated purely in terms of just the directed graph structure) for forming the tensor product of two crystal bases, thus the theory of crystals allows us to reduce some linear algebraic problems into combinatorial problems. For example, the number of times 퐿푞(휈) appears in the tensor product 퐿푞(휆)⊗퐿푞(휇) is equal to the number of times the connected component ℬ(휈) appears inside the graph ℬ(휆) ⊗ ℬ(휇).

There are many more 푈푞(Φ)-modules which admit crystal bases, for example the Verma modules (which are not integrable), and the extremal weight modules (which are integrable but not in general highest-weight). While the definition of crystal base (or at least of Kashiwara operators) needs to be modified slightly to includethese different cases, there is a combinatorial category of Φ-crystals into which all of the resulting crystal bases can be placed. These abstract crystals do not remember the original 푈푞(Φ) representation but instead have some extra data attached: the raising and lowering statistics 휀푖 and 휑푖 which (very roughly) allow a Φ-crystal to capture some non-semisimple behaviour, as one would expect from (say) the crystal of a Verma module. The combinatorial Φ-crystals are very convenient for working with crystals, as all of the data needed to compute tensor products, restrictions, morphisms and so on are right at our fingertips. However, the Φ-crystals also include many objects which do not arise from any 푈푞(Φ)-module, which is both a blessing and a curse: on one hand this + flexibility allows the crystal base associated to a Demazure module (which isonlya 푈푞 (Φ)-module) to be an honest Φ-crystal, while on the other hand one has to work very hard to show that a given Φ-crystal actually came from a 푈푞(Φ)-module and is not some ‘virtual’ crystal. So while the combinatorial axioms are convenient for working with crystals, they are not so suited to producing useful crystals in the first place. We begin this chapter by grounding ourselves in the theory of crystal bases of integrable highest-weight modules. Despite the fact we will really only use abstract crystals for the rest of the thesis, the author feels that without this grounding it is hard to see the explicit connection between bases of 푈푞(Φ)-modules and crystals. Next we spend some time discussing the category of Φ-crystals, giving many examples of crystals both coming and not coming from integrable 푈푞(Φ)-representations. We then come to the recognition theorems, which are the methods int by which we can show an abstract Φ-crystal does indeed come from the category 풪푞 (Φ). Finally we describe some interesting actions of the Weyl group and cactus group on these crystals.

4.1 Crystal bases of integrable modules

In this section we will give Kashiwara’s original [Kas91] definition of the crystal base of an integrable 푈푞(Φ)- module 푉 . A crystal base 퐵 is not quite a ℚ(푞)-basis of 푉 , but a ℚ-basis of a module obtained from a crystal lattice 퐿 ⊆ 푉 . In order to define what properties 퐿 and 퐵 should satisfy, we first introduce the Kashiwara operators, which are ℚ(푞)-linear endomorphisms of 푉 . For any root datum Φ of Cartan type 퐼 and a choice of vertex 푖 ∈ 퐼, we can consider the restricted root datum Φ{푖} = Φ푖 having the same weight and coweight lattices as Φ, but where we have forgotten all simple roots and ∨ coroots except for 훼푖, 훼푖 . The representation theory of 푈푞(Φ푖) looks very similar to the representation theory of 푈푞(SL2), just with a larger set of weights 푋(Φ푖) = 푋(Φ). The finite-dimensional weight representations of

31 4 Crystals

푈푞(Φ푖) are semisimple, each isomorphic to an integrable highest-weight module 퐿푞(휇) for some 푖-dominant weight ∨ 휇 ∈ 푋(Φ푖)+ = {휇 ∈ 푋(Φ) ∣ ⟨휇, 훼푖 ⟩ ≥ 0}. The irreducible 푈푞(Φ푖)-module 퐿푞(휇) is generated by a highest-weight [1] [푛] ∨ [푘] vector 푣휇 of weight 휇, and has a string basis (푣휇, 퐹푖 푣휇, … , 퐹푖 푣휇) where 푛 = ⟨휇, 훼푖 ⟩ and 퐹푖 are the quantum [푘] 푘 ! ∨ divided powers 퐹푖 = 퐹푖 / [푘]푖. Take for example any 휇 ∈ 푋(Φ) satisfying ⟨휇, 훼푖 ⟩ = 5, then we can picture the actions of 퐸푖 and 퐹푖 on the string basis as a diagram:

퐹푖 [1]푖 [2]푖 [3]푖 [4]푖 [5]푖 [5] 푣휇 퐹푖 푣휇

퐸푖 [5]푖 [4]푖 [3]푖 [2]푖 [1]푖

̃ The Kashiwara operators 푖̃푒 and 푓푖 are defined to be the ℚ(푞)-linear right and left-shift operators on this basis: ̃ 푓푖 1 1 1 1 1 [5] 푣휇 퐹푖 푣휇

푖̃푒 1 1 1 1 1

For example, we would have

̃ [2] [5] [3] [2] [5] [1] [4] 푓푖 (퐹푖 푣휇 + 퐹푖 푣휇) = 퐹푖 푣휇 + 0 and 푖̃푒 (퐹푖 푣휇 + 퐹푖 푣휇) = 퐹푖 푣휇 + 퐹푖 푣휇. (4.1.1)

Suppose that 푉 is an isotypic 푈푞(Φ푖) module, having direct summands isomorphic to 퐿푞(휇). We can define Kashi- [1] [푛] wara operators in a more coordinate-free way by restricting the operators 퐹푖 , … , 퐹푖 to the highest-weight subspace 푉휇, yielding isomorphisms 푉휇 → 푉휇−훼푖 , …, 푉휇 → 푉휇−푛훼푖 . These chosen isomorphisms may be inverted and composed to get a chain of isomorphisms 푉휇 → 푉휇−훼푖 → ⋯ → 푉휇−푛훼푖 , and along with the condition that ̃ ̃ 푓푖(푉휇−푛훼푖 ) = 0 this chain is precisely the graded decomposition of the Kashiwara operator 푓푖. The operator 푖̃푒 may be defined by inverting the chain of isomorphisms and adding the condition that 푖̃푒 (푉휇) = 0.

Now, suppose that 푉 is an integrable 푈푞(Φ)-module. Then for any 푖 ∈ 퐼 the module 푉 becomes a 푈푞(Φ푖)-module by restriction, and the integrability condition gives that it decomposes into a direct sum of finite-dimensional 푈푞(Φ푖)-modules. For each 푖-dominant 휇 ∈ 푋(Φ푖)+, let 푉 (푖, 휇) be the 휇-isotypic component: the sum of all subrep- ̃ resentations of 푉 isomorphic to 퐿푞(휇) as 푈푞(Φ푖)-modules. The Kashiwara operators 푖̃푒 (푣) and 푓푖(푣) are defined on this isotypic component, hence we get the 푖-Kashiwara operators defined on the whole of 푉 by taking the sum 1 over the 푈푞(Φ푖)-isotypic components of 푉 . Let 퐴 ⊆ ℚ(푞) be the subring consisting of rational functions without a pole at 푞 = 0. This is a discrete valuation ring: a principal ideal domain with unique nonzero maximal ideal 푞퐴. Its field of fractions is ℚ(푞), and its residue field is 퐴/푞퐴 ≅ ℚ via the isomorphism 푓 + 푞퐴 ↦ 푓 (0). Recall that an 퐴-lattice in a ℚ(푞)-vector space 푉 is a free 퐴-submodule 퐿 ⊆ 푉 such that 퐿 ⊗퐴 ℚ(푞) ≅ 푉 , and that we have an isomorphism of ℚ-vector spaces 퐿 ⊗퐴 ℚ ≅ 퐿/푞퐿.

Let 푉 be an integrable 푈푞(Φ)-module. An 퐴-lattice 퐿 ⊆ 푉 is called a crystal lattice if 퐿 is graded, meaning that ̃ 퐿 = ⨁휆∈푋(Φ) 퐿 ∩ 푉휆, and lattice 퐿 is invariant under the Kashiwara operators 푖̃푒 , 푓푖. This ensures that the ℚ-vector space is still graded into weight spaces, and that the Kashiwara operators descend to ℚ-linear operators on the ℚ-vector space 퐿/푞퐿. A pair (퐿, 퐵) is called a crystal basis of 푉 if:

1. 퐿 is a crystal lattice of 푉 , 2. 퐵 is a weight basis of 퐿/푞퐿,

1If we were defining crystals of 풪(Φ)-modules rather than integrable modules, the definition of the Kashiwara operators would have to change slightly (but everything following this point would remain the same). We cannot quite use 푖-isotypic components in this case, since 푉 may not be semisimple as a 푈푞(Φ푖)-module, but one can nevertheless define an appropriate direct sum decomposition by ‘pulling down’ 푖-highest weight vectors. See Section 3.5 of [Kas91] for details.

32 4.1 Crystal bases of integrable modules

̃ 3. 푖̃푒 퐵 ⊆ 퐵 ∪ {0} and 푓푖퐵 ⊆ 퐵 ∪ {0} for all 푖 ∈ 퐼, and ′ ′ ̃ ′ 4. For any 푏, 푏 ∈ 퐵 and 푖 ∈ 퐼 we have 푖̃푒 (푏) = 푏 if and only if 푏 = 푓푖(푏 ).

It is a priori unclear whether crystal bases exist for integrable highest-weight modules when |퐼| > 1. However, some properties of crystal bases are already visible, for example if (퐿1, 퐵1) and (퐿2, 퐵2) are crystal bases of 푉1 and 푉2 respectively then (퐿1 ⊕ 퐿2, 퐵1 ⊔ 퐵2) is a crystal basis of 푉1 ⊕ 푉2. Remarkably, something similar works for the tensor product: (퐿1 ⊗퐴 퐿2, 퐵1 ×퐵2) is a crystal base for 푉1 ⊗푉2. Furthermore, the action of the Kashiwara operators on 퐵1 × 퐵2 is given purely in terms of their actions on 퐵1 and 퐵2 individually, with a simple rule for which side to act on (see Definition 4.3.1 below). We could have attempted to formulate crystal bases for 픤(Φ)-modules rather than 푈푞(Φ)-modules, but we would have fallen flat attempting to get any nice behaviour out of the tensor product: something interesting is happening in the quantum world which makes this work. It is worth noting that just as the tensor product of 푈푞(Φ)-modules is asymmetric, so is the tensor product of crystals. The existence of crystal bases was shown by Kashiwara.

int 4.1.2 Theorem (Existence of crystal bases in 풪푞 (Φ))

(Theorem 2, [Kas91]). Let Φ be a root datum of type (퐼, ⋅) and let 휆 ∈ 푋(Φ)+ be a dominant weight. Then the integrable highest-weight module 퐿푞(휆) admits a crystal base, constructed as follows. Let 푣휆 ∈ 퐿푞(휆) be ̃ ̃ a highest-weight vector, and let ℱ 푣휆 be the set of vectors of the form 푓푖1 ⋯ 푓푖푙 푣휆. Define the lattice ℒ(휆) to be the 퐴-span of the set ℱ 푣휆, and ℬ(휆) to be the image of ℱ 푣휆 in ℒ(휆)/푞ℒ(휆) with zero removed. Then (ℒ(휆), ℬ(휆)) is a crystal base of 퐿푞(휆). There is a little more discussion of the theory of crystal bases at the end of this chapter, but from now on we will largely focus on the more combinatorial aspects of crystals.

33 4 Crystals

4.2 The category of crystals

The combinatorial nature of crystal bases suggests that they are governed by a combinatorially defined category. To the author’s knowledge, the first appearance of this category wasKas93 in[ ]. 4.2.1 Definition (Abstract crystal) Let Φ be a root datum of Cartan type (퐼, ⋅). An abstract crystal of type Φ, or Φ-crystal, is the data of

(퐵, wt, (휀푖)푖∈퐼 , (휑푖)푖∈퐼 , (푒푖)푖∈퐼 , (푓푖)푖∈퐼 ), (4.2.2)

where 퐵 is a set, wt ∶ 퐵 → 푋(Φ) and 휀푖, 휑푖 ∶ 퐵 → ℤ ⊔ {−∞} are functions which we call the raising and 99K lowering statistics, and 푒푖, 푓푖 ∶ 퐵 퐵 are partial functions called the crystal operators, such that for all 푏, 푏′ ∈ 퐵 and 푖 ∈ 퐼 the following axioms hold: ∨ 1. Balanced-strings: 휑푖(푏) = 휀푖(푏) + ⟨wt(푏), 훼푖 ⟩, 2. Raising: If 푒푖(푏) ≠ ⊥, then

wt(푒푖(푏)) = wt(푏) + 훼푖, 휀푖(푒푖(푏)) = 휀푖(푏) − 1, 휑푖(푒푖(푏)) = 휑푖(푏) + 1, (4.2.3)

3. Lowering: If 푓푖(푏) ≠ ⊥, then

wt(푓푖(푏)) = wt(푏) − 훼푖, 휀푖(푓푖(푏)) = 휀푖(푏) + 1, 휑푖(푓푖(푏)) = 휑푖(푏) − 1, (4.2.4)

′ ′ 4. Partial inverse: 푒푖(푏) = 푏 if and only if 푏 = 푓푖(푏 ), and 5. Infinity: If 휑푖(푏) = −∞, then 푒푖(푏) = 푓푖(푏) = ⊥.

In this definition, ℤ ⊔ {−∞} is understood to have the additive structure where 푥 + −∞ = −∞.

If (퐿, 퐵) is the crystal base of an integrable 푈푞(Φ)-module 푉 as defined above, then we can put an abstract Φ- ̃ crystal structure on the set 퐵 by taking the crystal 푒푖 and 푓푖 operators to be the Kashiwara operators 푖̃푒 and 푓푖, and 푛 푛 the raising and lowering statistics to be 휀푖(푏) = max{푛 ≥ 0 ∣ 푒푖 (푏) ≠ ⊥} and 휑푖(푏) = max{푛 ≥ 0 ∣ 푓푖 (푏) ≠ ⊥}. In this int way we get an abstract crystal ℬ(휆) for each integrable highest-weight module 퐿푞(휆) ∈ 풪푞 (Φ). 4.2.5 Example (Root strings) ∨ Let Φ be the root datum SL2 with 퐼 = {푖}, and 푋(Φ) = ℤ be the weight lattice, so that ⟨1, 훼푖 ⟩ = 1. Fix a dominant weight 푛 ≥ 0, and define the crystal ℬ(푛) = {푏푛, 푏푛−2, … , 푏−푛+2, 푏−푛} as a set, with crystal operators 1 푒푖(푏푛) = ⊥, 푓푖(푏−푛) = ⊥, 휀푖(푏푘) = (푛 − 푘), wt(푏푘) = 푘. 2 (4.2.6) 1 푒 (푏 ) = 푏 , 푓 (푏 ) = 푏 , 휑 (푏 ) = (푛 + 푘). 푖 푘 푘+2 푖 푘 푘−2 푖 푘 2 Then ℬ(푛) is a Φ-crystal, and it is obvious from the definition of crystal bases that it is the crystal arising from the integrable highest-weight module 퐿푞(푛). It can be pictured by its crystal graph, shown below for ℬ(5):

휑푖(푏1) = 3

푏5 푏3 푏1 푏−1 푏−3 푏−5

휀푖(푏1) = 2

The raising statistic 휀푖(푏1) = 2 counts the number of edges above 푏1, and 휑푖(푏1) = 3 is counting the number of edges below 푏1.

34 4.2 The category of crystals

If an abstract crystal comes from a crystal base of some integrable 푈푞(Φ)-module, then (by definition) the raising and lowering statistics behave in the simple way shown in Example 4.2.5, counting the number of arrows above and below an element on a root string. This is called seminormality, sometimes called normality or regularity in the literature. 4.2.7 Definition (Seminormality of crystals) A crystal is called 푘 1. Upper seminormal if 휀푖(푏) = max{푘 ≥ 0 ∣ 푒푖 (푏) ≠ ⊥} for all 푏 ∈ 퐵, 푘 2. Lower seminormal if 휑푖(푏) = max{푘 ≥ 0 ∣ 푓푖 (푏) ≠ ⊥} for all 푏 ∈ 퐵, and 3. Seminormal if it is both upper and lower seminormal.

The balanced-strings axiom implies that if 퐵 is lower or upper seminormal then the statistics 휀푖 and 휑푖 take only integer values. Furthermore, together with the partial inverse axiom it means that the data of an upper- seminormal crystal 퐵 is completely determined by (퐵, wt, (푒푖)푖∈퐼 ), and similarly the data of a lower-seminormal crystal is completely determined by (퐵, wt, (푓푖)푖∈퐼 ). (One still needs to remember whether the crystal was upper or lower-seminormal). We remark that there are many examples of abstract crystals which are not seminormal, for example those coming from Verma modules such as the crystal ℬ(∞) shown in Example 4.2.12, or those that do not come from 푈푞(Φ)- modules as all, such as the crystal 푇휆 of Example 4.2.11. The ad-hoc illustration of the crystal ℬ(5) in Example 4.2.5 can be formalised into the notion of a crystal graph. 4.2.8 Definition (Crystal graph) Let 퐵 be an abstract Φ-crystal. The crystal graph associated to 퐵 is the directed edge-labelled graph with 푖 ′ ′ ′ vertex set 퐵, with an edge 푏 −→푏 labelled 푖 if and only if 푓푖(푏) = 푏 , or equivalently if and only if 푏 = 푒푖(푏 ). If the underlying undirected graph of 퐵 is connected, we say that 퐵 is a connected crystal. A vertex with no incoming edges is called a primitive element. If a primitive element 푏휆 ∈ 퐵휆 generates 퐵 under the 푓푖, then we say that 퐵 is a highest-weight crystal of weight 휆.

The crystal operator 푒푖, the crystal operator 푓푖, and the crystal graph are all equivalent data. If the crystal is known to be upper or lower-seminormal, then the statistics 휀푖 and 휑푖 can also be inferred from the graph.

As an example for the reader to keep in mind, we present some SL푛-crystals.

4.2.9 Example (Some SL푛 crystals) The quantum group 푈푞(GL푛) has a representation on the ℚ(푞)-vector space 푉 with basis 푣1, … , 푣푛 where 퐸푖 acts as the coordinate matrix 푣푖+1 ↦ 푣푖, 퐹푖 acts as the coordinate matrix 푣푖 ↦ 푣푖+1, and the action of the 퐾휈 is determined by requiring that 푣푖 is a weight vector of weight 휖푖. We call 푉 the natural representation ̃ of 푈푞(GL푛), and in this special case the Kashiwara operators 푖̃푒 and 푓푖 on 푉 are equal to 퐸푖 and 퐹푖, therefore setting 퐵 = {푣1, … , 푣푛} and 퐿 = 퐴퐵 makes (퐿, 퐵) into a crystal base for 푉 . For example, when Φ = GL3 we can draw the crystal 퐵 as follows: 푣1 1 푣2 2 푣3

where the vector 푣푖 is in weight 휖푖. What we have drawn is exactly a crystal graph in the sense of Defini- tion 4.2.8: we are saying that 푓1(푣1) = 푣2 and 푓2(푣2) = 푣3, and 푓푖(푣푗) = ⊥ everywhere else. In order to grasp the shape of more complicated graphs, we will often colour the set 퐼, and then use those colours to label the edges. For example, if we let 퐼 = {1, 2} so that 1 is coloured blue and 2 is coloured orange, then we can draw the diagram more simply:

푣1 푣2 푣3

There is a morphism of A2-root data SL3 → GL3, and hence each GL3-crystal becomes an SL3-crystal by this restriction. Conveniently SL3 has a two-dimensional weight lattice (shown in Example 3.4.16), allowing

35 4 Crystals

us to draw crystals on top of it and picture the weights of a crystal element by its position on the drawing. Below are examples of two 픰픩3 representations, the one on the left corresponding to the dominant weight (2, 0, 0) , and the one on the right to 휃 = (2, 1, 0) , the highest root, making ℬ(휃) the crystal of the adjoint representation.

ℬ(2, 0, 0) ℬ(2, 1, 0)

The diagrams above show the crystal graph explicitly, and the weights of elements by their position on the weight space. This is almost enough information to specify the crystal completely, and once we declare both of these crystals to be seminormal then the 휀푖 and 휑푖 statistics can also be inferred from the graph. We remark that in the crystal for the adjoint representation, the two vertices in the zero weight space are far apart in terms of the crystal graph, despite having equal weight. The vertex set of the crystals in the previous example are not explicitly given (merely shown in the graph). It is often the case in Lie theory that we are able to find bases of themodules 퐿(휆) indexed by a set of combinatorial objects depending on 휆. Perhaps the most famous example of this is of Young tableaux. When 휆 is a partition with at most 푛 rows, it may be interpreted as a dominant weight of GL푛, and there is a basis of 퐿(휆) made of semistandard Young tableaux: fillings of the Young diagram 휆 using the letters 1, … , 푛 such that the filling strictly increases down columns and weakly increases to the right. A lovely account of this can be found in [Ful96]. We might then expect the existence of a crystal ℬ(휆) with vertices indexed by such semistandard tableaux, and moreover we should be able to interpret the crystal operators 푒푖, 푓푖 as partial functions from this set to itself. This is in fact the case, and furthermore we will see later on that one could use the theory of crystals to work the other way: starting only with a crystal structure on semistandard tableaux, it is possible to prove that the semistandard tableaux of shape 휆 must be a basis of 퐿(휆), using the tensor product of crystals and the notion of a closed family of crystals. But we are getting ahead of ourselves, let’s see some more examples of crystals. 4.2.10 Example (Crystals of tableaux)

The dominant polynomial weights of GL푛 are in bijection with the set of partitions of length at most 푛, and for a partition 휆 there exists a crystal basis ℬ(휆) of 퐿푞(휆) in bijection with the set of semi-standard tableaux of shape 푛 on the letters 1, … , 푛. The crystal operators 푒푖 and 푓푖 can be given on tableaux explicitly by relatively simple rules (see Chapter 7 of [HK12], or [BS17]). The weight of a semistandard tableau is determined by its entries, with a number 푖 contributing 휖푖 to the weight of the tableau.

Some examples for GL3 are shown below, in left-to-right order we have the trivial crystal, the crystal of the natural representation 푉 , the crystal of the representation Λ2(푉 ), and the crystal of the determinant representation Λ3(푉 ).

ℬ(0, 0, 0) ℬ(1, 0, 0) ℬ(1, 1, 0) ℬ(1, 1, 1) 1 1 1 2 1 2 3 2 2 3 3 3

The crystals ℬ(0, 0, 0) and ℬ(1, 1, 1) are identical except for their weights. Here are three more interesting

36 4.2 The category of crystals

GL3-crystals for the reader to gaze upon.

ℬ(2, 0, 0) ℬ(2, 1, 0) 2 2 1 1 1 2 2 3 3 3 1 1 2 1 3 1 2 1 1 2 3 ℬ(3, 0, 0)

1 1 1 1 3 1 2 2 3

1 1 2 1 1 3 1 3 2 2 3 3 1 2 2 1 2 3 1 3 3

2 3 2 2 2 2 2 3 2 3 3 3 3 3 3

We have a morphism of A2 root data SL3 → GL3, and so we may view each of the crystals above as SL3 crystals by restriction. While all the above crystals are non-isomorphic as GL3 crystals, the crystals ℬ(0, 0, 0) and ℬ(1, 1, 1) are isomorphic as SL3 crystals, due to the restricted weights (0, 0, 0) and (1, 1, 1) being equal in 푋(SL3). The SL3 restrictions of ℬ(2, 0, 0) and ℬ(2, 1, 0) both appeared in the previous Example 4.2.9. int The crystals we have seen so far have all come from 풪푞 (Φ) representations, however there are many Φ-crystals which do not come from any representation of 푈푞(Φ), let alone integrable representations. For example, this next 0 crystal could be thought of as a one-dimensional representation of 푈푞 (Φ) rather than 푈푞(Φ). 4.2.11 Example (Character crystal)

Given a weight 휆 ∈ 푋(Φ), let 푇휆 = {푡휆} with wt(푡휆) = 휆 ∈ 푋(Φ), the raising and lowering statistics 휑푖(푡휆) = 휀푖(푡휆) = −∞, and the crystal operators 푒푖(푏0) = ⊥ and 푓푖(푏0) = ⊥. int We could complain that the character crystal is a silly example of a crystal not coming from an 풪푞 (Φ) module, since the raising and lowering statistics take the special value −∞. However, even if they take integer values, we may have a crystal of a non-integrable module such as the ℬ(∞) crystal coming from the Verma module 푀푞(0). 4.2.12 Example (The rank-one infinity crystal)

Let Φ be the root datum of SL2. Define the set ℬ(∞) = {푥0, 푥1, 푥2, …} with the operators

푒푖(푥0) = ⊥, 푓푖(푥푘) = 푏푥+1, 휀푖(푥푘) = 푘, wt(푥푘) = −2푘, (4.2.13) 푒푖(푥푘) = 푥푘−1, 휑푖(푥푘) = −푘.

Then ℬ(∞) is a Φ-crystal. Its crystal graph looks as follows, with the 휀푖 and 휑푖 statistics drawn underneath: 푥0 푥1 푥2 푥3 푥4

휀푖(푥푘) 0 1 2 3 4 휑푖(푥푘) 0 −1 −2 −3 −4 The example of the ℬ(∞) crystal brings us to the definition of morphisms in the category of abstract Φ-crystals, which are slightly tricky to define.

37 4 Crystals

4.2.14 Definition (Morphisms of crystals) 99K A morphism 퐵1 → 퐵2 of Φ-crystals is a partial function 휓 ∶ 퐵1 퐵2 such that 1. 휓 commutes with wt, 휀푖, and 휑푖 on its domain of definition. 2. If both 푏, 푒푖(푏) ∈ 퐵1 are in the domain of definition of 휓, then 휓(푒푖(푏)) = 푒푖(휓(푏)). 3. If both 푏, 푓푖(푏) ∈ 퐵1 are in the domain of definition of 휓, then 휓(푓푖(푏)) = 푓푖(휓(푏)).

A crystal morphism is called strict if it commutes with all the 푒푖 and 푓푖. It is called an embedding if it is defined on the whole of 퐵1 and is an injective function. It is called an isomorphism if it is defined on the whole of 퐵1 and is a bijective function.

A non-strict crystal morphism need not quite commute with the 푒푖 and 푓푖 crystal operators. For example, there is a crystal morphism 휓 ∶ ℬ(0) → ℬ(∞) by taking the unique element 푏0 ∈ ℬ(0) of the trivial crystal to the element 푥0 ∈ ℬ(∞) of the same weight, which is an example of a non-strict embedding. However, if 휓 is an isomorphism, or both 퐵1 and 퐵2 are seminormal, then 휓 is automatically strict.

We will leave one final example here of a crystal coming froma 푈푞(Φ) module which is neither highest-weight nor integrable, but is nevertheless occasionally used in abstract arguments. 4.2.15 Example (The elementary crystal)

Let Φ be a root datum of type (퐼, ⋅). Fix a fixed vertex 푖 ∈ 퐼, define the elementary crystal ℬ푖 = {푏푖(푛) ∣ 푛 ∈ ℤ}, with structure wt(푏푖(푛)) = 푛훼푖 푓푖(푏푖(푛)) = 푏푖(푛 − 1) (4.2.16) 휀푖(푏푖(푛)) = −푛 휑푖(푏푖(푛)) = 푛,

with 휀푗 = 휑푗 = −∞ and 푒푗 = 푓푗 = ⊥ for 푗 ≠ 푖. The crystal graph is the following:

푏푖(2) 푏푖(1) 푏푖(0) 푏푖(−1) 푏푖(−2)

휀푖(푏푖(푛)) −2 −1 0 1 2 휑푖(푏푖(푛)) 2 1 0 −1 −2

38 4.3 Tensor product of crystals

4.3 Tensor product of crystals

The category of abstract Φ-crystals is equipped with a monoidal structure, the tensor product of crystals. There are two different definitions of the tensor product, each the reverse of the other. Here we use the “combinatorialist’s convention”, which plays the nicest with existing combinatorics such as the RSK algorithm. 4.3.1 Definition (Tensor product of crystals) Let 퐵 and 퐶 be two abstract Φ-crystals. The tensor product 퐵 ⊗ 퐶 has underlying set the Cartesian product 퐵 × 퐶, with elements denoted by 푏 ⊗ 푐 rather than (푏, 푐). We use the convention that 푏 ⊗ ⊥ = ⊥ = ⊥ ⊗ 푐, and equip the tensor product with the following crystal structure:

wt(푏 ⊗ 푐) = wt(푏) + wt(푐) ∨ 휀푖(푏 ⊗ 푐) = max {휀푖(푐), 휀푖(푏) − ⟨wt(푐), 훼푖 ⟩} ∨ 휑푖(푏 ⊗ 푐) = max {휑푖(푏), 휑푖(푐) + ⟨wt(푏), 훼푖 ⟩}

푒푖(푏) ⊗ 푐 if 휀푖(푏) > 휑푖(푐) (4.3.2) 푒푖(푏 ⊗ 푐) = { 푏 ⊗ 푒푖(푐) if 휀푖(푏) ≤ 휑푖(푐)

푓푖(푏) ⊗ 푐 if 휀푖(푏) ≥ 휑푖(푐) 푓푖(푏 ⊗ 푐) = { 푏 ⊗ 푓푖(푐) if 휀푖(푏) < 휑푖(푐)

It is routine to verify that this defines the structure of an abstract Φ-crystal on 퐵 ⊗ 퐶, and furthermore that if both 퐵 and 퐶 are both upper seminormal, then so is 퐵 ⊗ 퐶 (and similarly for lower seminormal).

The remarkable thing about the crystal tensor product is if ℬ(휆) and ℬ(휇) are the crystals of 퐿푞(휆) and 퐿푞(휇) respectively, then ℬ(휆)⊗ℬ(휇) is the crystal of 퐿푞(휆)⊗퐿푞(휇). This means that if we can find combinatorial models for ℬ(휆) and ℬ(휇) (such as the tableaux hinted at in Example 4.2.10), then we can compute the decomposition of 퐿푞(휆)⊗퐿푞(휇) by just finding the connected components (or highest-weight vertices) of thegraph ℬ(휆)⊗ℬ(휇).

We defined the crystal ℬ(∞) when |퐼| = 1 in Example 4.2.12. For general Φ, the crystal ℬ(∞) is a kind of limit of the ℬ(휆), which again plays the role of the Verma module 푀(0). The tensor product 푇휆 ⊗ ℬ(∞) plays the role of the Verma module 푀(휆) in the setting of crystals. 4.3.3 Example (Shadows of Verma modules)

Let Φ = SL2. Taking the tensor product of 푇휆 with ℬ(∞) gives an interesting crystal: the tensor product structure in Definition 4.3.1 simplifies to give the following crystal structure on 푇휆 ⊗ ℬ(∞):

wt(푡휆 ⊗ 푥푛) = 휆 − 푛훼푖 푒푖(푡휆 ⊗ 푥푛) = 푡휆 ⊗ 푒푖(푥푛) 휀푖(푡휆 ⊗ 푥푛) = 푛 ∨ (4.3.4) 푓푖(푡휆 ⊗ 푥푛) = 푡휆 ⊗ 푥푛+1 휑푖(푡휆 ⊗ 푥푛) = ⟨훼푖 , 휆⟩ − 푛.

For example, if we suppose that 휆 = 3 then we have the following picture of 푇휆 ⊗ ℬ(∞):

푡3 ⊗ 푥0 푡3 ⊗ 푥1 푡3 ⊗ 푥2 푡3 ⊗ 푥3 푡3 ⊗ 푥4

휀푖(푡3 ⊗ 푥푘) 0 1 2 3 4 휑푖(푡3 ⊗ 푥푘) 3 2 1 0 −1

Here we can see there is a non-strict crystal morphism ℬ(3) ↪ 푇3 ⊗ ℬ(∞) (recall that crystal morphisms must commute with wt, 휀푖, and 휑푖, but only need to commute with the 푒푖 and 푓푖 operators on their domain of definition).

39 4 Crystals

4.4 Recognition theorems

We know by now that the category of Φ-crystals is far more general than just those corresponding to crystals of int 풪푞 (Φ) modules, which form a full monoidal subcategory. Some abstract Φ-crystals we have seen so far which int are not crystals of 풪푞 (Φ)-modules are:

int • The ‘Verma crystal’ ℬ(∞), a Φ-crystal coming from the category 풪푞(Φ), but not from 풪푞 (Φ). • The ‘principal series’ crystals ℬ푖, which come from a representation of 푈푞(SL2) which is neither highest- weight nor integrable. • The ‘character crystal’ 푇휆, which does not come from a 푈푞(Φ)-representation at all.

The next counterexample in particular shows that the problem of classifying those abstract Φ-crystals coming int from 풪푞 (Φ) is quite subtle. 4.4.1 Example

The following are both abstract seminormal SL3-crystals, but only the one on the left comes from a 푈푞(Φ)- module.

ℬ(2, 1, 0) 퐵

int The question is: given an abstract Φ-crystal 퐵, how can one check that it is the crystal of a module from 풪푞 (Φ)? If the crystal is connected, one could hope to find a dominant weight 휆 and an existing model for ℬ(휆) and ∼ give a crystal isomorphism 퐵 −→ℬ(휆). However this does not help to ‘bootstrap’ the theory, since one needs to start with an existing model for ℬ(휆), and furthermore specifying an isomorphism is impossible if the exact decomposition of 퐵 into highest weights is unknown.

Fortunately, there are a number of recognition theorems available to us which we can use to check whether a Φ- int crystal 퐵 really does come from 풪푞 (Φ). The first of these is due to Kashiwara, allowing us to reduce theproblem to Cartan data of rank 2. As with 픤(Φ)-modules and 푈푞(Φ)-modules, an element 푏 of a crystal is called primitive if 푒푖(푏) = ⊥ for all 푖 ∈ 퐼, and a primitive element is called highest weight if it generates 퐵. 4.4.2 Theorem (Recognition by rank-2 restriction) (Proposition 2.4.4 of [Kan+92]). Let 퐵 be a Φ-crystal such that, for any subset 퐽 ⊆ 퐼 with at most two elements, any connected component of the restricted crystal 퐵퐽 containing a 퐽-primitive element is a crystal isomorphic to ℬ(Φ퐽 , 휆) for some 퐽-dominant 휆. Then any connected component of 퐵 containing a primitive element generates a subcrystal isomorphic to ℬ(휆), for some dominant 휆. The above condition on subsets 퐽 ⊆ 퐼 with one element is simply checking that 퐵 is seminormal, so checking the rank-2 restrictions is where all the work is. This theorem is very frequently used, and the other recognition theorems below rely on this one. We remark that if 퐵 satisfies the above condition then only the connected int components containing primitive vectors come from 풪푞 (Φ)-modules, there may be other connected components which are not highest-weight.

40 4.5 Weyl group action

Joseph [Jos95] has given a purely combinatorial method of checking whether a collection 풞 = {퐶(휆) ∣ 휆 ∈ 푋(Φ)+} of candidate Φ-crystals coincide with the ℬ(휆). Clearly, each 퐶(휆) should be a seminormal highest weight crystal of highest weight 휆. We need only one more condition on 풞 , ensuring that it behaves well with tensor products. 4.4.3 Definition (Closed families of crystals)

Let 풞 = {퐶(휆) ∣ 휆 ∈ 푋(Φ)+} be a family of highest-weight seminormal crystals, where 푐휆 ∈ 퐶(휆) is the highest-weight element of weight 휆. We say that 풞 is a closed family if the element 푐휆 ⊗ 푐휇 ∈ 퐶(휆) ⊗ 퐶(휇) generates a strict subcrystal of highest weight 휆 + 휇 isomorphic to 퐶(휆 + 휇).

It is immediate from the definition of the tensor product that 푐휆 ⊗ 푐휇 will be a primitive element of 퐶(휆) ⊗ 퐶(휇) of the correct weight, but it is far from clear that the subset generated by 푐휆 ⊗ 푐휇 under the 푓푖 operators is a strict subcrystal of 퐶(휆) ⊗ 퐶(휇), or even that it is stable under the 푒푖 operators. 4.4.4 Theorem (6.4.21 of [Jos95]). Let 풞 = {퐶(휆) ∣ 휆 ∈ 푋(Φ)+} be a closed family of highest-weight seminormal crystals. Then 퐶(휆) ≅ ℬ(휆) for all 휆 ∈ 푋(Φ)+.

It is very easy to show that the {ℬ(푛) ∣ 푛 ≥ 0} constructed in Example 4.2.5 is a closed family of SL2-crystals, and with some work (and proper definitions of the 푒푖 and 푓푖 operators) it can be shown that the crystals of semistandard tableaux hinted at in Example 4.2.10 form a closed family of GL푛 crystals. Given a closed family of Φ-crystals, we int have a way to work with the crystals of 풪푞 (Φ)-modules without the need to introduce the quantum enveloping algebra and the theory of crystal bases.

It can be shown that the Lakshmibai-Seshadri paths, with the Littelmann root operators, form a closed family and hence give an explicit model for all the ℬ(휆) in any type [Lit94]. In type A, these paths are in bijection with Young tableaux, making the LS paths a generalisation of Young tableaux to any type.

There is another kind of recognition theorem due to Stembridge, when (퐼, ⋅) is simply-laced. In [Ste03], Stembridge gives necessary and sufficient conditions for a directed graph with edges coloured by 퐼 to be the crystal graph of a int crystal from 풪푞 (Φ). After showing that the Stembridge axioms are necessary and permit at most one connected graph per highest weight, it is shown that these connected graphs are isomorphic to the corresponding crystal graphs given by the Littelmann paths, and hence must be the crystals of highest-weight integrable modules. This description only works for simply-laced type, since the uniqueness of the graphs only holds in simply-laced type, but crystals in other types can be obtained through a standard technique known as diagram folding. The recent text [BS17] defines crystals this way, as either Stembridge crystals or folded Stembridge crystals, without recourse to quantum groups.

4.5 Weyl group action

We often talk about crystals which come from crystal bases of integrable 푈푞(Φ)-modules. Throughout this section, we can use a slightly weaker property which appears in [Kas94]. 4.5.1 Definition A Φ-crystal 퐵 is called finite-normal if for any 퐽 ⊆ 퐼 of finite type, the restriction of 퐵 to Φ퐽 is isomorphic (as Φ퐽 -crystals) to the crystal base of an integrable 푈푞(Φ퐽 )-module. A finite-normal Φ-crystal is automatically seminormal, and furthermore if (퐼, ⋅) is finite type then the finite- int normality property is equivalent to being the crystal of a 풪푞 (Φ)-module.

For each 퐽 ⊆ 퐼 of finite type, a finite-normal crystal 퐵 decomposes into a disjoint union of finite Φ퐽 crystals, each of the form ℬ(Φ퐽 , 휆) for some 휆 ∈ 푋(Φ퐽 )+. It was first shown by Kashiwara that one can exploit this finiteness to obtain an action of the Weyl group 푊퐼 on the whole crystal 퐵.

41 4 Crystals

4.5.2 Theorem (Weyl action on finite-normal crystals)

(Section 7 of [Kas94]). Let 퐵 be a finite-normal Φ-crystal, and define for each 푖 ∈ 퐼 a map 푐푖 ∶ 퐵 → 퐵 by

∨ ⟨wt(푏),훼푖 ⟩ ∨ 푓 (푏) if ⟨wt(푏), 훼푖 ⟩ ≥ 0 푐푖(푏) = { ∨ (4.5.3) −⟨wt(푏),훼푖 ⟩ ∨ 푒 (푏) if ⟨wt(푏), 훼푖 ⟩ ≤ 0

Then: 2 1. Each 푐푖 is an involution: 푐푖 = id퐵. 2. Each 푐푖 acts by the reflection 푠푖 ∶ 푋(Φ) → 푋(Φ) on the weight of an element: wt(푐푖(푏)) = 푠푖(wt(푏)). 푚 3. The 푐푖 satisfy the braid relation (푐푖푐푗) 푖푗 = 1.

Hence the Weyl group 푊퐼 acts on 퐵 via the map 푠푖 ⋅ 푏 = 푐푖(푏).

Since 퐵 is a seminormal Φ-crystal, its restriction to Φ푖 breaks up into a disjoint union of balanced 푖-strings of finite length, and the operator 푐푖 acts by reversal on these strings. 4.5.4 Example

The following diagram shows the GL3 crystal of ℬ(2, 0, 0) from Example 4.2.10 on the left, with the computed actions of 푐1 and 푐2 on the right.

푓1 푓2 푐1 푐2

2 2 2 2 1 1 1 2 2 3 3 3 1 1 1 2 2 3 3 3 1 3 1 3

This makes the first two properties of Theorem 4.5.2 easy to verify: reversal is clearly an involution, and the balanced-strings axiom ensures that the reversal is taking place around zero. The third property is proved by reducing to the case where 퐽 ⊆ 퐼 is finite-type and has two elements, we refer the reader to Section 7of[Kas94] for the full proof.

4.6 Cactus group action

The Weyl group action in Theorem 4.5.2 may be extended to an action of the cactus group on a finite-normal crystal, via some involutions 푐퐽 where 푐{푖} = 푐푖 from before. 4.6.1 Theorem Let 퐵 be a finite-normal Φ-crystal, and let 퐽 ⊆ 퐼 be finite type and irreducible. Define amap 푐퐽 ∶ 퐵 → 퐵 to be the unique map preserving the connected components of the restricted crystal 퐵퐽 , and satisfying

(1) wt(푐퐽 (푏)) = 푤퐽 ⋅ wt(푏),

(2) 푒푗(푐퐽 (푏)) = 푐퐽 (푓휔퐽 (푗)(푏)) for all 푗 ∈ 퐽, (4.6.2)

(3) 푓푗(푐퐽 (푏)) = 푐퐽 (푒휔퐽 (푗)(푏)) for all 푗 ∈ 퐽.

Then the maps 푐퐽 ∶ 퐵 → 퐵 satisfy the cactus relations: 2 1. 푐퐽 = id퐵, 2. 푐퐽 푐퐾 = 푐퐾 푐퐽 if there are no edges between the vertices of 퐽 and 퐾, and

3. 푐퐽 푐퐾 = 푐퐾 푐휔퐾 (퐽) if 퐽 ⊆ 퐾.

Furthermore, the definition of 푐{푖} agrees with the one given in Theorem 4.5.2. Hence there is an action of 푚 the cactus group 퐶퐼 on 퐵 via the operators 푐퐽 , additionally satisfying the braid relations (푐푖푐푗) 푖푗 = 1.

42 4.7 Addendum

푚 The quotient of the cactus group 퐶퐼 by the braid relations (휏푖휏푗) 푖푗 has been called the reduced cactus group (3.4 of [Hal20]).

We remark that it is straightforward to compute the involution 푐퐽 if one knows both the involution 휔퐽 ∶ 퐽 → 퐽 and the whole crystal graph 퐵퐽 . Each element 푏 of a connected component of 퐵퐽 can be written (non-uniquely) as 푏 = 푓푗1 ⋯ 푓푗푘 푏high, where 푏high is the highest-weight element of the connected component and 푗1, … , 푗푘 ∈ 퐽. The equality 푐퐽 (푏high) = 푏low follows from (1), and applying condition (2) we get

푐퐽 (푏) = 푐퐽 (푓푗 ⋯ 푓푗 푏high) 1 푘 (4.6.3) = 푒휔퐽 (푗1) ⋯ 푒휔퐽 (푗푘 )푏low. So if we get from the highest-weight element to 푏 by following some arrows forwards, we get from the lowest- weight element to 푐퐽 (푏) by following the opposite arrows (as defined by 휔퐽 ) backwards. Consider the GL3-crystal ℬ(2, 1, 0) from earlier, which we show here: ℬ(2, 1, 0)

1 1 2

1 2 1 1 2 3

1 3 1 2 2 3

1 3 2 2 3 3

2 3 3

In this case (퐼, ⋅) = A2, and the involution 휔퐼 ∶ 퐼 → 퐼 swaps the blue and orange vertices 1 and 2 of the Dynkin diagram. So opposite arrow means the oppositely coloured arrow. Viewing ℬ(2, 1, 0) as a hexagon, the cactus involution 푐퐼 swaps opposite vertices. In particular, the two tableaux of weight (1, 1, 1) are swapped.

If ℬ(휆) is the GL푛-crystal of semistandard tableaux of shape 휆, then the involution 푐{1,…,푖} acts as the well-known Schützenberger involution on the sub-tableaux containing only entries {1, … , 푖 + 1}.

4.7 Addendum

Something that initially confused the author was the relationship between crystal bases and other bases of 푈푞(Φ), such as Lusztig’s canonical basis. Some people like to say (informally) that they are ‘the same’, which doesn’t make sense because they are not even objects of the same kind: the crystal basis of a representation 푉 is a ℚ- basis of a quotient lattice of 푉 , while Luztig’s canonical basis is an honest ℚ(푞)-basis of the whole algebra 푈푞(Φ). Something that was also unclear was the connection between Littelmann’s work on path models and Kashiwara’s on crystals. We have written this short section to hopefully be a small guide to the early literature explaining these things.

The story of crystal bases of 푈푞(Φ)-modules is due to Kashiwara. In [Kas90], he defined the notion of what we 2 now call an upper crystal base of a 푈푞(Φ)-module 푉 to be a pair (퐿, 퐵) of an 퐴-lattice 퐿 ⊆ 푉 together with a basis 퐵 of the ℚ-vector space 퐿/푞퐿, satisfying some axioms. He proved the existence and uniqueness of these crystal

43 4 Crystals bases in the classical types A푛, B푛, C푛, and D푛. In [Kas91], he further defined a lower crystal base and showed the − existence and uniqueness of these bases for the 퐿(휆) and the negative half 푈푞 (Φ) of the quantised enveloping algebra using a huge inductive argument called the grand loop. He also defined a global lower base, defined on an integral form 푉ℤ over the integral algebra 푈ℤ(Φ), to be an integral basis B which descends to a lower crystal basis in both the 푞 → 0 and 푞 → ∞ limits: for example, setting ℒ = ⨁푏∈B ℚ[푞]푏 and 퐵 = {푏 mod 푞ℒ ∣ 푏 ∈ B} should make (ℒ, 퐵) a lower crystal base. A good self-contained account of globalisation can be found in Chapter 6.2 of [Jos95]. In [Lit95a], Littelmann applied Kashiwara’s theory of crystal bases together with the theory of generalised Young tableaux (special cases of Lakshmibai-Seshadri paths) to give a combinatorial definition of the crystal graphs ℬ(휆) in the classical types A푛, B푛, C푛, D푛, E6, and G2. He then used these graphs together with the crystal tensor product rule to give a short proof of the generalised Littlewood-Richardson rule for computing the decomposition multiplicities of ℬ(휆)⊗ℬ(휇). He also conjectured that the extension-of-strings operators on crystal graphs could be used to construct crystal bases for Demazure modules. In [Kas93], Kashiwara proved Littelmann’s conjecture, giving a new proof of the Demazure character formula for symmetrisable Kac-Moody algebras. In this paper he also introduced the notion of an abstract Φ-crystal, so that ℬ(휆) and ℬ(∞) could be put on equal footing from a combinatorial point of view. Around the same time that Kashiwara defined crystal bases, Lusztig defined canonical bases[Lus90]. Lusztig’s canonical basis of 푈 −(Φ) may be defined purely algebraically however its existence is shown using geometric methods, namely realising the multiplication in the quantum group 푈푞(Φ) as a kind of convolution product on a Lusztig quiver variety. The canonical basis of Lusztig can be compared to the global (not crystal) basis of Kashiwara, and in fact the two have been shown to be equivalent [GL93].

2Recall that 퐴 ⊆ ℚ(푞) is the subring of rational functions without a pole at 푞 = 0, as in Section 4.1.

44 5 Demazure modules and crystals

Demazure modules are finite-dimensional subspaces of highest-weight representations: for each dominant weight + 휆 ∈ 푋(Φ)+ and Weyl group element 푤 ∈ 푊퐼 , the Demazure module 퐿푤 (휆) is a certain 푈 (Φ)-stable subspace of the integrable highest-weight module 퐿(휆). The Demazure modules give an inductive method to study the

퐿(휆), by analysing the successive embeddings 퐿푤 (휆) ⊆ 퐿푠푖푤 (휆) for the simple reflections 푠푖. In the case that 퐼 is finite type, a classic result due to Demazure is that the characters of these modules are relatedbythe Demazure operators 휋푖, with the inductive step being 휋푖(ch 퐿푤 (휆)) = ch 퐿푠푖푤 (휆), leading to the Demazure character formula 휆 ch 퐿푤 (휆) = 휋푤 (푒 ). There have been many proofs of the Demazure character formula when (퐼, ⋅) is not finite type, the most relevant to our work being the perspective of Demazure crystals. The theory of Demazure crystals was developed by Kashiwara and Littelmann, formalising the notion of a Demazure subcrystal ℬ푤 (휆) ⊆ ℬ(휆) as the analogue of a Demazure module 퐿푤 (휆) ⊆ 퐿(휆), and showing that the Demazure operators 휋푖 have combinatorial analogues on the level of crystals, the extension of strings operators 픇푖. Our analysis of the product monomial crystal in Chapter 7 follows along the same lines, breaking the crystal up into small pieces related by the extension of strings operators. We start this chapter with a digression into formal characters of representations and crystals, a topic we have not yet covered. The main point we want the reader to take away from this section is that the formal character gives a complete invariant of 풪int(Φ)-modules: two representations in this category with the same formal character are isomorphic. After this, we give the definition of Demazure modules and Demazure crystals, along withtheir formal characters called Demazure characters. We follow Kashiwara’s approach [Kas93] to the theory of Demazure crystals, the string property, and the extension-of-strings operators. Finally, we collect some history of Demazure modules and the Demazure character formula for the reader’s interest.

5.1 Formal Characters

Suppose that Φ is a root datum of finite type (퐼, ⋅), and we are working in the category of finite-dimensional weight representations of 픤(Φ). To every finite-dimensional weight representation 푉 we can associate its formal character recording the multiplicities of weight spaces:

휆 ch 푉 = ∑ (dim 푉휆)푒 ∈ ℤ[푋(Φ)]. (5.1.1) 휆∈푋(Φ)

Here ℤ[푋(Φ)] is the group algebra of the free abelian group 푋(Φ), written multiplicatively so that 푒휆푒휇 = 푒휆+휇. It is simple to verify that the characters of a direct sum add and the characters of a tensor product multiply, so we have ch(푈 ⊕ 푉 ) = ch 푈 + ch 푉 and ch(푈 ⊗ 푉 ) = (ch 푈 )(ch 푉 ). Furthermore, for any short exact sequence 0 → 푉 ′ → 푉 → 푉 ″ → 0 of 픤(Φ)-modules, we have ch(푉 ) = ch(푉 ′) + ch(푉 ″), and therefore the characters of isomorphic modules are equal. Characters give us useful invariants of modules, and in fact in the setting we are in (finite-dimensional represen- tions over a field of characteristic zero) they give us complete invariants: a finite-dimensional weight representation 푉 is determined up to isomorphism by its character ch 푉 . We will explain why this is the case.

The Weyl group 푊 = 푊퐼 acts on the space ℤ[푋(Φ)] of formal characters by permuting the standard basis: 푤 ⋅ 푒휆 = 푒푤휆, where 푤휆 is the usual action of the Weyl group on the weight lattice (Lemma 3.4.10). By considering the restriction of 푉 to each rank-one algebra 픤(Φ푖) and considering the classification of finite-dimensional 픰픩2

45 5 Demazure modules and crystals

modules, we see that 푉휆 and 푉푠푖휆 are isomorphic vector spaces. Since this works for all 푖 ∈ 퐼, we have that dim 푉휆 = dim 푉푤휆 for all 푤 ∈ 푊 , hence the character of a finite-dimensional representation 푉 is a member of the subring ℤ[푋(Φ)]푊 of Weyl-invariant characters. In fact, the characters of the 퐿(휆) for dominant 휆 form a basis for the subring of Weyl-invariant characters. 5.1.2 Lemma (A basis of the Weyl-invariant character ring)

Let Φ be a root datum of finite type (퐼, ⋅). Then the characters {ch 퐿(휆) ∣ 휆 ∈ 푋(Φ)+} of the highest-weight modules 퐿(휆) form a basis of the invariant ring ℤ[푋(Φ)]푊 . 5.1.3 Proof Since the Weyl orbit of a weight 휆 ∈ 푋(Φ) intersects the dominant weights 푋(Φ)+ exactly once, Weyl orbits can be parameterised by dominant weights. For each dominant weight 휆 define the orbit sum 푚휆 = 휇 푊 ∑휇∈푊 ⋅휆 푒 , then the set {푚휆 ∣ 휆 ∈ 푋(Φ)+} forms a basis of ℤ[푋(Φ)] . Since the highest-weight space of 퐿(휆) has dimension 1 and all other weights of 퐿(휆) are lower than 휆 in the partial ordering, we have

ch 퐿(휆) = 푚휆 + ∑ 푘휆,휇푚휇 for some 푘휆,휇 ∈ ℕ, (5.1.4) 휇<휆

1 showing that {ch 퐿(휆) ∣ 휆 ∈ 푋(Φ)+} is an alternative basis of the invariant character ring . The category of finite-dimensional weight modules of 픤(Φ) is semisimple because we are working over ℂ, hence the isomorphim class of 푉 is determined by the decomposition multiplicities [퐿(휆) ∶ 푉 ], which are given exactly by the expression of the character of 푉 in the basis of irreducible characters ch 푉 = ∑ [퐿(휆) ∶ 푉 ] ch 퐿(휆). 휆∈푋(Φ)+ Hence the character gives a complete invariant.

We now consider the case when (퐼, ⋅) is of arbitrary type. For a module 푉 ∈ 풪(Φ) the formal character Eq. (5.1.1) makes sense as a sum taking values in the completed group algebra ℤ[[푋(Φ)]], whose elements are infinite linear combinations of the 푒휆. This completion is no longer an algebra since the product of two elements may result in each coefficient being given by an infinite, rather than finite, sum. However,if 푈 , 푉 ∈ 풪(Φ) then the product (ch 푈 )(ch 푉 ) makes sense by the ‘weights bounded above’ condition on Category 풪(Φ) (Definition 3.5.3), and is equal to ch(푈 ⊗ 푉 ). The Weyl group still acts on the completed algebra, and the characters of integrable modules are Weyl-invariant, but we can no longer parametrise Weyl orbits in 푋(Φ) by dominant weights since not every orbit meets the dominant chamber. However, the characters of the integrable highest-weight modules 퐿(휆) are linearly independent, since if we have a sum ∑ 푎 ch 퐿(휆) = 0 we must have 푎 = 0 for any maximal 휆∈푋(Φ)+ 휆 휆 휆. Therefore the characters of modules in category 풪int(Φ) still give a complete invariant by semisimplicity, however we cannot package this up quite as nicely as in the finite case.

Modules over quantum groups and abstract Φ-crystals also have formal characters. In the case of abstract crystals, provided that the ‘weight spaces’ 퐵휆 = {푏 ∈ 퐵 ∣ wt(푏) = 휆} are finite we can define

휆 ch 퐵 = ∑ |퐵휆|푒 ∈ ℤ[푋(Φ)], (5.1.5) 휆∈푋(Φ) in exact analogy with Eq. (5.1.1). (Again, if (퐼, ⋅) is not finite type then ch 퐵 may be valued in the completed group algebra rather than the group algebra). The discussion above shows that if 퐵 is the crystal of a module from int 풪푞 (Φ), then 퐵 is determined up to isomorphism by ch 퐵, and hence characters also give a complete invariant of such crystals.

1 This fact would be immediate if (푋(Φ)+, ≤) were a finite partially ordered set, since Eq. (5.1.4) would show that the endomorphism 푚휆 ↦ ch 퐿(휆) is an upper-triangular matrix (in any total order refining ≤) with 1s along the diagonal, and hence has determinant 1. The analogous statement for infinite posets is not actually true in general, as one can see by considering the ℤ-module ℤ[ℤ] = −1 0 1 푖 푖 푖−1 spanℤ {… , 푒 , 푒 , 푒 ,…} and the linear map 푒 ↦ 푒 − 푒 . This satisfies the unitriangularity condition and is injective, but isnot surjective since every element of the image has its sum of coefficients equal to zero. The technical condition needed here isthat the poset (푋(Φ)+, ≤) satisfies the property: every non-empty subset contains a minimal element (as in VI.3.4, Lemma4of[BB02]). Alternatively, and perhaps more intuitively, one could take some filtration of 푋(Φ)+ with finite pieces and realise that if the mapis an isomorphism in each piece, so must it be in the limit.

46 5.2 Demazure modules and the character formula

5.2 Demazure modules and the character formula

The integrable highest-weight module 퐿(휆) is irreducible, and hence has no nontrivial 푈 (Φ)-submodules. How- ever, it may also be viewed as a 푈 +(Φ)-module, in which case it has many submodules. A particularly nice family of these submodules are the Demazure modules, each a finite-dimensional subspace of 퐿(휆) parameterised by the Weyl group 푊 . 5.2.1 Definition (Demazure module) Fix an irreducible integrable highest-weight module 퐿(휆) ∈ 풪int(Φ). The elements of the Weyl orbit 푊 ⋅ 휆 + are called the extremal weights of 퐿(휆). The Demazure module 퐿푤 (휆) ⊆ 퐿(휆) is defined as the 푈 (Φ)-orbit of the one-dimensional space 퐿(휆)푤휆 inside 퐿(휆). We say that 퐿푤 (휆) has Demazure lowest weight 푤휆. + For any dominant 휆, the Demazure module 퐿푒(휆) associated to the identity element 푒 ∈ 푊 is 푈 (Φ)⋅푉 (휆)휆 = 푉 (휆)휆, the one-dimensional highest-weight space. At the other extreme, when 퐼 is finite-type and 푤퐼 ∈ 푊 is the longest element, 퐿(휆)푤퐼 휆 is the lowest-weight space and hence the Demazure module 퐿푤퐼 (휆) is equal to the whole module 퐿(휆). If 퐼 is not finite-type then there is no longest element of the Weyl group, but we can still realise thefull representation 퐿(휆) as a limit of the finite-dimensional 퐿푤 (휆), since whenever 푥 ≤ 푦 in the Bruhat order we have 퐿푥 (휆) ⊆ 퐿푦 (휆).

For each 푖 ∈ 퐼 define the ℤ-linear Demazure operator 휋푖 ∶ ℤ[푋] → ℤ[푋] by the formula −훼 푓 − 푒 푖 (푠푖 ⋅ 푓 ) 휋푖(푓 ) = . (5.2.2) 1 − 푒−훼푖 This formula can be demystified a little by writing it out explicitly as a geometric series:

휆 휆−훼푖 푠푖(휆) ∨ ⎧푒 + 푒 + ⋯ + 푒 if ⟨휆, 훼푖 ⟩ ≥ 0, 휋 (푒휆) = 0 if ⟨휆, 훼∨⟩ = −1, (5.2.3) 푖 ⎨ 푖 휆+훼 휆+2훼 푠푖(휆)−훼 ∨ ⎩−(푒 + 푒 + ⋯ + 푒 ) if ⟨휆, 훼푖 ⟩ ≤ −2.

It is straightforward to verify that if 푓 ∈ ℤ[푋(Φ)] is symmetric in 푠푖, meaning 푠푖푓 = 푠푖, then 휋푖(푓 ) = 푓 . Together 2 푠푖 with the property 푠푖 ∘ 휋푖 = 휋푖, this implies that 휋푖 = 휋푖 is a projector to the subspace ℤ[푋] of 푠푖-symmetric characters. It takes considerably more work (a case-by-case analysis of the cases 푚푖푗 ∈ {2, 3, 4, 6}) to verify that the Demazure operators satisfy the braid relations (Eq. (3.3.2)), i.e. we have

⏟⏟⏟⏟⏟⏟⏟⏟⏟휋푖휋푗휋푖 ⋯ = 휋⏟⏟⏟⏟⏟⏟⏟⏟⏟푗휋푖휋푗 ⋯, (5.2.4)

푚푖푗 letters 푚푖푗 letters a fact we state but will not prove here. Since they do satisfy the braid relations, by Matsumoto’s theorem 휋푤 is well-defined for any 푤 ∈ 푊 by setting 휋푤 = 휋푠1 ⋯ 휋푠푘 where (푠1, … , 푠푘) is any reduced expression for 푤 (see Re- mark 5.2.10 for an interpretation of this in terms of zero-Hecke representations). We can now state the Demazure character formula, originally due to Demazure [Dem74] in finite type and numerous others (see Section 5.4) in arbitrary type. 5.2.5 Theorem (Demazure character formula)

Let Φ be a root datum of type (퐼, ⋅), 휆 ∈ 푋(Φ)+ a dominant weight, and 푤 ∈ 푊 an element of the Weyl group. Then the character of the Demazure module 퐿푤 (휆) is

휆 ch 퐿푤 (휆) = 휋푤 (푒 ). (5.2.6)

휆 Furthermore, if (퐼, ⋅) is finite type then ch 퐿(휆) = ch 퐿푤퐼 (휆) = 휋푤퐼 (푒 ).

Each Demazure module is parametrised by a pair (휆, 푤) of a dominant weight 휆 ∈ 푋(Φ)+ and a Weyl element 푤 ∈ 푊 , but this parametrisation has some redundancy. If 휆 is not a regular weight, then the stabiliser 푊휆 = Stab푊 (휆) is nontrivial, and the Demazure submodules 퐿푥 (휆) and 퐿푦 (휆) are equal whenever 푥 = 푦 in the quotient 푊 /푊휆. When (퐼, ⋅) is finite type and we are working in the category of finite-dimensional 픤(Φ)-modules, we can remove this redundancy by parametrising Demazure modules instead by their Demazure lowest weights.

47 5 Demazure modules and crystals

5.2.7 Lemma (The Demazure basis of the character ring) Let Φ be a root datum of finite type (퐼, ⋅), and let 퐷(휇) be the Demazure module of Demazure lowest weight 휇 ∈ 푋(Φ). Then the formal characters {ch 퐷(휇) ∣ 휇 ∈ 푋(Φ)} form a basis of the character ring ℤ[푋(Φ)]. 5.2.8 Proof The definition of 퐷(휇) is not ambiguous, since if we take any two Demazure modules of the same Demazure lowest weight 휇, they must be submodules of 퐿(휆) where 휆 is that unique dominant weight in the orbit 푊 ⋅휇. 휇 휈 We have ch 퐷(휇) = 푒 + ∑휈>휇 푑휇,휈 푒 for some 푑휇,휈 ∈ ℕ, and therefore the Demazure characters are linearly independent inside ℤ[푋(Φ)] via a triangularity argument. The fact that they form a basis can be seen by taking a filtration of ℤ[푋(Φ)] by suitable finite-rank spaces, such as sums supported only over the convex hull of the Weyl orbit of finitely many weights. In the case that (퐼, ⋅) is not finite type, we need to be a little more careful about which weights we permit inthe definition of 퐷(휇) as the ‘Demazure module of Demazure lowest weight 휇’. Since we only want to be capturing Demazure submodules of the 퐿(휆) for dominant 휆, we need to ensure that 휇 is in the Weyl orbit of some dominant weight. Define the Tits cone as the Weyl orbit of the dominant chamber: 퐾(Φ) = ⋃푤∈푊 푤푋(Φ)+ ⊆ 푋(Φ). Then 퐾(Φ) is the whole of the weight lattice if and only if (퐼, ⋅) is finite type (Proposition 1.4.2 of[Kum02]), and so indeed outside of finite type there are weights 휇 which are not in the Weyl orbit of any dominant weight. The dominant chamber 푋ℝ(Φ)+ is a fundamental domain for the action of 푊 on 퐾ℝ(Φ) however, and so for any 휇 in 퐾(Φ) there is a unique dominant weight in its Weyl orbit. So we can speak of the Demazure module with Demazure lowest weight 휇 provided that we limit ourselves to those weights 휇 in the Tits cone. 5.2.9 Lemma (Demazure characters of highest-weight modules are linearly independent) Let Φ be a root datum of arbitrary type (퐼, ⋅), and for any weight 휇 in the Tits cone 퐾(Φ), let 퐷(휇) be the Demazure module with lowest weight 휇. Then the formal characters {ch 퐷(휇) ∣ 휇 ∈ 퐾(Φ)} are linearly independent in the completed algebra ℤ[[푋(Φ)]].

2 We conclude this section with a discussion of the relations 휋푖 = 휋푖 and the braid relations into a broader frame- work, that of Hecke algebras. 5.2.10 Remark (Zero-Hecke actions) Let (푊 , 푆) be the Coxeter system associated to the Cartan datum (퐼, ⋅), as defined in Definition 3.3.1. Given a commutative ring 푅 and two parameters 휆, 휇 ∈ 푅, we may define the Hecke algebra ℋ (휆, 휇) to be the associative 푅-algebra generated by {푇푖 ∣ 푖 ∈ 퐼}, with the two relations

2 Quadratic relation: 푇푖 = 휆푇푖 + 휇, Braid relation: ⏟⏟⏟⏟⏟⏟⏟⏟⏟푇푖푇푗푇푖 ⋯ =⏟⏟⏟⏟⏟⏟⏟⏟⏟ 푇푗푇푖푇푗 ⋯ . (5.2.11)

푚푖푗 푚푖푗

For each 푤 ∈ 푊 , define 푇푤 = 푇푖1 ⋯ 푇푖푟 for some reduced expression (푖1, … , 푖푟 ) of 푤. Since the 푇푖 satisfy the braid relations, Matsumoto’s theorem implies that the 푇푤 are independent of the choice of reduced expression. After much work (See Chapter 7of[Hum90] for a full proof, or Exercise 2.23 in Chapter IV of [BB02] if you don’t want the fun spoiled), it turns out that ℋ (휆, 휇) is free as an 푅-module on the basis {푇푤 ∣ 푤 ∈ 푊 }, with left multiplication by a generator 푇푖 given by the rule

푇푠푖푤 if ℓ(푠푖푤) > ℓ(푤) 푇푖푇푤 = { (5.2.12) 휆푇푤 + 휇푇푠푖푤 if ℓ(푠푖푤) < ℓ(푤). The choice of 휆 and 휇 (the choice of quadratic relation) gives different algebras:

2 1. The quadratic relation 푇푖 = 1 makes ℋ (0, 1) isomorphic to the group algebra 푅푊 . 2 2. The quadratic relation 푇푖 = 0 makes ℋ (0, 0) an algebra called the nil-Hecke ring. The cell ordering on this ring with respect to the standard basis {푇푤 ∣ 푤 ∈ 푊 } is precisely the Bruhat ordering. 2 3. The quadratic relation 푇푖 = 푇푖 makes ℋ (1, 0) an algebra called the zero-Hecke ring.

48 5.2 Demazure modules and the character formula

2 The Demazure operators we have just defined satisfy the braid relation and the quadratic relation 휋푖 = 휋푖, and hence they define an action of the zero-Hecke ring (defined over 푅 = ℤ) on the space ℤ[푋(Φ)] of formal characters by sending 푇푖 to 휋푖. This is useful to know, since we re-use the multiplication rule above to see that

휋푠푖푤 if ℓ(푠푖푤) > ℓ(푤) 휋푖휋푤 = { (5.2.13) 휋푤 if ℓ(푠푖푤) < ℓ(푤). We will eventually make use of this to argue that our character formula for truncations of the product monomial crystal gives a formula for the whole crystal in the case that (퐼, ⋅) is finite type.

49 5 Demazure modules and crystals

5.3 Demazure crystals

When working with modules over the quantum group 푈푞(Φ) we can define Demazure modules in the same way + as Definition 5.2.1: for any dominant 휆 ∈ 푋(Φ)+ and Weyl group element 푤 ∈ 푊퐼 , let 퐿푞,푤 (휆) = 푈푞 (Φ) ⋅ 퐿푞(휆)푤휆 + be the orbit of the one-dimensional extremal weight space 퐿푞(휆)푤휆 under the positive half 푈푞 (Φ) of the quantum group. It was shown in [Kas93] that the quantum Demazure module admits a crystal base, which can be taken to be a subset of the crystal base ℬ(휆) of 퐿푞(휆). In order to specify this subset, we introduce the following operators. 5.3.1 Definition (Extension of strings)

Let 퐵 be a Φ-crystal. For each 푖 ∈ 퐼, define the extension of 푖-strings operator 픇푖, which takes a subset 푍 ⊆ 퐵 to the set

푛 푛 픇푖푍 = ⋃{푓푖 (푥) ∣ 푧 ∈ 푍} = {푏 ∈ 퐵 ∣ 푒푖 (푏) ∈ 푍 for some 푛 ∈ ℕ}. (5.3.2) 푛≥0

These operators satisfy 푍 ⊆ 픇푖푍 ⊆ 퐵, and 픇푖픇푖푍 = 픇푖푍. It is unclear to the author whether they satisfy the braid relations on arbitrary subsets of the crystal, however they do braid when 푍 = {푏휆} the highest-weight element. 5.3.3 Example

The picture below is the GL3 crystal ℬ(2, 0, 0), previously seen in Examples 4.2.10 and 4.5.4. If the subset 푍 ⊆ ℬ(2, 0, 0) is the two middle elements 푍 = {(2, 2), (1, 3)}, then 픇1(푍) = 푍 ∪ {(2, 3)} and 픇2(푍) = 푍 ∪ {(2, 3), (3, 3)}. 푓1 푓2 2 2 1 1 1 2 2 3 3 3 1 3

We can now give the definition of the Demazure crystal ℬ푤 (휆). 5.3.4 Definition (Demazure crystal)

Fix a dominant weight 휆 and a Weyl group element 푤 ∈ 푊 . Let (푖1, … , 푖푟 ) be a reduced expression for 푤. The Demazure crystal is the set

ℬ푤 (휆) = 픇푖1 ⋯ 픇푖푟 {푏휆}, (5.3.5) equipped with the canonical upper-seminormal crystal structure restricted from ℬ(휆).

An interesting difference is that while the quantum Demazure module 퐿푞,푤 (휆) is defined from the bottom up- + wards, by taking a lower weight and orbiting it under the positive half 푈푞 (Φ) of the quantum group, the Demazure crystal is defined from the top down, starting with a highest weight and taking just enough extensions of strings in the correct order. Something that might help the reader bridge this mental gap is Corollary 3.2.2 from [Kas93]: if (푖1, … , 푖푟 ) is a reduced expression for 푤, then 푘 퐿 (휆) = ∑ ℚ(푞)퐹 푘1 ⋯ 퐹 푟 푣 , (5.3.6) 푞,푤 푖1 푖푟 휆 푘1,…,푘푟 ≥0 where 푣휆 ∈ 퐿푞(휆) is a highest-weight vector. This shows that the quantum Demazure module 퐿푞,푤 (휆) can also be generated from the top downwards by ‘extending root strings’. 5.3.7 Example (An example of Demazure crystals)

Let Φ be the root datum of SL3, of Cartan type A2. The highest root is 휃 = 훼1 + 훼2 = 휛1 + 휛2, and 퐿(휃) is the 2 2 adjoint representation of the semisimple Lie algebra 픤(Φ) = 픰픩3. The Weyl group 푊 = ⟨푠, 푡 ∣ 푠 = 푡 = 1, 푠푡푠 = 푡푠푡⟩ has six elements, and since 휃 is a regular weight there are six distinct Demazure subcrystals of ℬ(휃).

50 5.3 Demazure crystals

Each of these Demazure subcrystals ℬ푤 (휃) are shown in the following diagram, with the Demazure lowest weight element 푤 ⋅ 푏휃 marked with a double circle (recall the action of 푊 on ℬ(휃) from Theorem 4.5.2).

픇 2 픇1

ℬid(휃) = {푏휃 }

ℬ2(휃) ℬ1(휃)

픇1 픇2

ℬ12(휃) ℬ21(휃)

ℬ푤퐼 (휃) = ℬ(휃) 픇2 픇1

In the two crystals ℬ12(휃) and ℬ21(휃) there is an element which cannot be reached by following arrows backwards from the Demazure lowest weight element. Hence Demazure crystals really do need to be defined from the top down, rather than from the bottom up. Kashiwara gave an alternative proof of the Demazure character formula, using an interesting equivariance property of the extension-of-strings operator 픇푖 and the Demazure operator 휋푖. Before we introduce this equivariance, we need to state what we mean by the string property, a definition first introduced in[Kas93].

51 5 Demazure modules and crystals

5.3.8 Definition (String property) Let Φ be a root datum of type (퐼, ⋅), and let 퐵 be a seminormal Φ-crystal. We say that a subset 푍 ⊆ 퐵 has the string property if for any 푖-string 푆 ⊆ 퐵, either 푆 ∩ 푍 = 푆, 푆 ∩ 푍 = ∅, or 푆 ∩ 푍 = 푆top, where 푆top ∈ 푆 is that top unique element at the top of the 푖-string, satisfying 푒푖(푆 ) = ⊥. We have encountered 푖-strings before, when defining the action of the Weyl group on a crystal (Theorem 4.5.2). Shown below is the SL3-crystal ℬ(2, 0, 0) , its 1-strings, its 2-strings, and a certain subset 푍 ⊆ ℬ(2, 0, 0) of cardinality 5 indicated by the circled points. The subset 푍 does not satisfy the string property, since the 2-string with 3 elements intersects 푍 in two elements.

ℬ(2, 0, 0) 1-strings 2-strings A ’bad’ subset 푍

On the other hand, all of the Demazure crystals shown in Example 5.3.7 satisfy the string property — this turns out to be a general property of Demazure crystals, though is quite difficult to prove, and indeed is the ‘deepest’ result in Kashiwara’s proof of the Demazure character formula. We will not reproduce this proof here as it requires − appealing to the theory of global bases and the crystal ℬ(∞) of the Verma module 푈푞 (Φ), we merely state the result. 5.3.9 Theorem (Demazure crystals satisfy the string property) (Proposition 3.3.5 of [Kas93]). Let Φ be a root datum of type (퐼, ⋅), 휆 a dominant weight, and 푤 ∈ 푊 a Weyl group element. The Demazure crystal ℬ푤 (휆) satisfies the string property.

The last ingredient in Kashiwara’s proof is relating the extension of strings operator 픇푖 to the Demazure operator 휋푖. Consider a seminormal Φ-crystal 퐵 and any 푖-string 푆 ⊆ 퐵. By seminormality the string 푆 is finite, having some 푖-highest element 푆top and some 푖-lowest element 푆bot. The weight 휆 of the 푖-highest element 푆top must be ∨ 푖-dominant, meaning ⟨휆, 훼푖 ⟩ ≥ 0, and the balanced-strings axiom implies that the weight of the 푖-lowest element bot ∨ 푆 is 푠푖휆 = 휆 − ⟨휆, 훼푖 ⟩훼푖. The situation is pictured below: the reader should imagine that this is plotted inside ∨ the weight lattice 푋(Φ), with the reflecting hyperplane 퐻푖 = ker⟨−, 훼푖 ⟩ straight down the centre.

휆 휆 − 훼푖 푠푖휆 + 훼푖 푠푖휆

top bot 푆 퐻푖 푆

Now we want to consider the action of the Demazure operator on the character ch 푆. Recall from Eq. (5.2.3) that we can treat the Demazure operator as a geometric series. Since 휆 is 푖-dominant and 푠푖휆 is 푖-antidominant, we have 휆 휆−훼 푠 휆 휋푖(휆) = 푒 + 푒 푖 + ⋯ + 푒 푖 = ch 푆 (5.3.10) 휆−훼 푠 휆+훼 휆−훼 휋푖(푠푖휆) = − (푒 푖 + ⋯ + 푒 푖 푖 ) = −휋푖(푒 푖 ). These two series can be pictured as follows:

휋푖(휆)

휆 휆 − 훼푖 푠푖휆 + 훼푖 푠푖휆

−휋푖(푠푖휆)

52 5.4 History: The Demazure character formula

휆 We can see now that 휋푖(ch 푆) = ch 푆, by noting that almost every weight contributed by 휋푖(푒 ) is later removed 푠 휆 휆 푠 휆 휆 푠 휆 by 휋푖(푒 푖 ), leaving 휋푖(푒 + 푒 푖 ) = 푒 + 푒 푖 . Working inwards from there shows that 휋푖(ch 푆) = ch 푆, or we could have appealed to the general fact that 휋푖 acts as the identity on 푠푖-invariant characters. The key observation is 휆 that ch 푆 = 휋푖(푒 ). We are now ready to state and prove the equivariance property. 5.3.11 Lemma Let Φ be a root datum of type (퐼, ⋅), and 퐵 a seminormal Φ-crystal. If 푍 ⊆ 퐵 is a subset with the string property, then ch 픇푖(푍) = 휋푖(ch 푍) for all 푖 ∈ 퐼. 5.3.12 Proof Fix an 푖 ∈ 퐼. By the additivity of characters on one hand, and the fact that 픇푖 only depends on the 푖-strings on the other, it suffices to show that the theorem holds on each intersection 푍 ∩ 푆 with the 푖-strings 푆 ⊆ 퐵. Since 푍 satisfies the string property, for each 푖-string 푆 ⊆ 퐵 there are only three cases to check: 1. If 푆 ∩ 푍 = ∅, then the theorem holds since 0 = 0. top top 2. If 푆 ∩ 푍 = {푆 }, let 휆 = wt 푆 . Then 픇푖(푆 ∩ 푍) = 푆, and by the above discussion we have ch 푆 = 휆 top 휋푖(푒 ) = 휋푖(ch 푆 ), so the theorem holds. 3. If 푆 ∩ 푆 = 푆, then 픇푖(푆) = 푆 and hence the theorem holds. It seems that this theorem would immediately imply the Demazure character formula, but one really does need to prove Theorem 5.3.9 somehow. It is not true in general that if 푍 satisfies the string property then 픇푖(푍) does also — see Section 13 of [BS17] for a counterexample. However, it is now clear to see that the Demazure character formula is a direct consequence of Theorem 5.3.9 and Lemma 5.3.11, and the fact that ℬ푤 (휆) is a crystal base of 퐿푞,푤 (휆).

5.4 History: The Demazure character formula

Demazure modules were historically considered in a different way to how they have been presented above. Aroot datum Φ of finite type (퐼, ⋅) determines a reductive group scheme 퐺 = 퐺(Φ) equipped with a pinning 푇 ⊆ 퐵 ⊆ 퐺, defined as a scheme over ℤ and hence over any field 푘 via base change. The quotient of the group 퐺 by its Borel subgroup 퐵 is called the flag variety, a projective scheme. The category of 퐺-equivariant line bundles on 퐺/퐵 is equivalent to the one-dimensional representations of 퐵, which are seen (via a factorisation 퐵 = 푇 푈 into a product of the torus 푇 , and the unipotent group 푈 which must act trivially) to simply be the weights 푋(Φ). For each weight 휆 ∈ 푋(Φ) there is a line bundle ℒ휆 over 퐺/퐵, whose vector space of sections Γ(퐺/퐵, ℒ휆) is nonzero precisely when 휆 is dominant.

When the base field 푘 = ℂ and 휆 ∈ 푋(Φ)+ is dominant, the vector space Γ(퐺/퐵, ℒ휆) of sections is precisely the highest-weight module 퐿(휆) we have been considering thus far, with the 퐺-action canonically given by 퐺- equivariance of ℒ휆. Where the Demazure modules enter the picture is by considering the restriction of the line bundle ℒ휆 to the Schubert variety 푋(푤) = 퐵 ̇푤퐵 , the closure of the Bruhat cell 퐵 ̇푤퐵. (In the setting of reductive group schemes, the Weyl group may be realised as the quotient 푊 ∶= 푁퐺(푇 )/푇 of the normaliser of the maximal torus 푇 inside 퐺. We use the notation ̇푤∈ 퐺 to denote any lift of the Weyl group element 푤 ∈ 푊 back to 퐺). The space of sections Γ(푋(푤), ℒ휆) is the Demazure module 퐿푤 (휆), which is a 퐵-module, but not a 퐺-module unless 푋(푤) = 퐺, i.e. 푤 = 푤퐼 the longest element. A formula for the characters of the Demazure modules (in the setting above, with 퐺 semisimple and 푘 = ℂ) was first given by Demazure [Dem74]. Later on in 1983, Victor Kac was attempting to generalise this formula to infinite root systems and found that the proof contained a gap, in an argument due to Verma. Thiscreateda ‘spate’ of correct proofs in the following few years [Jos95], in which many new techniques were developed.

Mehta and Ramanathan [MR85] introduced the notion of a Frobenius split variety, for which the higher coho- mologies of ample line bundles vanish, and showed that the Schubert varieties are Frobenius split. Ramanan and Ramanathan [RR85] and Seshadri proved the projective normality of Schubert varieties, and improved the result of Mehta and Ramanathan to include effective line bundles. This work culminated with Andersen [And85] also

53 5 Demazure modules and crystals contributing a proof of the Demazure character formula, valid over reductive algebraic groups in any characteristic. This problem also kicked off the systematic study of 퐵-modules, a good survey of which is [KI93]. Joseph [Jos85; Jos86] defined functors (now called Joseph functors) 퐻푤 ∶ Rep 퐵 → Rep 퐵 taking a 퐵-module 푀 to the space of sections Γ(푋(푤), ℒ(푀)) of the vector bundle ℒ(푀) over the Schubert variety 푋(푤) ⊆ 퐺/퐵, and was able to prove Demazure’s original result for 휆 sufficiently large. Later, both Kumar [Kum87] and Mathieu [Mat88] independently proved the Demazure character formula, this time in the setting of an arbitrary Kac-Moody algebra and an integrable highest-weight module. Our passage through this chapter has been following the much later work of both Littelmann and Kashiwara, from the perspective of crystals. Littelmann was involved in the development of path models for representations, bases indexed by certain piecewise-linear paths through the weight lattice, with root operators (crystal operators) given by operations on paths, and a positive combinatorial rule for the tensor product multiplicities given in terms of paths [Lit95b; Lit90]. It became clear that these paths could in fact index crystal bases, with the tensor product rule immediately following from the crystal tensor product, a fact which was shown in [Lit95a]. In this paper, Littelmann conjectured the Demazure character formula for Demazure crystals, which was later proven by Kashiwara in [Kas93].

54 6 Monomial crystals

In this chapter we introduce the monomial crystal ℳ(Φ) associated to a root datum Φ, first defined by Nakajima. The monomial crystal is very large, and inside of it we can find infintely many copies of ℬ(휆) for any dominant weight 휆, as well as more exotic crystals in the case where (퐼, ⋅) is not finite type. The monomial crystal has been used to study extremal weight crystals of affine algebras, each connected component being isomorphic to a subcrystal of such an extremal weight crystal. As the underlying set of the monomial crystal ℳ(Φ) is an abelian group, the group operation being multiplication of monomials, it is possible to form a monomial-wise product (rather than crystal tensor product) of subcrystals. If we are careful about the kind of subcrystals we start with, the resulting set of products is again a subcrystal of ℳ(Φ) called the product monomial crystal. The proof we know of this fact is quite involved and only works when (퐼, ⋅) is simply-laced, as it goes via the geometry of Nakajima quiver varieties. We defer this proof to Chapter 8. In this chapter we first state the definition of the monomial crystal and give a feel for how it works, before moving on to define the product monomial crystal and start introducing terminology which will help us analyse it.

6.1 Nakajima’s Monomial Crystal

The monomial crystal is originally due to Nakajima, first appearing in Section 3 of[Nak02] and Section 3 of [Kas02b]. A later definition appearing in [HN06] is a straightforward modification of the original to make the crystal make sense for an arbitrary root datum Φ rather than a simply connected one. We adopt this later definition since it will make the arguments in Chapter 10 a lot more pleasant, allowing us to work with weights of GL푛 rather than SL푛, but we will point out the simplification in simply-connected type. The other thing to notice about this definition is that we require the Cartan datum (퐼, ⋅) to be bipartite: the crystal structure does not work otherwise (see Example 6.1.10 for a counterexample). 6.1.1 Definition (The monomial crystal) Let (퐼, ⋅) be a bipartite Cartan datum with a fixed two-colouring 휁 ∶ 퐼 → ℤ/2ℤ. For a Kac-Moody root datum Φ of type (퐼, ⋅), let 퐴(Φ) be the product 푋(Φ) × ℤ[퐼 × ℤ] of abelian groups, each written multiplicatively so that a typical element 푝 ∈ 퐴(Φ) is of the form

휔(푝) 푝[푖,푐] 푝 = 푒 ⋅ ∏ 푦푖,푐 for some 휔(푝) ∈ 푋 and 푝[푖, 푐] ∈ ℤ, (6.1.2) (푖,푐)∈퐼×ℤ where 푝[푖, 푐] ≠ 0 for only finitely many (푖, 푐) ∈ 퐼 × ℤ. For each (푖, 푘) ∈ 퐼 × ℤ, define the auxiliary monomial

푎 훼푖 푗푖 푧푖,푘 = 푒 ⋅ 푦푖,푘 ⋅ 푦푖,푘+2 ⋅ ∏ 푦푗,푘+1. (6.1.3) 푗≠푖 Let ℳ(Φ) ⊆ 퐴(Φ) be the subgroup defined by the two conditions

∨ ⟨휔(푝), 훼푖 ⟩ = ∑ 푝[푖, 푙] for all 푖 ∈ 퐼, (6.1.4) 푙∈ℤ

푝[푖, 푐] = 0 if 푐 ≢ 휁 (푖) (mod 2). (6.1.5) For each monomial 푝 ∈ ℳ(Φ), define

55 6 Monomial crystals

푘 1. 휑푖 (푝) = ∑푙≥푘 푝[푖, 푙], the upper column sum. 푘 2. 휑푖(푝) = max푘 휑푖 (푝), the largest upper column sum. 푘 3. 휀푖 (푝) = − ∑푙≤푘 푝[푖, 푙], the negated lower column sum. 푘 4. 휀푖(푝) = max푘 휀푖 (푝), the largest negated lower column sum. 푘 푘 5. 푛푓 ,푖(푝) = max{푘 ∈ ℤ ∣ 휑푖 (푝) = 휑푖(푝)}, the largest 푘 maximising the upper column sum 휑푖 (푝). 푘 푘 6. 푛푒,푖(푝) = min{푘 ∈ ℤ ∣ 휀푖 (푝) = 휀푖(푝)}, the smallest 푘 maximising the negated lower column sum 휀푖 (푝).

Note that 푛푓 ,푖(푝) is undefined if 휑푖(푝) = 0, and 푛푒,푖(푝) is undefined if 휀푖(푝) = 0. We set wt(푝) = 휔(푝), and define the crystal operators 0 if 휀푖(푝) = 0 푒푖(푝) = { (6.1.6) 푝푧푖,푛푒,푖(푝) otherwise,

0 if 휑푖(푝) = 0 푓 (푝) = { (6.1.7) 푖 푝푧−1 otherwise. 푖,푛푓 ,푖(푝)−2

The monomial crystal is the set ℳ(Φ), equipped with the crystal structure (wt, 푒푖, 푓푖, 휀푖, 휑푖) given above. The definition can be simplified a little in the case that Φ is simply-connected and of finite type, by erasing the 푒휆 term in each monomial and taking wt(푦푖,푘) to be the fundamental weight 휛푖. When allowing arbitrary root data, 휆 we instead enforce the condition that in any monomial 푒 ⋅ 푧, the number of 푦푖,• appearing in 푧 must be equal ∨ to ⟨휆, 훼푖 ⟩ — this is condition (1) above. The parity condition (2) is required in order to satisfy the partial inverse axiom of a crystal, and relies on the existence of a two-colouring 휁 ∶ 퐼 → ℤ/2ℤ.

The definition of ℳ(Φ) can be understood pictorially. Let 퐼 ×ℤ̇ ⊆ 퐼 ×ℤ denote the set of parity-respecting pairs1

퐼 ×̇ ℤ = {(푖, 푐) ∈ 퐼 × ℤ ∣ 휁 (푖) = 푐 in ℤ/2ℤ} . (6.1.8)

Forgetting the ‘tag’ 푒휆 for a moment, a monomial 푝 ∈ ℳ(Φ) can be thought of as a finitely-supported function 푝 ∶ 퐼 ×̇ 푍 → ℤ. We can draw the set 퐼 ×̇ ℤ in the plane and label it with the values 푝[푖, 푐]. The monomial

휆 8 −8 8 5 6 −4 7 푝 = 푒 ⋅ 푦2,−2 ⋅ 푦2,0 ⋅ 푦2,2 ⋅ 푦3,1 ⋅ 푦4,−2 ⋅ 푦4,2 ⋅ 푦5,−1 (6.1.9)

−2 in Cartan type A5 = {1, 2, 3, 4, 5} is pictured below, with example computations of the upper column sum 휑2 (푝) 0 and the negated lower column sum 휀4 (푝), and the largest integer 푛푓 ,2(푝) maximising the upper column sum • 휑2(푝).

ℤ

3 −2 8 −4 휑2 (푝) = 8 2 5 1 −8 0 0 휀4 (푝) = −6 7 −1 8 6 −2 푛푓 ,2(푝) = 2 −3

퐼 = A 5 1 2 3 4 5

1The definition of 퐼 ×̇ ℤ really depends on the choice of 2-colouring 휁 .

56 6.1 Nakajima’s Monomial Crystal

This monomial diagram captures the 푦푖,푐 terms but not 휆, but due to the weight constraint Eq. (6.1.4) all of the ∨ inner products ⟨휆, 훼푖 ⟩ are determined by the 푦푖,푐. Even though we have not yet specified a root datum Φ we can ∨ be sure that 휆 must satisfy ⟨휆, 훼4 ⟩ = 6 − 4 = 2. If Φ is the simply-connected datum Φ = SL6 then we must have 휆 = 8휛2 + 5휛3 + 2휛4 + 7휛5 as a sum of fundamental weights.

Next, we look at the action of the crystal operator 푓2 on our monomial 푝. The maximum value of the upper 푘 −2 2 column sum 휑2 (푝) in column 2 is achieved at 휑2 (푝) = 휑2(푝) = 8, hence 푛푓 ,2(푝) = max {2, −2} = 2 as indicated −1 on the diagram. By Eq. (6.2.4) the action of the crystal operator 푓2 on the monomial 푝 is multiplication by 푧2,0, −1 so we have 푓2(푝) = 푝 ⋅ 푧2,0. Recalling the definition of the auxiliary monomial 푧푖,푘 from Eq. (6.1.3), we can picture this multiplication as follows:

푓2

−1 −1 푝 푧2,0 푝 ⋅ 푧2,0 ℤ 3 8 −4 −1 7 −4 2 5 1 1 1 6 1 −8 −1 −9 0 7 7 −1 8 6 8 6 −2 −3

The reader could now check that the 푒푖 operator will act at the correct spot, so that we have 푒푖(푓푖(푝)) = 푝. Also remember that we have excluded the weight wt(푝) = 휆 from our diagrams, but one can simply imagine that each monomial is a pair consisting of a compatible weight 휆 and a diagram determined by the 푦푖,푘, and remember that 푧푖,푘 has weight 훼푖.

We present a few more examles of the 푧푖,푘 monomials in types A3, B3, and D4. In each picture the auxiliary monomial 푧푖,푘 is shown with the point (푖, 푘) circled in green. It is clear that 푧푖,푘 has a straightforward interpretation in terms of the Dynkin diagram.

1 1 1 1 −1 −1 1 −2 −1 −1 −1 −1 −1 1 −1 1 1 1 1

A3 A3 B3 B3 D4

We now resume our formal discussion of the monomial crystal. We first point out that without the parity condition Eq. (6.1.5), the set ℳ(Φ) would not be a Φ-crystal as it would fail the partial inverse axiom. Therefore we really do need to have the Cartan datum (퐼, ⋅) bipartite.

57 6 Monomial crystals

6.1.10 Example (Necessity of parity condition)

This counterexample appears in Example 3.3 of [Kas02b]. Let Φ a root datum of Cartan type (퐼, ⋅) = A1. 훼 Here 퐼 = {푖} is a single vertex, and the monomial 푧푖,푘 takes the simple form 푧푖,푘 = 푒 푖 푧푖,푘푧푖,푘+1. Consider the 0 −1 monomial 푝 = 푒 ⋅ 푦푖,2 ⋅ 푦푖,1 acted on first by 푓푖 and then by 푒푖:

푓푖 푒푖 0 −1 −훼푖 −1 −1 0 −1 푝 = 푒 ⋅ 푦푖,2 ⋅ 푦푖,1 −→푒 ⋅ 푦푖,1 ⋅ 푦푖,0 −→푒 ⋅ 푦푖,3 ⋅ 푦푖,0 . (6.1.11)

We find that even though 푓푖 is defined on 푝, we have 푒푖(푓푖(푝)) ≠ 푝 which is a violation of the partial inverse axiom of a crystal. In terms of a monomial diagram, the sequence above appears as: 1 3 1 2 푓푖 푒푖 −1 −1 1 −1 −1 0

(The above picture is not a crystal graph, since we would usually omit the 푒푖 from a crystal graph for the very reason that 푒푖 and 푓푖 are inverse partial functions). We now check directly that ℳ(Φ) is an abstract seminormal Φ-crystal, paying close attention to the partial inverse axiom in light of the counterexample above. 6.1.12 Theorem The set ℳ(Φ) together with the maps wt, 휀푖, 휑푖, 푒푖, 푓푖 for 푖 ∈ 퐼 is an abstract seminormal Φ-crystal. 6.1.13 Proof We first show that ℳ(Φ) is an abstract crystal, by checking the axioms. 푘 ∨ 푘+1 1. Balanced-strings: we have the property that 휀푖 (푝) + ⟨wt(푝), 훼푖 ⟩ = 휑푖 (푝). Then ∨ 푘 ∨ 푘+1 휀푖(푝) + ⟨wt(푝), 훼푖 ⟩ = max (휀푖 (푝) + ⟨wt(푝), 훼푖 ⟩) = max 휑푖 (푝) = 휑푖(푝). (6.1.14) 푘∈ℤ 푘∈ℤ

2. Raising operators: Suppose that 푒푖(푝) ≠ 0. Set 푣 = 푛푒,푖(푝), so by definition we have 푘 푣 푙 휀푖 (푝) < 휀푖 (푝) ≥ 휀푖 (푝) for all 푘 < 푣 < 푙. (6.1.15)

푎푗푖 Since 푒푖(푝) = 푝 ⋅ 푧푖,푣 = 푝 ⋅ 푦푖,푣 ⋅ 푦푖,푣+2 ⋅ ∏푗≠푖 푦푗,푣+1, we have 푘 푘 휀푖 (푒푖(푝)) = 휀푖 (푝), 푣 푣 휀푖 (푒푖(푝)) = 휀푖 (푝) − 1, (6.1.16) 푙 푙 휀푖 (푒푖(푝)) = 휀푖 (푝) − 2, for all 푘 < 푣 < 푙, and hence 푘 푣 푙 휀푖 (푒푖(푝)) ≤ 휀푖 (푒푖(푝)) > 휀푖 (푒푖(푝)) for all 푘 < 푣 < 푙. (6.1.17) Therefore 푘 푣 휀푖(푒푖(푝)) = max 휀푖 (푒푖(푝)) = 휀푖 (푒푖(푝)) = 휀푖(푝) − 1. 푘

The property 휑푖(푒푖(푝)) = 휑푖(푝) + 1 then follows from both this, the fact that wt(푒푖(푝)) = wt(푝) + 훼푖, and the balanced-strings property. We will also check the partial inverse property now: applying the 푘 ∨ 푘+1 identity 휀푖 (푝) + ⟨wt(푝), 훼푖 ⟩ = 휑푖 (푝) to the above equation gives 푘+1 푣+1 푙+1 휑푖 (푒푖(푝)) ≤ 휑푖 (푒푖(푝)) > 휑푖 (푒푖(푝)) for all 푘 < 푣 < 푙. (6.1.18)

58 6.1 Nakajima’s Monomial Crystal

푘 푘+1 Now we apply the parity condition, which implies that 휑푖 (푝) = 휑푖 (푝) for all 푝 ∈ ℳ(Φ) and integers 푘 ∈ ℤ. This implies

푘+2 푣+2 푙+2 휑푖 (푒푖(푝)) ≤ 휑푖 (푒푖(푝)) > 휑푖 (푒푖(푝)) for all 푘 < 푣 < 푙. (6.1.19) −1 Hence 푛푓 ,푖(푒푖(푝)) = 푣 + 2, and 푓푖 acts on 푒푖(푝) = 푝 ⋅ 푧푖,푣 as multiplication by 푧푖,푣 .

The proof for the lowering operators 푓푖 is similar, and hence ℳ(Φ) is an abstract crystal. Seminormality follows from the fact that the statistics 휀푖 and 휑푖 take values in ℕ, and that 푒푖(푝) = ⊥ if and only if 휀푖(푝) = 0 (and similarly for 푓푖(푝) and 휑푖(푝)). int We now know that ℳ(Φ) is a seminormal Φ-crystal, but we do not know if it is a disjoint union of 풪푞 (Φ) crystals. The following theorem is due to Kashiwara. 6.1.20 Theorem The monomial crystal ℳ(Φ) is finite-normal. 6.1.21 Proof The proof is given in Proposition 3.1 of [Kas02b]. There it is stated that we need to find a ‘good’ subset ℳgood ⊆ ℳ satisfying certain properties: in our setup above, this is our subset ℳ(Φ) ⊆ 퐴(Φ). Condition (i) of a ‘good’ subset is that 푝[푖, 푐] > 0 should imply 푝[푖, 푐] ≥ 0, which is true because of the parity condition, and condition (ii) of a ‘good’ subset is that it is stable under the 푒푖 and 푓푖 operators, which we have shown in Theorem 6.1.12. Hence if (퐼, ⋅) is finite type, then every connected component of ℳ(Φ) is isomorphic to ℬ(휆) for some dominant weight 휆. This is no longer the case if (퐼, ⋅) is not finite type: the subcrystal generated by a primitive element 푏휆 ∈ ℳ(Φ) of some dominant weight 휆 is still isomorphic to ℬ(휆), but there are other subcrystals of ℳ(Φ) which are not of this form. A theorem of Hernandez and Nakajima classifies those remaining. 6.1.22 Theorem (Due to [HN06], Theorem 2.2). Let 푝 ∈ ℳ(Φ) be any monomial. Then the subcrystal of ℳ(Φ) generated by 푝 is isomorphic to some connected component of an extremal weight module. As the product monomial crystal (our main crystal of study, introduced in Section 6.3) is assembled from subcrystals of ℳ(Φ) generated by highest-weight monomials of domainant weight, we will not concern ourselves further with the study of extremal weight modules or extremal weight crystals.

59 6 Monomial crystals

6.2 A variation on the monomial crystal

Kashiwara gives a different crystal structure ℳ푐(Φ) on a set of monomials in Section 4 of [Kas02b], defined for all Cartan data (퐼, ⋅) rather than just for bipartite type. The definition depends on a choice 푐 = (푐푖푗)푖,푗∈퐼 of integers satisfying 푐푖푗 + 푐푗푖 = 1 for 푖 ≠ 푗. We introduce this crystal since it appears in the literature much more frequently than ℳ(Φ). The main results of this thesis pertain to the product monomial crystal defined as a subcrystal of ℳ(Φ), but in fact by Corollary 6.2.10 (an observation we believe to be novel) our results apply to similar subcrystals of ℳ푐(Φ). 6.2.1 Definition (A variation on the monomial crystal) Let Φ be a root datum of any type (퐼, ⋅), and let 퐴(Φ) be the set of monomials defined in Definition 6.1.1. Choose a set (푐푖푗)푖,푗∈퐼 of integers satisfying 푐푖푗 +푐푗푖 = 1 for all 푖 ≠ 푗 (this function need only be defined on those pairs (푖, 푗) ∈ 퐼 which are connected in the Dynkin diagram). Let ℳ푐(Φ) ⊆ 퐴(Φ) be the subset of monomials satisfying the compatible weight condition Eq. (6.1.4) (this subset does not depend on the choice of 푐). We will now define an abstract Φ-crystal structure on the set ℳ푐(Φ). Introduce the auxiliary monomial

푎 푎 = 푒훼푖 ⋅ 푦 ⋅ 푦 ⋅ ∏ 푦 푗푖 . (6.2.2) 푖,푘 푖,푘 푖,푘+1 푗,푘+푐푗푖 푗≠푖

For each monomial 푝 ∈ ℳ푐(Φ), define

푘 1. 휑푖 (푝) = ∑푙≤푘 푝[푖, 푙], the lower column sum. 푘 2. 휑푖 (푝) = max푘 휑푖 (푝), the largest lower column sum. 푘 3. 휀푖 (푝) = − ∑푙>푘 푝[푖, 푙], the negated strict upper column sum. 푘 4. 휀푖 (푝) = max푘 휀푖 (푝), the largest negated strict upper column sum. 푘 푘 5. 푛푓 ,푖 (푝) = min{푘 ∈ ℤ ∣ 휑푖 (푝) = 휑푖 (푝)}, the smallest 푘 maximising the lower column sum 휑푖 (푝). 푘 6. 푛푒,푖 (푝) = max{푘 ∈ ℤ ∣ 휀푖 (푝) = 휀푖 (푝)}, the largest 푘 maximising the negated strict upper column sum 푘 휀푖 (푝).

Note that 푛푓 ,푖 (푝) is undefined if 휑푖 (푝) = 0, and 푛푒,푖 (푝) is undefined if 휀푖 (푝) = 0. We set wt(푝) = 휔(푝), and define the crystal operators 0 if 휀 (푝) = 0 푒 (푝) = { 푖 (6.2.3) 푖 푝 ⋅ 푎 otherwise, 푖,푛푒,푖 (푝)

0 if 휑푖 (푝) = 0 푓 (푝) = { (6.2.4) 푖 푝 ⋅ 푎−1 otherwise. 푖,푛푓 ,푖 (푝)

The varied monomial crystal is the set ℳ푐(Φ), equipped with the crystal structure (wt, 푒푖 , 푓푖 , 휀푖 , 휑푖 ) given above. 푘 푘 The statistics 휑푖 and 휀푖 are essentially (modulo an off-by-one shift on the 휀) the composition of 푦푖,푘 ↦ 푦푖,−푘 with 푘 푘 the previous statistics 휑푖 and 휀푖 . However, the auxiliary monomial 푎푖,푘 is quite different to 푧푖,푘, and furthermore depends on the integers 푐푖푗 chosen. Nevertheless, one proves in a similar way to Theorem 6.1.12 that the varied monomial crystal ℳ푐(Φ) is a seminormal abstract Φ-crystal, and furthermore by Corollary 4.4 of [Kas02b] it is shown that ℳ푐(Φ) is a finite-normal crystal, regardless of the integers 푐푖푗 chosen. 퐼 There is an action of the abelian group ℤ on the set of possible parameters (푐푖푗)푖∼푗 satisfying 푐푖푗 + 푐푗푖 = 1, where 퐼 푚 ∈ ℤ acts by (푚 ⋅ 푐)푖푗 = 푐푖푗 + 푚푖 − 푚푗, and the isomorphism of abelian groups defined by 푦푖,푘 ↦ 푦푖,푘+푚푖 gives ∼ 퐼 an isomorphim of crystals ℳ푐(Φ) −→ℳ푚⋅푐(Φ). When the graph 퐼 is acyclic, the action of ℤ on the parameters is transitive, and therefore in this case the isomorphism class of ℳ푐(Φ) does not depend on 푐 (this is stated in [Kas02b]).

60 6.2 A variation on the monomial crystal

The next lemma shows that if (퐼, ⋅) is bipartite, then depending on the two-colouring 휁 ∶ 퐼 → ℤ/2ℤ defining the ∼ monomial crystal ℳ(Φ), we may find 푐푖푗 such that there is an isomorphism of crystals ℳ푐(Φ) −→ℳ(Φ). 6.2.5 Lemma (Isomorphism of monomial and varied monomial crystals) Let Φ be a root datum of bipartite Cartan type (퐼, ⋅), and let 휁 ∶ 퐼 → {0, 1} be a 2-colouring defining the monomial crystal ℳ(Φ). Whenever 푖 and 푗 are connected in the Dynkin diagram set 푐푖푗 = 휁 (푖), then we have 푐푖푗 + 푐푗푖 = 휁 (푖) + 휁 (푗) = 1 by the fact that 휁 is a 2-colouring. Hence the choice of (푐푖푗)푖,푗∈퐼 defines a varied monomial crystal ℳ푐(Φ).

Define the map Γ∶ ℳ푐(Φ) → ℳ(Φ) of abelian groups by sending the generator 푦푖,푘 to the generator Γ(푦푖,푘) = 휆 휆 푦푖,−2푘+휁 (푖), and leaving Γ(푒 ) = 푒 . Then Γ is an isomorphism of crystals. In fact, the proof of Lemma 6.2.5 will be valid in greater generality. 6.2.6 Remark The function 휁 can be taken to be any map 휁 ∶ 퐼 → ℤ such that |휁 (푖) − 휁 (푗)| = 1 for connected 푖, 푗 ∈ 퐼, 휁 in which case taking the parity of each 휁 (푖) makes 퐼 −→ℤ ↠ ℤ/2ℤ a 2-colouring defining the monomial crystal ℳ(Φ). For such a 휁 we should set 푐 = 휁 (푖)−휁 (푗)+1 . Then for connected 푖, 푗 we have 푐 ∈ {0, 1} and 푖푗 2 푖푗 푐푖푗 + 푐푗푖 = 1, and if 휁 takes values in {0, 1} then this choice of 푐푖푗 agrees with the one defined above. We remark that the proof is made a little hard to follow because of the reversal of the ℤ-indices of the monomials, 푘 along with the fact that the 휀푖 statistics are off-by-one between the crystals. However, we wanted to leave the definition of ℳ(Φ) as close to its definition in [Kam+19a] as possible, and likewise for ℳ(Φ) and its definition in [Kas02b] (this will also help up match up yet another crystal appearing in Chapter 8), deciding instead to complicate the isomorphism a little. Before reading the proof, one might like to skip past it and see what this map looks like in terms of the monomial diagrams. 6.2.7 Proof Γ is an isomorphism of abelian groups, commuting with the weight functions. We have that for the 휑 statistics 푘 −2푘+휁 (푖) 푘 −2푘−2+휁 (푖) that 휑푖 (푝) = 휑푖 (Γ(푝)) and for the 휀 statistics that 휀푖 (푝) = 휀푖 (Γ(푝)), showing that Γ commutes with the crystal raising and lowering statistics 휀푖 and 휑푖. Furthermore we have that −2푛푓 ,푖 (푝) + 휁 (푖) = 푛푓 ,푖(Γ(푝)) and −2푛푒,푖 (푝) − 2 + 휁 (푖) = 푛푒,푖(Γ(푝)) whenever 푛푓 ,푖 and 푛푒,푖 are defined. We now perform straightforward comparisons of the auxiliary monomials:

푎 Γ(푎 ) = Γ (푒훼푖 ⋅ 푦 ⋅ 푦 ⋅ ∏ 푦 푗푖 ) 푖,푘 푖,푘 푖,푘+1 푗,푘+푐푗푖 푖≠푗 푎 = 푒훼푖 ⋅ 푦 ⋅ 푦 ⋅ ∏ 푦 푗푖 (6.2.8) 푖,−2푘+휁 (푖) 푖,−2푘−2+휁 (푖) 푗,−2푘−2푐푗푖+휁 (푗) 푗≠푖 푎 훼푖 푗푖 푧푖,−2푘−2+휁 (푖) = 푒 ⋅ 푦푖,−2푘−2+휁 (푖) ⋅ 푦푖,−2푘+휁 (푖) ⋅ ∏ 푦푗,−2푘−1+휁 (푖). 푗≠푖

These two monomials are equal if and only if 휁 (푖) − 1 = 휁 (푗) − 2푐푗푖 for all connected pairs (푖, 푗), which is true by our definition of 푐푖푗. Hence we have Γ(푎푖,푘) = 푧푖,−2푘−2+휁 (푖).

Now examining the rules for the crystal operators in ℳ푐(Φ) and ℳ(Φ), we have

−1 −1 −1 푓푖 (푝) = 푝 ⋅ 푎푖,푛 (푝) ⟺ 푓푖(Γ(푝)) = Γ(푝) ⋅ 푧푖,−2푛 (푝)−2+휁 (푖) = Γ(푝) ⋅ Γ(푎푖,푛 (푝)), 푓 ,푖 푓 ,푖 푓 ,푖 (6.2.9) 푒 (푝) = 푝 ⋅ 푎 ⟺ 푒 (Γ(푝)) = Γ(푝) ⋅ 푧 = Γ(푝) ⋅ Γ(푎 ). 푖 푖,푛푒,푖 (푝) 푖 푖,−2푛푒,푖 (푝)−2+휁 (푖) 푖,푛푒,푖 (푝)+1

Hence Γ commutes with the crystal operators 푒푖, 푓푖 on their domain of definition, and so we conclude that Γ is an isomorphism ℳ푐(Φ) → ℳ(Φ) of crystals.

The integers 푐푖푗 defining the crystal ℳ푐(Φ) in the statement of Lemma 6.2.5 are dependent on the function 휁 ∶ 퐼 →

61 6 Monomial crystals

퐼 ℤ chosen. However, we can combine Lemma 6.2.5 and Kashiwara’s action of ℤ on the space of parameters (푐푖푗)푖∼푗 to get the following result, which does not appear in the literature as far as we can tell. 6.2.10 Corollary Let Φ be a root datum of type (퐼, ⋅), where 퐼 is both bipartite and acyclic. For any two-colouring 휁 ∶ 퐼 → ℤ/2ℤ and any choice of parameters (푐푖푗)푖∼푗 satisfying 푐푖푗 + 푐푗푖 = 1, the crystal ℳ(Φ) defined by 휁 and the crystal

ℳ푐(Φ) defined by 푐 are isomorphic, via an isomorphism ℳ푐(Φ) → ℳ(Φ) of the form 푦푖,푘 ↦ 푦푖,−2푘+푛푖 for some integers (푛푖)푖∈퐼 .

Here is an example of the crystal isomorphism Γ∶ ℳ푐(Φ) → ℳ(Φ) in type (퐼, ⋅) = A5, with the standard choice of 휁 (푖) being 1 if 푖 is odd and 0 if 푖 is even. Note that we have drawn the ℤ-coordinates of the monomials in ℳ푐(Φ) increasing down the page rather than up the page.

Γ(푝) 푝 ℤ 3 −3 8 −4 2 −2 5 1 8 −4 −1 −8 0 −8 5 0 7 −1 8 6 7 1 8 6 −2 2 −3 3 ℤ

휁 1 0 1 0 1

Another example is shown below, this time in type D6 with a 휁 ∶ 퐼 → ℤ which has more general values than just {0, 1}.

Γ(푝) 푝 ℤ 3 3 −3 8 −4 2 −2 5 1 −4 −1 −8 0 3 8 5 7 0 7 −1 −8 6 1 8 6 −2 8 2 −3 3 ℤ 휁 3 3 2 1 0 −1

We conclude this section with some remarks about various monomial crystals, how they are related, and how each is related to this thesis. The first monomial crystal ℳ(Φ) we introduced first appeared in [Nak02] in simply- laced types A and D, and was later generalised to arbitrary bipartite types in Section 3 of [Kas02b]. The definition we use comes from [HN06], which is a slight generalisation to permit arbitrary root data Φ rather than only

62 6.3 The product monomial crystal simply-connected root data. The product monomial crystal (the main object of study) is defined in [Kam+19a] as a subcrystal of ℳ(Φ), and so this monomial crystal is the most important one for most of our work.

The varied monomial crystal ℳ푐(Φ) first appeared in Section 4 of[Kas02b], although one can see from the discussion in Section 8.5 that this crystal more-or-less already appeared in symmetric type in Section 8 of [Nak01b]. The varied monomial crystal is the one more commonly seen in the literature, thus Lemma 6.2.5 makes the work appearing in the rest of this thesis applicable to a much wider class of crystals. The authors of [KS14] use monomial multiplication in ℳ푐(Φ) in finite type as a model of the tensor product of crystals, where (similarly toour product monomial crystal) as long as the subcrystals are taken far enough apart vertically, the monomial-wise product is isomorphic to the tensor product of crystals. The authors of [AN18] focus on the crystal ℳ푐(Φ) in type A푛, and show that the monomial-wise product of subcrystals is indeed a subcrystal, and give an explicit decomposition theorem. Both of these results are implied (in symmetric type) by our work, and Lemma 6.2.5.

There is a third crystal appearing in the literature under the name modified Nakajima monomials in [KKS07] which is commonly denoted by ℳ̂푐(Φ). This crystal is certainly different in nature to ℳ(Φ) and ℳ푐(Φ), as the subcrystal generated by a primitive element of weight 휆 is isomorpic to the crystal 푇휆 ⊗ ℬ(∞), which is the int crystal of the Verma module 푀(휆) rather than being a crystal of an 풪푞 (Φ)-module. The techniques in our work could probably be adapted to analyse this crystal, but there is nothing we can say about the relation between our work and this crystal directly, and so we will not mention it further.

6.3 The product monomial crystal

In light of the isomorphism given in Lemma 6.2.5, we could define the product monomial crystal inside either ℳ(Φ) or the variation ℳ푐(Φ). We choose to go with the first crystal ℳ(Φ), following its original definition in [Kam+19a]. The product monomial crystal will be a monomial-wise product of various subcrystals of ℳ(Φ), with each subcrystal generated by a certain dominant monomial. We introduce some necessary notation for this. 6.3.1 Definition (Dominant monomials and fundamental subcrystals)

For each multiset R based in 퐼 ×̇ ℤ, let 푦R ∈ 퐴(Φ) be the monomial 푦R = ∏(푖,푐)∈R 푦푖,푐.A dominant pair is a 휆 pair (휆, R) of a weight 휆 ∈ 푋(Φ) and a multiset R based in 퐼 ×̇ ℤ such that 푒 ⋅ 푦R ∈ ℳ(Φ). This condition is ∨ equivalent to ⟨휆, 훼푖 ⟩ = ∑푙∈ℤ R[푖, 푙] for all 푖 ∈ 퐼. 휆 If (휆, R) is a dominant pair then the monomial 푒 ⋅ 푦R is primitive of highest weight 휆, and hence generates 휆 a subcrystal of ℳ(Φ) isomorphic to ℬ(휆) by finite-normality of ℳ(Φ) and Theorem 4.4.2. Let ℳ(푒 , 푦R) denote this subcrystal. Suppose that R is concentrated in a single entry, so that R = {(푖, 푐)푛} for some 푛 > 0. In this case we call a 휆 휆 푛 dominant pair (휆, R) a fundamental pair, and call the crystal generated by 푒 ⋅ 푦R = 푒 ⋅ 푦푖,푐 a fundamental subcrystal ℳ(푒휆, R). 푛 푛 If Φ is simply-connected, then a fundamental pair is always of the form 휆 = 휛푖 and R = {(푖, 푐) }. We now define the product monomial crystal as a monomial-wise product over fundamental subcrystals. 6.3.2 Definition (Product monomial crystal) ∨ For a dominant pair (휆, R), fix a decomposition 휆 = ∑(푖,푐)∈R 휆푖,푐 such that ⟨훼푖 , 휆푖,푐⟩ = R[푖, 푐] for all (푖, 푐) ∈ R. The product monomial crystal is the set defined as the monomial-wise product of the fundamental subcrystals: 휆푖,푐 R[푖,푐] ℳ(휆, R) = ∏ ℳ(푒 ⋅ 푦푖,푐 ). (6.3.3) (푖,푐)∈Supp R The product monomial crystal does not depend on the decomposition of 휆. We remark that here we have chosen a definition where the product Eq. (6.3.3) has as few terms as possible. We will later see that product of fundamental subcrystals concentrated over the same vertex (푖, 푐) is again a

63 6 Monomial crystals fundamental subcrystal: 휆 푛 휇 푚 휆+휇 푛+푚 ℳ(푒 ⋅ 푦푖,푐) ⋅ ℳ(푒 ⋅ 푦푖,푐) = ℳ(푒 ⋅ 푦푖,푐 ), (6.3.4) reconciling our definition of the product monomial crystal with the definition of[Kam+19a] in the simply- connected case. This fact will be implied by our character formula, and also by the proof of Theorem 6.3.5. As defined, the product monomial crystal ℳ(휆, R) is only a subset of ℳ(Φ), and it is unclear whether it is closed under the crystal operators 푒푖 and 푓푖. The following theorem justifies the name crystal. 6.3.5 Theorem When Φ is a root datum whose type (퐼, ⋅) is symmetric, bipartite, and without cycles, then ℳ(휆, R) is a subcrystal of ℳ(Φ). The proof of this theorem is sketched in Section 7 of [Kam+19a], and the purpose of Chapter 8 will be to explain this proof. The restriction on 퐼 being bipartite comes from the fact that the monomial crystal ℳ(Φ) (Defini- tion 6.1.1) is only defined in bipartite type. The restriction on 퐼 being symmetric comes from the method of proof, using Nakajima quiver varieties. Various computer experiments suggest that the product monomial crystal ℳ(휆, R) is still a crystal even when 퐼 is of arbitrary finite type. 6.3.6 Remark Our notation for the product monomial crystal differs from its original definition in[Kam+19a] in three ways. Firstly, we use the symbol ℳ(휆, R) for the crystal rather than ℬ(R). Secondly, they work only in the simply-connected case making the weight term 푒휆 in the monomials unnecessary. Thirdly, they use a collection of multisets (R푖)푖∈퐼 , where R푖 is a multiset based in 2ℤ + 휁 (푖): to go between the two notations set R[푖, 푘] = R푖[푘]. It was noted that in the simply-connected case, there exist embeddings of crystals (Theorem 2.2 of [Kam+19a])

ℬ(휆) ↪ ℳ(휆, R) ↪ ⨂ ℬ(휛푖), (6.3.7) (푖,푐)∈R and that furthermore by varying R while keeping 휆 fixed, both extremes ℳ(휆, R) ≅ ℬ(휆) and ℳ(휆, R) ≅ 휆 ⨂(푖,푐)∈R ℬ(휛푖) can be achieved. The first embedding ℬ(휆) ↪ ℳ(휆, R) is clear to see: the monomial 푒 ⋅ R ∈ ℳ(휆, R) is highest weight and hence generates a subcrystal isomorphic to ℬ(휆). The second embedding is much more subtle, and we will explain it in Section 7.3. 6.3.8 Remark There is a case where the isomorphism ℳ(휆, R) ≅ ⨂(푖,푐) ℬ(휛푖) is easy to see, which is when (퐼, ⋅) is finite type, and the parameter R is sufficiently ‘generic’. By this we mean that the elements of R are spread out enough in the ℤ-direction so that the fundamental subcrystals ℳ(푒휛푖 , (푖, 푐)) for (푖, 푐) ∈ R do not interact. For example, suppose that R = {(푖1, 푐1), … , (푖푁 , 푐푁 )} with 푐1 > ⋯ > 푐푁 , and suppose further that a vertical line can 휛푖 휛푖 be drawn on the monomial diagram, separating any element of ℳ(푒 푟 , (푖푟 , 푐푟 )) from ℳ(푒 푟+1 , (푖푟+1, 푐푟+1)), for all 푟 ∈ 1, … , 푁 − 1. (Since (퐼, ⋅) is finite type, each fundamental subcrystal ℳ(푒휛푖 , (푖, 푐)) is finite, and hence this can always be done by taking the 푐푟 to be far enough apart). It follows that the map of sets

푖 푖 mult ℳ(푒 1 , (푐1, 푟1)) ⊗ ⋯ ⊗ ℳ(푒 푁 , (푐푁 , 푟푁 )) −−−−→ℳ(휆, R) (6.3.9)

from the Cartesian product to the product monomial crystal is a bijection (as each subcrystal is vertically separated from the others, we can uniquely factorise the product), and it is straightforward to see that the crystal structure inherited from ℳ(Φ) actually makes mult into a crystal isomorphism.

64 6.4 Labelling elements of the crystal

6.4 Labelling elements of the crystal

Let R and S be finite multisets based in 퐼 ×̇ ℤ, and define the auxiliary monomials

푎 훼푖 푗푖 −1 −1 푦R ∶= ∏ 푦푖,푐, 푧S ∶= ∏ 푧푖,푘 = ∏ 푒 ⋅ 푦푖,푘 ⋅ 푦푖,푘+2 ⋅ ∏ 푦푗,푘+1, 푧S = (푧S) . (6.4.1) (푖,푐)∈R (푖,푘)∈S (푖,푘)∈S 푗≠푖

휆 휆 The fundamental subcrystal ℳ(푒 ⋅ 푦푖,푐) is generated by the highest-weight element 푒 ⋅ 푦푖,푐, and so every element 휆 −1 of the crystal is of the form 푒 ⋅ 푦푖,푐 ⋅ 푧S for some finite multiset S based in 퐼 ×̇ ℤ. It follows by definition that 휆 −1 every element of the product monomial crystal ℳ(휆, R) is of the form 푒 ⋅ 푦R ⋅ 푧S for some finite multiset S. Since the set {푧푖,푘 ∣ (푖, 푘) ∈ 퐼 ×̇ ℤ} of auxiliary monomials is linearly independent in ℳ(Φ) (which can be seen by using a triangularity argument from the 푦푖,푘), a monomial 푝 ∈ ℳ(휆, R) is uniquely determined by the S-multiset 휆 −1 appearing in the expression 푝 = 푒 ⋅푦R ⋅푧S , and we will call this the S-labelling of an element. Under this labelling, the exponent 푝[푖, 푘] of a monomial 푝 ∈ ℳ(휆, R) may be expressed as

푝[푖, 푘] = R[푖, 푘] − S[푖, 푘 − 2] − S[푖, 푘] − ∑ 푎푗푖S[푗, 푘 − 1]. (6.4.2) 푗≠푖

In the type A case, this S labelling has a direct interpretation in terms of Young diagrams. 6.4.3 Remark Consider the simply-connected root datum Φ = SL5 of Cartan type A4. The fundamental subcrystal gener- 휛 ated by the monomial ℳ(푒 2 ⋅ 푦2,0) ≅ ℬ(휛2) has 10 elements, indexed by the Young diagrams fitting within a 2×3 rectangle. The empty partition corresponds to the highest-weight element 푦2,0, and in general a mono- −1 mial 푦2,0푧S corresponds to a Young diagram drawn in ‘Russian’ style, where the box with bottom corner (푗, 푘) is present if and only if (푗, 푘) ∈ S. Below the partition corresponding to the lowest-weight element of 휛 휛 ℳ(푒 2 ⋅ 푦2,0) is shown on the left, with some of the crystal ℳ(푒 2 ⋅ 푦2,0) shown on the right. In each picture, the point (2, 0) is indicated with a circle.

The crystal operators on the right are easy to remember: the operator 푓푖 adds a box in column 푖, if there is an addable box in that position. 2 Now let 휆 = 휛2 + 휛3 + 2휛4, R = {(2, 0), (3, −1), (4, 0) }, and consider the problem of deciding whether a 휆 −1 monomial 푒 ⋅ 푦R ⋅ 푧S is an element of ℳ(휆, R). We may picture the multiset R by circling points on the monomial diagram, and the multiset S by placing multiplicities in their corresponding box positions. Below is shown a potential S on the left, with the figure on the right showing a valid ‘covering’ of S by partitions fitting within boxes. The partitions used are (2, 1) above the vertex 2, the partition (2, 1) above the vertex 3, and the two partitions (1) and (1, 1, 1) above the vertex 4.

65 6 Monomial crystals

1 2 1 2 1 3 1 3 2 1 2 1

휆 −1 The monomial 푒 ⋅푦R⋅푧S is an element of ℳ(휆, R) if and only if there exists a valid covering of S by partitions hung from the pegs described by R. The diagram on the right is not the only way to resolve the multiset S into overlapping partitions hung from the pegs R. Each resolution corresponds to some factorisation of a monomial back into a product of monomials coming from the fundamental crystals making up ℳ(휆, R). This interpretation of the product monomial crystal in type A was originally described in Section 6 of [Kam+19a], and Section 2.5.3 of [WWY17]. Although we will not use it further in the paper, we found this observation invaluable in making the initial connection to generalised Schur modules, which eventually led us to our general Demazure character formula for ℳ(휆, R).

6.5 A partial order

Define a partial order ≤ on the set 퐼 ×̇ ℤ as the transitive closure of (푖, 푘) ≤ (푖, 푘 + 2) and (푖, 푘) ≤ (푗, 푘 + 1) for all 푗 ∼ 푖. (6.5.1)

(Recall that 푗 ∼ 푖 means that the vertices 푖 and 푗 are connected in the Dynkin diagram, or equivalently that 푎푖푗 ≠ 0). A subset 퐽 ⊆ 퐼 ×̇ ℤ is called upward-closed if whenever 푥 ∈ 퐽 and 푦 ∈ 퐼 ×̇ ℤ satisfy 푥 ≤ 푦, then 푦 ∈ 퐽. (This condition is sometimes called being an upper set). A minimal element in an upward closed set 퐽 is an element 푥 ∈ 퐽 such that for all 푦 ∈ 퐽, either 푥 ≤ 푦, or 푥 and 푦 are incomparable. When 퐼 is connected, we will say that an upward-closed set 퐽 ⊆ 퐼 ×̇ ℤ is proper if it is a proper nonempty subset. For general 퐼, we say an upward-closed set is proper if it its restriction to each connected component is proper. For any subset 퐽 ⊆ 퐼 ×̇ ℤ, let up(퐽) = {푦 ∈ 퐼 ×̇ ℤ ∣ 푥 ≤ 푦 for some 푥 ∈ 퐽} be the upward-closed set generated by 퐽. Every proper upward-closed set is a union of the upward-closed sets generated by its finitely many minimal elements. We define downward-closed sets and down(퐽) similarly. 6.5.2 Example

The following diagram shows an example of a proper upward-closed set in type A5, and another in type D5 (or we could also view both diagrams as a single proper upward-closed set in type A5 × D5). The minimal elements have been marked with a circle.

A5 D5 Define the boundary of the upward-closed set 퐽 to be 휕퐽 = {(푖, 푘) ∈ 퐽 ∣ (푖, 푘 − 2) ∉ 퐽}. (6.5.3)

66 6.6 Supports of monomials

An upward-closed set 퐽 is proper if and only if |휕퐽| = |퐼|. The figure in Example 6.5.2 contains 10 boundary points in total, but only 4 minimal points: a minimal point is always a boundary point, but not conversely.

휆 푛 The reason we have introduced this partial order on 퐼 ×̇ ℤ is that the fundamental crystals ℳ(푒 ⋅ 푦푖,푐) ‘grow downwards’ with respect to the order. 6.5.4 Lemma (Fundamental subcrystals grow downwards) 푛 휆 푛 −1 휆 푛 Let (휆, 푦푖,푐) be a fundamental pair. Then if 푒 ⋅ 푦푖,푐 ⋅ 푧S ∈ ℳ(푒 ⋅ 푦푖,푐), then 푥 ≤ (푖, 푐 − 2) for all 푥 ∈ Supp S. 6.5.5 Proof 휆 푛 The claim is vacuous for the highest-weight element 푒 ⋅ 푦푖,푐, since its associated S-multiset is empty. As the 휆 푛 fundamental subcrystal ℳ(푒 ⋅푦푖,푐) is connected, it suffices to show that the crystal 푓푖 operators preserve the above property. 휆 푛 −1 휆 Suppose that 푝 = 푒 ⋅ 푦푖,푐 ⋅ 푧S ∈ ℳ(푒 푦푖,푐) satisfies Supp S ≤ (푖, 푐 − 2), meaning that 푥 ≤ (푖, 푐 − 2) for all 푥 ∈ Supp S. Fix a vertex 푗 ∈ 퐼. If 푓푗(푝) = ⊥ then there is nothing to prove, so assume instead that −1 푓푗(푝) = 푝 ⋅ 푧푗,푘−2 (we seek to prove that (푗, 푘 − 2) ≤ (푖, 푐 − 2)). In particular, this means that the largest upper column sum 휑푗(푝) > 0 was maximised at (푖, 푘), and hence 푝[푗, 푘] > 0. Applying this inequality to Eq. (6.4.2) gives

R[푗, 푘] + ∑|푎푗푘|S[푗, 푘 − 1] > S[푗, 푘] + S[푗, 푘 − 2] ≥ 0 (6.5.6) 푘≠푗 where R is the multiset {(푖, 푐)푛}. If R[푗, 푘] = 푛, then (푗, 푘) = (푖, 푐) and so (푗, 푘 − 2) = (푖, 푐 − 2). Otherwise, we must have R[푗, 푘] = 0 since R is concentrated in a single element. This means that ∑푘≠푗|푎푗푘|S[푗, 푘 − 1] is strictly positive, and hence there exists an 푙 ∼ 푗 such that (푙, 푘 − 1) ∈ S, so we have found an upward neighbour of (푗, 푘 − 2) already contained in Supp S. So the claim follows by the inductive assumption and the transitivity of ≤.

We give two illustrations of Lemma 6.5.4 in types A5 and D5. In each picture, the point (푖, 푐) has been circled, and the set {(푗, 푘) ∣ (푗, 푘) ≤ (푖, 푐 − 2)} has been shaded.

A5 D5

6.6 Supports of monomials

In light of Lemma 6.5.4 we will define the based support of a monomial, a certain “shadow” it makes on the 휆 −1 underlying set 퐼 ×̇ ℤ. For a monomial 푝 = 푒 ⋅ 푦R ⋅ 푧S ∈ ℳ(휆, R), define its based support to be SuppR 푝 = Supp R ∪ Supp S. Note that the based support of a monomial 푝 relies in a fundamental way on R, since the 휆 −1 multiset R determines the factorisation 푝 = 푒 ⋅푦R ⋅푧S and hence determines S. The based support of a monomial can be quite large compared to its support in terms of the 푦푖,푐.

67 6 Monomial crystals

6.6.1 Example 휆 −1 Let Φ = SL4, 휆 = 휛1 +휛3, R = {(1, 1), (3, 5)}, S = {(1, 1), (2, 2), (3, 3)}, and 푝 = 푒 ⋅푦R ⋅푧S . Then as a monomial, 푝 = 1, and hence SuppR(1) = {(1, 1), (2, 2), (3, 3), (3, 5)}. The based support is additive, in the sense that if we have two different monomials from two different product monomial crystals 푝 ∈ ℳ(휆, R) and 푞 ∈ ℳ(휇, Q), then we have SuppR+Q(푝 ⋅ 푞) = SuppR(푝) + SuppQ(푞), which follows from the identity of monomials

휆 −1 휇 −1 휆+휇 −1 (푒 ⋅ 푦R ⋅ 푧S ) ⋅ (푒 ⋅ 푦Q ⋅ 푧T ) = 푒 ⋅ 푦R+Q ⋅ 푧S+T. (6.6.2) Applying this additivity property to the definition of the product monomial crystal in terms of fundamental subcrystals, together with Lemma 6.5.4, we find that for all 푝 ∈ ℳ(휆, R) the based support SuppR(푝) is contained in the downward-closed set generated by Supp R. Examining Eq. (6.4.1) shows that if 푝[푖, 푘] ≠ 0, then (푖, 푘) ≥ 푥 for some point 푥 ∈ SuppR(푝). The next lemma shows that a monomial whose based support extends below up(R) cannot be highest weight. 6.6.3 Lemma (Raising) 휆 −1 푛 Let 푝 = 푒 ⋅ 푦R ⋅ 푧S ∈ ℳ(휆, R) and suppose that (푖, 푘) is a minimal point of SuppR(푝). Let 푞 = 푒푖 (푝) be the element at the top of 푝‘s 푖-string, so 푛 = 휀푖(푝). If (푖, 푘) ∉ R then 푛 > 0 and SuppR(푞) ⊆ SuppR(푝) ⧵ (푖, 푘). 6.6.4 Proof 휆 −1 휆 −1 Let 푝 = 푒 ⋅푦R ⋅푧S and 푞 = 푒 ⋅푦R ⋅푧T . We have T ⊆ S by definition of the crystal raising operator 푒푖, and hence SuppR(푞) ⊆ SuppR(푝). By the minimality of (푖, 푘) we have 푝[푖, 푟] = 푞[푖, 푟] = 0 for 푟 < 푘, and by the assumption 푘 푘 that R[푖, 푘] = 0 we have for the lower negated column sums 휀푖 (푝) = −S[푖, 푘] and 휀푖 (푞) = −T[푖, 푘]. However, 푘 푘 푛 = 휀푖(푝) ≥ 휀푖 (푝) = S[푖, 푘] > 0 shows that 푛 > 0, and 0 = 휀푖(푞) ≥ 휀푖 (푞) = T[푖, 푘] shows that (푖, 푘) ∉ SuppR 푞. Hence we get a useful necessary condition for highest-weight monomials in the product monomial crystal. 6.6.5 Corollary

If 푝 ∈ ℳ(휆, R) is highest-weight, then SuppR(푝) ⊆ up(R).

68 7 Truncations and the character formula

The product monomial crystal ℳ(휆, R) is bounded between two extremes: on one hand it can be isomorphic to the irreducible highest-weight crystal ℬ(휆), and on the other hand it can be isomorphic to a large tensor product ℬ(휆1) ⊗ ⋯ ⊗ ℬ(휆푛) where 휆 = 휆1 + ⋯ + 휆푛. We can see the product monomial crystal as interpolating between these two extremes (we make this precise in type A in Chapter 10, where we show that the product monomial crystal is the crystal of a generalised Schur module). The question remains: what can we say about the other cases? In this chapter we define truncations, a family of subsets of the product monomial crystal parametrised by upward- closed sets. These truncations are somewhat like Demazure crystals as they are finite, closed under the crystal raising operators 푒푖, have containment compatible with containment of upward-closed sets, and in their limit we recover the whole product monomial crystal. In fact it turns out that these truncations are disjoint unions of Demazure crystals, but this is not obvious from their definition. By relating ‘nearby’ truncations to each other in purely crystal-theoretic terms, we give an inductive character formula (Theorem 7.2.3) for any of the truncations, specialising in the finite type case to a character formula for the whole product monomial crystal (Corollary 7.3.9). Since ℳ(휆, R) is the crystal associated to the categorical 픤(Φ)-representation 풱 (휆, R) introduced in Chapter 1, our formula has implications for the study of the truncated 휆 shifted Yangian algebras 푌휇 (R).

7.1 Truncations defined by upward sets

We saw in Corollary 6.6.5 at the end of the last section that a highest-weight monomial 푝 ∈ ℳ(휆, R) satisfies SuppR(푝) ⊆ up(R). Consider the subset ℳ(휆, R, up(R)) of the product monomial crystal ℳ(휆, R) consisting of those monomials whose based support is contained within the upward-closed set up(R). This subset is closed under the crystal raising operators, because each such operator will only ever remove elements from the based support of a monomial. Furthermore, this subset contains every highest-weight element of ℳ(R). It is the pro- totypical example of one of our truncations. 7.1.1 Definition Let 퐽 ⊆ 퐼 ×̇ ℤ be an upward-closed set containing1R. The truncation of ℳ(휆, R) by 퐽 is the subset

ℳ(휆, R, 퐽) = {푝 ∈ ℳ(휆, R) ∣ SuppR(푝) ⊆ 퐽}. (7.1.2) The subset ℳ(휆, R, 퐽) is closed under the crystal raising operators, and hence we may equip it with its unique upper-seminormal crystal structure coming from the restriction of the (푒푖)푖∈퐼 and wt functions on the original crystal. (This agrees with the crystal operators defined on ℳ(휆, R, 퐽) coming from the monomial crystal ℳ(Φ)). Each of our truncations ℳ(휆, R, 퐽) satisfies up(R) ⊆ 퐽 by definition, and therefore by Corollary 6.6.5 contains every highest-weight element of the product monomial crystal ℳ(휆, R). As the product monomial crystal decomposes into highest-weight crystals, this means that knowing its highest-weight elements determines it up to isomorphism. Therefore in order to determine the isomorphism class of the product monomial crystal ℳ(휆, R), it is enough to determine the highest-weight elements of any truncation ℳ(휆, R, 퐽).

1We use the terminology “퐽 contains R” to mean that Supp R ⊆ 퐽. In particular, elements of R may still have multiplicity greater than one.

69 7 Truncations and the character formula

However, only knowing the character ch ℳ(휆, R, 퐽) of a truncation is not enough information to recover its highest-weight elements, unless the character satisfies some special properties. We will eventually end up showing that each truncation is a disjoint union of Demazure crystals, hence by the linear independence of Demazure characters (Lemma 5.2.9) the character of any truncation will determine the isomorphism class of the product monomial crystal.

An observation we will use repeatedly is that if 푝 ∈ ℳ(휆, R, 퐽) then 푝[푖, 푐] = 0 for all (푖, 푐) ∉ 퐽. For instance, it plays a part in the next lemma which shows that using a single crystal lowering operator on a monomial repeatedly can only push it outside the truncation by a single element. 7.1.3 Lemma (Lowering) 푛 Let 푝 ∈ ℳ(휆, R, 퐽) be an element of a truncation, 푖 ∈ 퐼, and (푖, 푘) ∈ 휕퐽. Then SuppR(푓푖 (푝)) ⊆ 퐽 ∪ {(푖, 푘 − 2)} for all 0 < 푛 ≤ 휑푖(푝). 7.1.4 Proof By definition of the monomial crystal we have that 푛푓 ,푖(푓푖(푞)) ≥ 푛푓 ,푖(푞) for all 푞 ∈ ℳ(Φ). Since the monomial 푝 lies in the truncation ℳ(휆, R, 퐽) we have 푝[푖, 푟] = 0 for all 푟 ≤ 푘 −2, and hence 푛푓 ,푖(푝) ≥ 푘. This means that 푛 when applying 푓푖 repeatedly, the monomial 푝 will be multiplied by 푧푖,푟 for 푟 ≥ 푘−2. Therefore SuppR(푓푖 (푝)) ⊆ 퐽 ∪ {(푖, 푘 − 2)} for all 0 < 푛 ≤ 휑푖(푝). The next lemma shows that when 퐽 and 퐽 ′ are upward-closed sets which differ by a single minimal element, then the associated truncations are related in purely crystal-theoretic terms: the larger truncation will be an extension of strings of the smaller truncation. 7.1.5 Lemma (Minimal points and extensions of strings) Suppose that 퐽 are 퐽 ′ are two upward-closed sets containing R, which differ in a single element 퐽 ′ = 퐽 + ′ {(푖, 푘)}. Then ℳ(휆, R, 퐽 ) = 픇푖ℳ(휆, R, 퐽), where 픇푖 is the extension of 푖-strings (Definition 5.3.1) operator. 7.1.6 Proof Since both 퐽 and 퐽 ′ are upward-closed and contain R, the point (푖, 푘) is minimal in 퐽 ′ and (푖, 푘) ∉ R. (If it were not minimal, its removal from 퐽 ′ would result in a non-upward-closed set). Hence if a monomial ′ 푞 ∈ ℳ(휆, R, 퐽 ) satisfies (푖, 푘) ∈ SuppR(푞), the conditions of Lemma 6.6.3 are met and so there exists a 푝 ∈ 푛 ′ ℳ(휆, R, 퐽) and 푛 > 0 with 푞 = 푓푖 (푝). If (푖, 푘) ∉ SuppR(푞) then 푞 ∈ ℳ(휆, R, 퐽) already. Hence ℳ(휆, R, 퐽 ) ⊆ 픇푖(휆, R, 퐽). ′ Conversely, applying Lemma 7.1.3 to this special case gives ℳ(휆, R, 퐽 ) ⊇ 픇푖(휆, R, 퐽). Lemma 7.1.5 is one of the two key pieces we need for the character formula, which will give is the ‘Demazure’ part of our ‘Demazure character formula’. It allows us to relate any two trunctions defined by nested upward- closed sets 퐽 ⊆ 퐽 ′, for a fixed choice of (휆, R). The second piece of our character formula will be a fact which allows us to relate truncations defined by the same upward-closed set, where (휆, R) is varied along the boundary of that set. 7.1.7 Lemma (Factorisation along a boundary) Suppose (휆, R) and (휇, Q) are two dominant pairs, and that 퐽 is an upward-closed set containing both R and Q, and further that Q is supported only along the boundary of 퐽, i.e. Supp Q ⊆ 휕퐽. Then ℳ(휆 +휇, R+Q, 퐽) = 휇 푒 ⋅ 푦Q ⋅ ℳ(휆, R, 퐽), where ⋅ denotes a product of monomials. 7.1.8 Proof By the definition of the product monomial crystal and the additivity of based support, we havethat

ℳ(휆, R, 퐽) ⋅ ℳ(휇, Q, 퐽) = ℳ(휆 + 휇, R + Q, 퐽). (7.1.9)

휇 Since Q is concentrated along the boundary 휕퐽, Lemma 6.5.4 gives that ℳ(휇, Q, 퐽) = {푒 ⋅ 푦Q}. Using only Lemma 7.1.5 and Lemma 7.1.7, we may already compute decompositions of small product monomial crystals quite quickly, as we will show in the following example.

70 7.2 A Demazure character formula

7.1.10 Example

Consider the Φ = SL4 crystal ℳ(휆, R) defined by 휆 = 휛1 + 2휛3 and R = {(1, 3), (3, 1), (3, 3)}. Since Φ is 휆 simply-connected, we may omit the 푒 terms from the monomial crystal, instead declaring that wt(푦푖,푐) = 휛푖, the 푖th fundamental weight. We will determine the isomorphism class of ℳ(휆, R) by calculating a suitable truncation ℳ(휆, R, 퐽3). 1. Begin with 퐽0 = up((2, 2)) and R0 = ∅. By definition, ℳ(R0, 퐽0) = {1}. 2. Let 퐽1 = 퐽0 and R1 = {(1, 3), (3, 3)}. Since R1 − R0 is concentrated along 휕퐽1, Lemma 7.1.7 applies and we get ℳ(R1, 퐽1) = 푦1,3 ⋅ 푦3,3 ⋅ ℳ(R0, 퐽0) = {푦1,3 ⋅ 푦3,3}. 3. Let R2 = R1 and 퐽2 = up((3, 1)). Since 퐽2 differs from 퐽1 by adding the minimal element (3, 1), Lemma 7.1.5 applies and we get ℳ(R2, 퐽2) = 픇3ℳ(R1, 퐽1). Computing this extension of 3-strings is easy since 푦1,3 ⋅ 푦3,3 is the highest-weight element of a crystal isomorphic to ℬ(휛1 + 휛3), so we get a 푓 3 −1 −1 3-string with two elements: 푦1,3⋅푦3,3 −−→푦1,3⋅푦3,3⋅푧3,1. We find that ℳ(R2, 퐽2) = {푦1,3⋅푦3,3, 푦1,3⋅푦3,3⋅푧3,1}. 4. Let R3 = R2 ∪ {(3, 1)} and 퐽3 = 퐽2. Since (3, 1) ∈ 휕퐽3, we apply Lemma 7.1.7 again to find that −1 ℳ(R3, 퐽3) = 푦3,1 ⋅ ℳ(R2, 퐽2) = {푦1,3 ⋅ 푦3,3 ⋅ 푦3,1, 푦1,3 ⋅ 푦3,3 ⋅ 푦3,1 ⋅ 푧3,1}.

We can represent this process graphically by drawing a monomial diagram, with each truncating set 퐽푖 overlaid, with the elements of the multiset R푖 shown by circling points. Beneath each diagram, we draw the elements of the subset ℳ(R푖, 퐽푖).

⋅푦1,3 ⋅ 푦3,3 픇3 ⋅푦3,1

1 푦1,3 ⋅ 푦3,3 푦1,3 ⋅ 푦3,3 푦1,3 ⋅ 푦3,3 ⋅ 푦3,1

푓3 −1 푦1,3 ⋅ 푦3,3 ⋅ 푦3,1 ⋅ 푧3,1 −1 푦1,3 ⋅ 푦3,3 ⋅ 푧3,1

The two elements of ℳ(R3, 퐽3) = ℳ(R, 퐽3) are both highest-weight, of weights 휛1 + 2휛3 and 휛1 + 휛2 respectively. Since the truncation ℳ(R, 퐽3) contains all highest-weight elements of ℳ(R, 퐽), we have

ℳ(휆, R) ≅ ℬ(휛1 + 2휛3) ⊕ ℬ(휛1 + 휛2). (7.1.11)

7.2 A Demazure character formula

At this point the reader should re-familiarise themselves with the content of Section 5.3, specifically the string property (Definition 5.3.1) and what it means for the equivariance of the extension-of-strings operator 픇푖 with the Demazure operator 휋푖 (Lemma 5.3.11). As should be clear from Example 7.1.10 above, we can build characters of truncations by starting from the subset 푍 = {1} containing the trivial monomial, and repeatedly applying steps of the form

휆 1. 푍 ↦ 푝 ⋅ 푍, for some dominant monomial 푝 = 푒 ⋅ 푦R, or 2. 푍 ↦ 픇푖(푍), where 픇푖 is the extension-of-strings operator.

71 7 Truncations and the character formula

휆 휆 The first of these operations is straightforward on the level of characters, wehave ch(푒 ⋅ 푦R ⋅ 푍) = 푒 ch 푍. For the second operation we would like to have the equivariance ch 픇푖(푍) = 휋푖(ch 푍) where 휋푖 is the Demazure operator, but in order for this to be true the subset 푍 must satisfy the string property. As we have remarked previously, 푍 satisfying the string property does not imply that 픇푖(푍) does. Furthermore, even if 푍 satisfies the string property, the product 푝⋅푍 may not. (For a counterxample, let ℳ(푦1,1) ≅ ℬ(휛1) be the SL2-crystal generated 2 by the monomial 푦1,1. Then 푦1,1 ⋅ ℳ(푦1,1) is a string with two elements inside ℳ(푦1,1) ≅ ℬ(2휛1), a violation of the string property). We get around this problem by using the fact that all of our subsets 푍 have a specific form: they are truncations ℳ(휆, R, 퐽). (The surprising fact here is Lemma 7.1.5, showing that the abstract extension-of-strings operation takes a truncation to another truncation, rather than some arbitrary subset of the product monomial crystal). We can show directly that each of our truncations has the string property. 7.2.1 Lemma (Truncations have the string property) If 퐽 is an upward-closed set containing R, then ℳ(휆, R, 퐽) has the string property. 7.2.2 Proof Since ℳ(R, 퐽) is closed under the crystal raising operators 푒푖, it suffices to show that for any 푝 ∈ ℳ(R, 퐽) such 휆 −1 휆 −1 that 푓푖(푝) ∉ ℳ(R, 퐽)⊔{⊥} that 푒푖(푝) = ⊥. Suppose we have such a 푝 = 푒 ⋅푦R⋅푧S with 푓푖(푝) = 푒 ⋅푦R⋅푧S+{(푖,푘−2)}, then by Lemma 7.1.5 we must have (푖, 푘) ∈ 휕퐽. By the definition of 휑푖 we know that 푘 is largest such that 푘 푙+2 푘 휑푖 (푝) = 휑푖(푝) and hence 휑푖 (푝) < 휑푖 (푝) for all 푙 ≥ 푘. But since 푝[푖, 푟] = 0 for all 푟 < 푘 we have that 푙 푙+2 푘 휀푖 (푝) = 휑푖 (푝) − 휑푖 (푝) < 0 for all 푙 ≥ 푘, and hence 휀푖(푝) = 0 and so 푒푖(푝) = ⊥. We arrive at our first main result, an inductive character formula for any truncation ℳ(휆, R, 퐽) by interpreting Lemmas 7.1.5 and 7.1.7 in terms of characters. 7.2.3 Theorem (A character formula for truncations) The following rules give an inductive character formula for any truncation ℳ(휆, R, 퐽): 1. If (휆, ∅) is a dominant pair, then ch ℳ(휆, ∅, 퐽) = 푒휆 for any upward-closed set 퐽. 2. If (휆, R) and (휇, Q) are dominant pairs, 퐽 an upward-closed set containing R, and Q is contained in the boundary 휕퐽, then ch ℳ(휆 + 휇, R + Q, 퐽) = 푒휇 ch ℳ(휆, R, 퐽). 3. If (휆, R) is a dominant pair, 퐽 an upward-closed set containing R, and (푖, 푘) ∉ 퐽 an element such that 퐽 ∪ {(푖, 푘)} is upward-closed, then ch ℳ(휆, R, 퐽 ∪ {(푖, 푘)}) = 휋푖 ch ℳ(R, 퐽).

An algorithm for applying the above rules to the data (휆, R, 퐽) defining a truncation is the following:

1. If R = ∅ then terminate with the result 푒휆. 2. Otherwise, if there is an element of R along the boundary 휕퐽 then define Q to be the multiset supported along 휕퐽 with multiplicities Q[푖, 푐] = R[푖, 푐] for (푖, 푐) ∈ 휕퐽, and choose a weight 휆 such that (휆, Q) is a dominant pair. Recursively compute 푒휇ℳ(휆 − 휇, R − Q, 퐽). 3. Otherwise, choose a minimal element (푖, 푘) ∈ 퐽 − up(R) satisfying (푖, 푘) ∈ down(R) and recursively compute 휋푖 ch ℳ(휆, R, 퐽 − {(푖, 푘)}). If the Dynkin diagram 퐼 is connected, the third step of the algorithm can be simplified a little: any minimal point of 퐽 − up(R) will do. We need to prove that each inductive rule is valid, and that the algorithm given terminates after finitely many steps without ‘getting stuck’ somewhere. 7.2.4 Proof Each inductive step is true: 휆 휆 휆 1. If (휆, ∅) is a dominant pair, then ℳ(휆, ∅) = {푒 } which has character 푒 . Since Supp∅ 푒 = ∅, every upward-closed set 퐽 is a valid truncation, with ch ℳ(휆, ∅, 퐽) = ch {푒휆} = 푒휆. 휇 −1 휇 2. From Lemma 7.1.7 we have that ch ℳ(휆 + 휇, R + Q, 퐽) = ch(푒 ⋅ 푧Q ⋅ ℳ(휆, R, 퐽)) = 푒 ch ℳ(휆, R, 퐽).

72 7.3 Truncations are Demazure crystals

3. From Lemma 7.1.5 we have that ch ℳ(휆, R, 퐽 ∪ {(푖, 푘)}) = ch 픇푖(ℳ(휆, R, 퐽)), which by the fact that each truncation satisfies the string property (Lemma 7.2.1) and the equivariance property of 픇푖 and 휋푖 (Lemma 5.3.11) gives that ch ℳ(휆, R, 퐽 ∪ {(푖, 푘)}) = 휋푖(ℳ(휆, R, 퐽)).

It remains to be seen that the given algorithm terminates, which amounts to proving that step 3 can always progress to a point where step 2 can be applied, since we decrease the size of the finite multiset R on each application of step 2. Fix the parameters (휆, R, 퐽), and suppose we have reached step 3 of the algorithm. Since we did not stop at step 1, the multiset R is nonempty, and since we did nothing at step 2, we have Supp R ∩ 휕퐽 = ∅. The upward-closed set 퐽 is proper, therefore there exists a minimial point (푖, 푐) ∈ 퐽 such that (푖, 푐) ∈ down(R). This minimal point must be a boundary point, and hence lies outside of Supp R so it may be removed. This decreases the size of the finite set down(R) ∩ 퐽, completing the proof. To make the theorem more concrete, we will return to a previous example and use the theorem to compute its character. 7.2.5 Example We will use Theorem 7.2.3 to determine the character of the truncated crystal we computed previously in Example 7.1.10. The crystal we ended up with was obtained through applying multiplications and extension- of-strings operators: ℳ(R3, 퐽3) = 푦3,1 ⋅ 픇3(푦1,3 ⋅ 푦3,3 ⋅ 1). (7.2.6) Now that we know that each intermediate result satisfies the string property, we can take characters by replacing all the monomials by their weights, and all of the string extension operators by Demazure operators: 휛 휛 +휛 ch ℳ(R3, 퐽3) = 푒 3 휋3(푒 1 3 ). (7.2.7) 3 As explained in Example 3.4.16, the weight lattice 푋(SL3) is isomorphic to ℤ /(1, 1, 1), and hence we have an isomorphism of algebras ℤ[푋(SL3)] ≅ ℤ[푥1, 푥2, 푥3]/(푥1푥2푥3 − 1) taking the weight 휆 = (푎, 푏, 푐) to the 푎 푏 푐 element 푥1 푥2푥3. Together with 휛1 = (1, 0, 0) and 휛2 = (1, 1, 0), the character formula becomes 2 ch ℳ(R3, 퐽3) = 푥1푥2푥3휋3(푥1 푥2푥3) 3 2 = 푥1 푥2 푥3휋3(푥3) 3 2 (7.2.8) = 푥1 푥2 푥3(푥3 + 푥4) 3 2 2 2 = 푥1 푥2 푥3 + 푥1 푥2.

We cannot yet say for sure what the isomorphism class of the truncation ℳ(R3, 퐽3) is from this character, since truncations are not necessarily a highest-weight crystal (and indeed this one is not). If we did know that the truncation was a Demazure crystal however, a fact we will later prove, we could decompose the character above in the basis of Demazure characters and find that it is ch 퐷(휛1 + 2휛3) + ch 퐷(휛1 + 휛2), as expected from the crystal computed in Example 7.1.10.

7.3 Truncations are Demazure crystals

As we have now pointed out several times, knowing the character of an abstract Φ-crystal 퐵 is not very useful. If int we happened to know that 퐵 was a crystal coming from the category 풪푞 (Φ), then its isomorphism class would be determined by the expression of ch 퐵 into the basis {ch ℬ(휆) ∣ 휆 ∈ 푋(Φ)+}, as discussed in Section 5.1. If we int knew that 퐵 was a disjoint union of Demazure subcrystals coming from crystals of 풪푞 (Φ), then its isomorphism class would be determined by decomposing its character into the basis of Demazure characters. In this section we will show that every truncation ℳ(휆, R, 퐽) is a disjoint union of Demazure crystals. The property of being a Demazure crystal is preserved by the extension-of-strings operations 픇푖, more or less by definition of a Demazure crystal (Definition 5.3.4). In light of our character formula (or rather the implicit computation of

73 7 Truncations and the character formula the truncation which lies behind it), it suffices to show that the ‘multiplication along a boundary’ operation of Lemma 7.1.7 preserves the property of being a Demazure crystal. We begin by reformulating this ‘boundary multiplication’ in purely crystal-theoretic terms. 7.3.1 Lemma Let ℳ(휆, R, 퐽) be a truncation, and (휇, Q) a dominant pair such that Supp Q ⊆ 휕퐽. There is a bijective, weight-preserving map

휆+휇 −1 휆+휇 −1 Φ∶ ℳ(휆 + 휇, R + Q, 퐽) → ℳ(R, 퐽) ⊗ ℬ(휇), 푒 ⋅ 푦R+Q ⋅ 푧S ↦ 푒 ⋅ 푦R ⋅ 푧S ⊗ 푏휇, (7.3.2)

which is equivariant under the crystal raising operators 푒푖 for all 푖 ∈ 퐼. Hence ℳ(R+Q, 퐽) ≅ ℳ(R, 퐽)⊗ℬ(휇) as upper-seminormal crystals. 7.3.3 Proof The map is defined as a consequence of Lemma 7.1.7 and is bijective and weight-preserving, hence all that 휇 휆 −1 remains to be seen is the 푒푖-equivariance. Let 푞 = 푒 ⋅ 푦Q, fix an 푖 ∈ 퐼 and a 푝 = 푒 ⋅ 푦R ⋅ 푧S ∈ ℳ(휆, R, 퐽) so that Φ(푝푞) = 푝 ⊗ 푏휇. The tensor product rule for applying 푒푖 to a tensor product of two crystal elements gives ∨ 푒푖(푝) ⊗ 푏휇 if 휀푖(푝) > ⟨휇, 훼푖 ⟩, 푒푖(푝푞) = { ∨ (7.3.4) ⊥ if 휀푖(푝) ≤ ⟨휇, 훼푖 ⟩. Fix an 푖 ∈ 퐼 and let (푖, 푘) ∈ 휕퐽. We have (푝푞)[푖, 푙] = 푝[푖, 푙] + 푞[푖, 푙] for all 푙, but 푞 is concentrated at 푘 and hence ∨ 푞[푖, 푙] = 훿푙푘⟨휇, 훼푖 ⟩. Since (푝푞)[푖, 푙] = 0 for all 푙 < 푘 we then have

푙 0 for 푙 < 푘, 휀푖 (푝푞) = { 푙 ∨ (7.3.5) 휀푖 (푝) − ⟨휇, 훼푖 ⟩ for 푙 ≥ 푘.

푙 ∨ If 푒푖(푝푞) = ⊥ then 휀푖(푝푞) = 0 and hence 휀푖 (푝) ≤ ⟨휇, 훼푖 ⟩ for all 푙 ≥ 푘. Therefore we are in the second case of Eq. (7.3.4) and hence 푒푖(푝 ⊗ 푏휇) = ⊥ also. 푙 If 푒푖(푝푞) = 푝푞푧푖,푙 then we must have 푙 ≥ 푘 and also 0 < 휀푖(푝푞) = 휀푖 (푝푞). Applying Eq. (7.3.5) gives that ∨ 푙 ⟨휇, 훼푖 ⟩ < 휀푖 (푝) and so 푒푖(푝 ⊗ 푏휇) = 푒푖(푝) ⊗ 푏휇, so all we have remaining to check is that 푒푖(푝) = 푝푧푖,푙. This is clear however, since 푙 0 for 푙 < 푘, 휀푖 (푝) = { 푙 (7.3.6) 휀푖 (푝) for 푙 ≥ 푘, 푙 푙 and so the point 푙 where 휀푖 (푝푞) first attains the positive value 휀푖(푝푞) is the same as the point 푙 where 휀푖 (푝) ∨ first attains the positive value 휀푖(푝) = 휀푖(푝푞) + ⟨휇, 훼푖 ⟩. Repeated application of the raising operators followed by Lemma 7.3.1 leads to an embedding of ℳ(휆, R) into the tensor product ⨂(푖,푐)∈R ℬ(휛푖), as promised in Section 6.3.

We now appeal to the general fact that the tensor product of a Demazure module with a highest-weight vector is again a Demazure module. In crystal terms, this is the main result of [Jos03]: the tensor product 푋 ⊗ 푏휇 of a Demazure crystal 푋 with the highest-weight element 푏휇 ∈ ℬ(휇) is again a Demazure crystal. 7.3.7 Theorem (Truncations are Demazure crystals) Let (휆, R) be a dominant pair, and 퐽 an upward-closed set containing R. The truncation ℳ(휆, R, 퐽) is a Demazure crystal. 7.3.8 Proof 휆 For each dominant pair (휆, R) the set {푒 ⋅ 푦R} is a Demazure crystal, isomorphic to ℬ푒(휆). By Proposition 3.2.3 of [Kas93], the property of being Demazure is preserved under extension of 푖-strings. By the main theorem of [Jos03] and Lemma 7.3.1, multiplication at the boundary 휕퐽 of a truncation ℳ(휆, R, 퐽) by a monomial of the form 푒휇 ⋅ Q for Supp Q ⊆ 휕퐽 also preserves the property of being Demazure. Since every

74 7.3 Truncations are Demazure crystals

truncation can be built out of these operations (Theorem 7.2.3), every truncation is Demazure. The upshot of this result is twofold. The fact that the truncations ℳ(휆, R, 퐽) are Demazure is quite interesting, since the truncations are defined straightfowardly, and globally in terms of the monomials (rather than interms of any crystal-theoretic operations). The second is that our character formula is now useful for truncations, as it uniquely determines the isomorphism class of the truncation as a Demazure crystal. As a consequence of this, we obtain in finite type a character formula for the full crystal ℳ(휆, R) in terms of the character of any truncation of it. 7.3.9 Corollary (A character formula in finite type)

Let Φ be a root datum of finite type (퐼, ⋅), 푤퐼 ∈ 푊퐼 the longest element of the Weyl group, and 퐽 ⊇ Supp R any upward-closed set containing R. Then a formula for the character of ℳ(휆, R) is

ch ℳ(휆, R) = 휋푤퐼 ch ℳ(휆, R, 퐽). (7.3.10) 7.3.11 Proof As it is a disjoint union of highest-weight crystals, the character of the product monomial crystal is

ch ℳ(휆, R) = ∑ ch ℬ(wt ℎ), (7.3.12) ℎ∈퐻

where 퐻 ⊆ ℳ(휆, R) is the set of highest-weight elements. Since the truncation ℳ(휆, R, 퐽) has the same set of primitive elements 퐻 and is a Demazure crystal, there exists some function 푤 ∶ 퐻 → 푊퐼 into the Weyl group such that ℳ(휆, R, 퐽) ≅ ⨁ ℬ푤(ℎ)(wt ℎ), (7.3.13) ℎ∈퐻 which on the level of characters gives the sum of Demazure characters

wt ℎ ch ℳ(휆, R, 퐽) = ∑ 휋푤(ℎ)(푒 ). (7.3.14) ℎ∈퐻

Since the Demazure operators give a zero-Hecke action (Remark 5.2.10) on the character ring ℤ[푋(Φ)], we have wt ℎ 휋푤퐼 ch ℳ(휆, R, 퐽) = ∑ 휋푤퐼 (푒 ), (7.3.15) ℎ∈퐻 which is precisely the character of the product monomial crystal ℳ(휆, R), since by the Demazure character wt ℎ formula Theorem 5.2.5 we have 휋푤퐼 (푒 ) = ch ℬ(wt ℎ). We now have two of our main results: the general inductive formula Theorem 7.2.3 for the character of a truncation, and the finite-type specialisation Corollary 7.3.9 to the character of the whole crystal finite type. Where should we go from here? We still have not proven that the product monomial crystal is actually a crystal (Theo- rem 6.3.5), and that will be the purpose of the next chapter. After that we will move on to putting this character formula to good use, showing in type A that the product monomial crystal is the crystal of a particular module called a generalised Schur module.

8 Nakajima quiver varieties

Nakajima has defined varieties 픐(휆, 휇), now called Nakajima quiver varities, which geometrise the 휇-weight space of the integrable highest-weight representation 퐿(휆). By this we mean that there is symplectic structure on 픐(휆, 휇) and a Lagrangian subvariety 픏(휆, 휇) ⊆ 픐(휆, 휇) such that the top homology of the Lagrangian subvariety has the same dimension as 퐿(휆)휇. The top homology (with complex coefficients) of the variety 픏(휆) ∶= ⋃휇≤휆 픏(휆, 휇) posesses an action of the Kac-Moody algebra 픤(Φ), making the homology into a module isomorphic to 퐿(휆). Furthermore, the set Irr 픏(휆) of irreducible components can be given a crystal structure, isomorphic to the crystal ℬ(휆). The goal of this chapter is to give a relatively self-contained explanation of why the product monomial crystal, which was defined as a subset of the monomial crystal ℳ(Φ), is actually a subcrystal — this was Theorem 6.3.5 whose proof we had deferred. The proof will use a subvariety of a quiver variety called a graded quiver variety, and a construction (again by Nakajima) of a Φ-crystal on the set of connected components of the graded quiver variety. We will show that Nakajima’s crystal structure agrees with the analogue of the product monomial crystal inside the variation ℳ푐(Φ) of the monomial crystal. The method of proof we use is not our own, instead following Section 7 of [Kam+19a]. One caveat to this proof is that Nakajima quiver varieties only work when the Cartan datum (퐼, ⋅) is symmetric and bipartite, meaning that out of the finite and affine type Cartan matrices this proof will only be valid forfinite (1) types A, D, and E, as well as their untwisted affinisations (excluding those A푛 which are an odd cycle). Various computer experiments suggest that the product monomial crystal is always a genuine subcrystal of ℳ(Φ), and all of the results in this thesis still hold true in this generality, assuming one could somehow prove Theorem 6.3.5 for any bipartite type (퐼, ⋅).

8.1 Representations of quivers

A quiver is a generalisation of a directed graph, which is allowed to have arbitrarily many doubled edges and self- loops. The quivers we deal with will be finite in both the number of vertices and the number of edges, however we give the general definition below. 8.1.1 Definition (Quiver) A quiver is a quadruple 푄 = (퐼, 퐸, tail, head), where 1. 퐼 is a set, called the vertex set, 2. 퐸 is a set, called the edge set, 3. tail∶ 퐸 → 퐼 is a function, giving the tail vertex of each edge, and 4. head∶ 퐸 → 퐼 is a function, giving the head vertex of each edge. Quivers are drawn in the same way as directed graphs, where vertices are represented by dots, and the edge 푒 ∈ 퐼 푒 is represented as an arrow from the tail of 푒 to the head of 푒, i.e. tail(푒) −→ head(푒). For example, below is a quiver on the vertex set 퐼 = {푎, 푏, 푐} and the edge set 퐸 = {푤, 푥, 푦, 푧}.

푤 푎 푏 푦 푐

푥 푧

77 8 Nakajima quiver varieties

A representation of a quiver is an assignment of a vector space to each vertex, and a linear map to each arrow from the tail vector space to the head vector space. 8.1.2 Definition (Quiver representation) A representation (푉 , 휑) of the quiver 푄 = (퐼, 퐸, tail, head) over the field 한 is the data of: 1. For each vertex 푖 ∈ 퐼 a 한-vector space 푉푖, and 푒 2. for each edge 푒 = (푖 −→푗) ∈ 퐸 a 한-linear map 휑푒 ∶ 푉푖 → 푉푗.

Let (푉 , 휑) and (푊 , 휓) be two 한-representations of the same quiver 푄.A morphism of quiver representations 푒 푇 ∶ (푉 , 휑) → (푊 , 휓) is a collection 푇 = (푇푖)푖∈퐼 of 한-linear maps 푇푖 ∶ 푉푖 → 푊푖 such that for each edge 푒 = (푖 −→ 푗) ∈ 퐸 in the quiver, the obvious square 푇푗휑푒 = 휓푒푇푖 commutes.

Let 푄 be a quiver and 한 a field. Define Rep한 푄 to be the category whose objects are representations of the quiver 푄 over the field 한, and whose morphisms are morphisms of quiver representations.

It is a pleasant exercise with the axioms to show that the category Rep한 푄 is an abelian category. An alternative way of seeing this fact is to find an associative algebra 한푄 (commonly called the path algebra) whose representation category is equivalent to Rep한 푄.

8.2 Moduli spaces of quiver representations

Fix a quiver 푄 = (퐼, 퐸) and an 퐼-graded vector space 푉 = ⨁푖∈퐼 푉푖. Let Rep(푄, 푉 ) denote the set of representations of 푄 with underlying vector spaces 푉 . This is simply a direct product of morphism spaces:

푒 Rep(푄, 푉 ) = {(휑푒)푒∈퐸 ∣ 푒 = (푖 −→푗) ∈ 퐸 and 휑푒 ∈ Hom한(푉푖, 푉푗)}. (8.2.1)

Hence Rep(푄, 푉 ) is a 한-vector space. (The vector space structure of Rep(푄, 푉 ) is not so easy to see from the 한푄- module perspective). Furthermore, it carries a 퐺푉 = ∏푖∈퐼 GL(푉푖)-action by base-change, where the action on the 푒 −1 × component 휑푒 for (푖 ←−푗) is given by 푔 ⋅ 휑푒 = 푔푖휑푒푔푗 . The subgroup 한 ↪ 퐺푉 of diagonally embedded scalars acts trivially on Rep(푄, 푉 ). 8.2.2 Example Consider the quiver 푄 consisting of two vertices 퐼 = {1, 2} and a single edge 1 → 2. Suppose we choose the 2 2 퐼-graded vector space 푉 with 푉1 = ℂ and 푉2 = ℂ . Then the space Rep(푄, 푉 ) ≅ 픸ℂ since we may identify a 푥 map 휑 ∶ ℂ → ℂ2 with the column vector 휑 (1) = ( ). The group 퐺 = GL × GL acts by 1→2 1→2 푦 푉 1 2

푥 푥 (푔 , 푔 ) ⋅ ( ) = 푔 ( ) 푔−1. (8.2.3) 1 2 푦 2 푦 1

78 8.3 Nakajima quiver varieties

8.3 Nakajima quiver varieties

In this section we will go over the basics of Nakajima’s construction of the quiver variety 픐(휆) for any dominant weight 휆. Let (퐼, ⋅) be a symmetric Cartan datum — for our purposes, we will also assume that it is bipartite and hence admits a 2-colouring 휁 ∶ 퐼 → {0, 1}. Define a quiver by taking 퐼 as the vertex set, and adding |푖 ⋅ 푗| oriented edges between 푖 and 푗, pointing from the odd vertex to the even vertex: call the collection of all these edges Ω. Similarly, add another |푖⋅푗| edges between 푖 and 푗, this time pointing from the even vertex to the odd vertex: call the collection of all these edges Ω . Then 푄 = (퐼, Ω ⊔ Ω) is a quiver. For example, if (퐼, ⋅) is A3, the resulting quiver 푄 looks like

Fix a root datum Φ of type (퐼, ⋅), and a pair of weights 휆, 휇 such that 휆 is dominant and 휇 ≤ 휆. In order to construct the quiver variety 픐(휆, 휇) we choose 퐼-graded vector spaces 푊 and 푉 which will ‘represent’ the weights 휆 and 휇. ∨ The framing space 푊 is any 퐼-graded vector space such that ⟨휆, 훼푖 ⟩ = dim 푊푖 for all 푖 ∈ 퐼, while the other vector space 푉 is any 퐼-graded vector space such that 휆 − 휇 = ∑푖∈퐼 (dim 푉푖)훼푖. Now that we have our fixed choice of 푊 and 푉 , define the large vector space

M(푉 , 푊 ) = Repℂ(푄, 푉 ) ⊕ Homℂ퐼 (푊 , 푉 ) ⊕ Homℂ퐼 (푉 , 푊 ). (8.3.1) ♡ ♡ This space can be thought of as a kind of Repℂ(푄 , 푉 , 푊 ) for a framed quiver 푄 . The notation used in [Nak01b] is (퐵, 푖, 푗) ∈ M(푉 , 푊 ) for a typical element, so 퐵 ∶ 푉 → 푉 are the horizontal maps, 푖∶ 푊 → 푉 goes downwards, and 푗 ∶ 푉 → 푊 goes upwards. In the 퐴3 example the framed quiver looks like this, with a ‘summary schematic’ on the right.

푊1 푊2 푊3 푊

= 푖 푗

퐵 푉1 푉2 푉3 푉

Since we really want to keep to Nakajima’s notation here, in this chapter only we will use 푖, 푗 to mean these maps of quivers, and switch to using the letters 푘, 푙 ∈ 퐼 to index vertices of the Dynkin diagram.

The reductive group 퐺푉 = ∏푘∈퐼 GL(푉푘) acts on the left of ℳ(푉 , 푊 ) by base change automorphisms: 푔 ⋅ (퐵, 푖, 푗) = (푔퐵푔−1, 푔푖, 푗푔−1). Choosing any function 휖 ∶ Ω → ℂ× satisfying 휖(푒) + 휖(푒) = 0 defines a symplectic form on the vector space M(푉 , 푊 ) by 휔((퐵, 푖, 푗), (퐵′, 푖′, 푗′)) = tr((휖퐵)퐵′) + tr(푖푗′ − 푖′푗), (8.3.2) where 휖퐵 means to scale each map 퐵푒 ∶ 푉tail(푒) → 푉head(푒) by the scalar 휖(푒). The action of 퐺푉 preserves the symplectic form: 휔(푔 ⋅ (퐵, 푖, 푗), 푔 ⋅ (퐵′, 푖′, 푗′)) = 휔((푔퐵푔−1, 푔푖, 푗푔−1), (푔퐵′푔−1, 푔푖′, 푗′푔−1)) = tr(푔(휀퐵)퐵′푔−1) + tr(푔푖푗′푔−1 − 푔푖′푗푔−1) (8.3.3) = 휔((퐵, 푖, 푗), (퐵′, 푖′, 푗′)).

The moment map (which we stress is nonlinear) associated to the action of 퐺푉 is defined up to an additive constant. The moment map 휇 which vanishes at the origin is given by

∗ 휇 ∶ M(푉 , 푊 ) → (Lie 퐺푉 ) , 휇(퐵, 푖, 푗) = (휖퐵)퐵 + 푖푗, (8.3.4)

79 8 Nakajima quiver varieties where Lie 퐺푉 has been identified with its dual via the trace pairing, so thatfor 퐶 ∈ Lie 퐺푉 = ∏푘∈퐼 Endℂ(푉푘) we have 휇(퐵, 푖, 푗)(퐶) = tr(퐶((휖퐵)퐵 + 푖푗)). (8.3.5) The preimage 휇−1(0) of the moment map is an affine algebraic variety, not necessarily reduced. The first kind of quiver variety is the categorical quotient

−1 −1 퐺 픐0(푉 , 푊 ) = 휇 (0) //퐺푉 = Spec ℂ[휇 (0)] 푉 . (8.3.6)

−1 This is an affine algebraic variety, whose underlying set is identified with the setofclosed 퐺푉 -orbits in 휇 (0). A point (퐵, 푖, 푗) ∈ 휇−1(0) is stable if the only 퐵-invariant 퐼-graded subspace 푆 ⊆ 푉 contained in ker 푗 is 0. We write −1 푠 휇 (0) for the set of stable points, which is a 퐺푉 -invariant set. Define the second kind of quiver variety asthe GIT (geometric invariant theory) quotient

−1 푠 픐(푉 , 푊 ) = 휇 (0) /퐺푉 . (8.3.7)

−1 푠 The underlying set of points of the GIT quotient is the quotient of the 퐺푉 -set 휇 (0) of stable points by the group 퐺푉 . The 퐺푉 action is free, and 픐(푉 , 푊 ) is a smooth projective variety inheriting a symplectic form from M(푉 , 푊 ). There is a projective morphism

휋 ∶ 픐(푉 , 푊 ) → 픐0(푉 , 푊 ), (8.3.8) sending the equivalence class [퐵, 푖, 푗] to that unique closed orbit contained in the orbit closure 퐺푉 ⋅ (퐵, 푖, 푗) .

8.3.9 Example (Nakajima quiver varieties in type A1) ♡ In type A1, the Coxeter graph consists of a single vertex, and hence the framed quiver 푄 has only two vertices. Let 푊 be the framing vector space and 푉 the one corresponding to the unique vertex of the original quiver 푄. The linear space simplifies since 푄 contains no edges:

M(푉 , 푊 ) = Homℂ(푊 , 푉 ) ⊕ Homℂ(푉 , 푊 ). (8.3.10)

−1 The 퐺푉 = GL(푉 )-action on the point (푖, 푗) ∈ 픐(푉 , 푊 ) is by 푔 ⋅ (푖, 푗) = (푔푖, 푗푔 ), with the moment map −1 휇(푖, 푗) = 푖푗 ∈ Endℂ(푉 ). The condition (푖, 푗) ∈ 휇 (0) is precisely 푖푗 = 0 ∈ Endℂ(푉 ), or in other words im 푗 ⊆ ker 푖. The stability condition is equivalent to requiring that 푗 is injective, and hence we have

휇−1(0) = {(푖∶ 푊 → 푉 , 푗 ∶ 푉 → 푊 ) ∣ 푗 is injective, and 푖푗 = 0}. (8.3.11)

Let Gr(푉 , 푊 ) = {푈 ⊆ 푊 ∣ 푈 ≅ 푉 } be the Grassmannian of 푉 -planes in 푊 . Alternatively, we can think ∘ of Gr(푉 , 푊 ) as the quotient of the space Homℂ(푉 , 푊 ) of injective linear maps by the right action of 퐺푉 , identifying all injective maps with the same image. Fix a splitting 푊 = 푉 ⊕ 푉 ′, then any point 푈 ∈ Gr(푉 , 푊 ) 푗푉 ′ may be written as the image of the map ( ) ∶ 푉 → 푉 ⊕ 푉 = 푊 , where 푗푉 ∶ 푉 ↪ 푊 is the inclusion 휎푈 ′ and 휎푈 ∶ 푉 → 푉 . This defines an affine open neighbourhood of the point 푉 ∈ Gr(푉 , 푊 ) isomorphic to Hom(푉 , 푉 ′). Correspondingly, the cotangent space at the point 푉 is the affine space Hom(푉 ′, 푉 ) ≅ {푗 ∈ Hom(푊 , 푉 ) ∣ 푗(푉 ) = 0}. The cotangent bundle of the Grassmannian is precisely the quotient of 휇−1(0) by the action of 퐺푉 , hence we have

−1 ∗ 휇 (0)/퐺푉 = 픐(푉 , 푊 ) ≅ 푇 Gr(푉 , 푊 ). (8.3.12)

We find that 픐(푉 , 푊 ) is nonempty if and only if dim 푉 ≤ dim 푊 , and that if dim 푉 = 0 or dim 푉 = dim 푊 then 픐(푉 , 푊 ) is a point.

Define the Nakajima quiver variety associated to 푊 to be 픐(푊 ) = ⨆푉 픐(푊 , 푉 ), where we let 푉 vary over a fixed set of vector spaces, one for each possible 퐼-graded dimension. As we have already seen in Example 8.3.9, there will typically be many 푉 for which 픐(푊 , 푉 ) is empty. It is important to remember that whenever a 푉 appears

80 8.4 Vector bundles on quiver varieties in the discussion of the variety 픐(푊 ), we are implicitly working inside the piece 픐(푊 , 푉 ). This is obvious once pointed out, but might trip up the casual reader otherwise. Lastly, recall that the 퐼-graded dimensions of 푊 and 푉 were determined by a dominant weight 휆 and a weight 휇 ≤ 휆. We define 픐(휆, 휇) = 픐(푊 , 푉 ) and 픐(휆) = 픐(푊 ) when we prefer to speak of weights rather than graded vector spaces.

8.4 Vector bundles on quiver varieties

For a given pair (푊 , 푉 ), both 푊 and 푉 can be considered as left 퐺푉 representations, where 푉 is the defining −1 푠 representation of 퐺푉 , and 푊 is a trivial representation. As the action of 퐺푉 on the set 휇 (0) of stable points is −1 푠 free, the projection map 휇 (0) ↠ 픐(푉 , 푊 ) is a left principal 퐺푉 -bundle. We may perform the associated bundle construction (Chapter 4.5 of [Hus94]) yielding two 퐼-graded vector bundles 풱 and 풲 over 픐(푊 , 푉 ).

We have vector bundles Rep(푄, 풱 ), Homℂ퐼 (풲 , 풱 ) and Homℂ퐼 (풱 , 풲 ) analogously as before.

8.5 Graded quiver varieties

In [Nak01b], Nakajima constructs representations of tensor products 퐿(휆1) ⊗ ⋯ ⊗ 퐿(휆푛) by considering a suitable subvariety inside 픐(휆1 + ⋯ + 휆푛), defined by picking a grading on the framing space 푊 which separates it into pieces of ‘sizes’ 휆1, … , 휆푛. Later in the same paper (Section 8), a more general grading is considered, defining a subvariety which we will call a graded Nakajima quiver variety. A crystal structure is defined on the connected components of this graded quiver variety, which we will show is isomorphic to the product monomial crystal. We will now consider 푊 to be not just an 퐼-graded vector space, but an (퐼, ℤ)-bigraded vector space, meaning that we put a ℤ-grading on each 퐼-homogeneous piece 푊푘. We will continue to use lower indices for the 퐼-grading, 푝 and start using upper indices for the ℤ-grading, so that 푊푘 is the subspace of the 퐼-graded piece 푊푘 which has ℤ-grading 푝. Up to conjugacy in GL퐼 (푊 ), the extra data of such a ℤ-grading is determined by a finite multiset Q 푝−1 based in 퐼 × ℤ, where dim 푊푖 = Q[푖, 푝] (the indexing shift 푝 − 1 we use will make the morphism to the varied × monomial crystal ℳ푐(Φ) more pleasant later on). Let 휌Q ∶ ℂ → GL퐼 (푊 ) be a morphism of algebraic groups giving the grading Q (again, 휌Q is determined by Q only up to conjugacy, so we fix any such morphism for the remainder of this section). From this point we are closely following Section 8 of [Nak01b], albeit choosing notation we find more clear. Using this extra ℤ-grading on 푊 , we define a ℂ×-action on M(푊 , 푉 ) by the formulas

퐵푒 if 푒 ∈ Ω, −1 푡 ⋄Q 퐵푒 = { 푡 ⋄Q 푖 = 푖휌Q(푡) , 푡 ⋄Q 푗 = 푡휌Q(푡)푗, (8.5.1) 푡퐵푒 if 푒 ∈ Ω,

−1 푠 which together define an action 푡⋄Q(퐵, 푖, 푗). The ⋄Q action preserves the set 휇 (0) of stable points and commutes with the 퐺푉 ⋅ action, and hence descends to an action on the quiver variety 픐(푊 ). The graded quiver variety is the Q set of fixed points 픐(푊 ) ⊆ 픐(푊 ) under the ⋄Q action, and we will soon see how to equip its set of connected components with a crystal structure. Consider a fixed point [퐵, 푖, 푗] ∈ 픐(푊 , 푉 )Q. For each representative (퐵, 푖, 푗) ∈ M(푊 , 푉 ) of the fixed point [퐵, 푖, 푗], × there exists a unique map 휌 ∶ ℂ → 퐺푉 of algebraic groups satisfying

−1 푡 ⋄Q (퐵, 푖, 푗) = 휌 (푡) ⋅ (퐵, 푖, 푗). (8.5.2)

The conjugacy class of 휌 is independent of the choice of representative (퐵, 푖, 푗), and hence we obtain a map

Q × 퐹Q ∶ 픐(푊 , 푉 ) → HomAlgGrp(ℂ , 퐺푉 )/퐺푉 , (8.5.3)

81 8 Nakajima quiver varieties taking a fixed point [퐵, 푖, 푗] to the conjugacy class [휌]. It turns out that the fibres of this map are precisely the connected components of the graded quiver variety 픐(푊 , 푉 )Q, and so we have a map

× Q −1 HomAlgGrp(ℂ , 퐺푉 )/퐺푉 → 휋0(픐(푊 , 푉 ) ), [휌] ↦ 퐹Q ([휌]). (8.5.4)

Technically, in order to construct a crystal structure on the set of connected components, one should consider a larger Lagrangian subvariety containing the graded quiver variety, of points whose 푡 → ∞ ⋄Q-limit lands inside the graded quiver variety. This gives ‘enough information’ to determine where the crystal operators should go. However, it turns out that all of this information is implied by the conjugacy class [휌] indexing a connected component, and so we will not mention these attracting sets further.

The crystal structure on the set of connected components is spelled out in Proposition 8.3 and Lemma 8.4 of [Nak01b]. We will write it down here in compatible notation. Our main change is replacing a conjugacy class × [휌] ∈ HomAlgGrp(ℂ , 퐺푉 )/퐺푉 by a finite multiset T휌 ∶ 퐼 × ℤ → ℕ based in 퐼 × ℤ, noting that the conjugacy class 푝 of 휌 in 퐺푉 is entirely determined by the dimensions T휌[푖, 푝] = dim 푉푖 , where the ℤ-grading on 푉 is coming from 휌. Therefore we have that each connected component of the graded quiver variety 픐(푊 , 푉 )Q is indexed by a finite multiset T.

−1 Over the connected component 퐹Q (T) indexed by the multiset T, there is a complex of (퐼, ℤ)-graded vector bundles, whose (푘, 푝)-graded component is

푝 푝 푝−1 푝−1 푝 푝−1 퐶푘 (T) ∶ 푉푘 → 푊푘 ⨁ 푉푙 ⨁ 푉푙 → 푉푘 , (8.5.5) 푒 푒 (푘−→푙)∈Ω (푘−→푙)∈Ω

푝 where the middle term is in homological degree zero (recall that the dimensions of the 푉푘 appearing in the com- 푝 plex above are determined by T, via T[푖, 푝] = dim 푉푖 ). The rank of this complex (alternating sum of dimensions) features in the definition of the crystal structure, so we compute it explicitly here. Recall that our partition Ω ⊔ Ω was determined by a two-colouring 휁 ∶ 퐼 → {0, 1}, where we declared that Ω contains edges going from odd to even vertices, while Ω goes from even to odd vertices, where ‘odd’ or ‘even’ is given by the two-colouring 휁 . This means that for each vertex, one of the two summations in Eq. (8.5.5) is zero, depending on the parity of the 푝 푝−1 vertex. For even vertices, only the 푉푙 term contributes, while for odd vertices only the 푉푙 term contributes. 푝 Together with our contrived choice of indexing shift dim 푊푘 = Q[푘, 푝 + 1] we get

푝 rank 퐶푘 (T) = Q[푘, 푝] − T[푘, 푝] − T[푘, 푝 − 1] − ∑ 푎푙푘 T[푙, 푝 − 휁 (푙)]. (8.5.6) 푙≠푘

Q We can now give the crystal structure on the set 휋0(픐(푊 ) ) of connected components of the graded Nakajima quiver variety. 8.5.7 Definition (Crystal structure on the graded quiver variety) 1 This is Lemma 8.4 of [Nak01b] . Let Φ be a root datum of symmetric bipartite type (퐼, ⋅), 휆 ∈ 푋(Φ)+ a ∨ dominant weight, 푊 an 퐼-graded vector space with dim 푊푖 = ⟨휆, 훼푖 ⟩, and Q a multiset giving the (퐼, ℤ)- 푝−1 −1 Q Q bigrading on 푊 via dim 푊푖 = Q[푖, 푝]. Over the connected component 퐹Q (T) ⊆ 픐(푊 , 푉 ) ⊆ 픐(푊 ) , define 푝 푞 푝 푞 휀푘 (T) = − ∑ rank 퐶푘 (T), 휑푘 (T) = ∑ rank 퐶푘 (T), 푞>푝 푞≤푝 (8.5.8) 푝 푝 휀푘(T) = max 휀 (T), 휑푘(T) = max 휑 (T), 푝∈ℤ 푘 푝∈ℤ 푘 and let 푞 푞 푛푒,푘(T) = max {푞 ∣ 휀푘 (T) = 휀푘(T)} , 푛푓 ,푘(T) = min {푞 ∣ 휑푘 (T) = 휑푘(T)} . (8.5.9)

82 8.5 Graded quiver varieties

Define the weight on the connected component indexed by T to be

wt(T) = 휆 − ∑(dim 푉푘)훼푘 푘∈퐼 (8.5.10) = 휆 − ∑ ∑ T[푘, 푝]훼푘. 푘∈퐼 푝∈ℤ

Define the crystal operators by

0 if 휀푘(T) = 0, 0 if 휑(T) = 0, 푒푘(T) = { 푓푘(T) = { (8.5.11) T − {(푘, 푛푒,푘(T))} if 휀푘(T) > 0, T + {(푘, 푛푓 ,푘(T))} if 휑푘(T) > 0.

Q Then (wt, 휀푘, 휑푘, 푒푘, 푓푘) gives the set of connected components 휋0(픐(휆) ) the structure of a seminormal crystal.

Q We can give an embedding of the crystal 휋0(픐(휆) ) into the varied monomial crystal ℳ푐(Φ) of Definition 6.2.1 with parameters 푐푘푙 = 휁 (푘). Recall the auxiliary monomial from Definition 6.2.1: 푎 푎 = 푒훼푘 ⋅ 푦 ⋅ 푦 ⋅ ∏ 푦 푙푘 . (8.5.12) 푘,푝 푘,푝 푘,푝+1 푙,푝+푐푙푘 푙≠푘

휆 −1 Extracting the exponent of 푦푘,푝 from the monomial 푒 ⋅ 푦Q ⋅ 푎T we get

휆 −1 (푒 ⋅ 푦Q ⋅ 푎T )[푘, 푝] = Q[푘, 푝] − T[푘, 푝] − T[푘, 푝 − 1] − ∑ 푎푙푘T[푝 − 푐푙푘], (8.5.13) 푙≠푘

푝 precisely the same expression as rank 퐶푘 (T) in Eq. (8.5.6) since 푐푙푘 = 휁 (푙). Define the map of sets Q −1 휆 −1 휙 ∶ 휋0(픐(푊 ) ) → ℳ푐(Φ), 퐹Q (T) ↦ 푒 ⋅ 푦R ⋅ 푧T . (8.5.14) It is straightforward to verify that 휙 is a strict inclusion of crystals, by directly comparing Definitions 6.2.1 and 8.5.7 using Eq. (8.5.13) to convert between monomials and multisets.

We now wish to show that the image of 휙 is the product monomial crystal ℳ푐(휆, Q), by which we mean the analogue of the product monomial crystal ℳ(휆, Q), but defined inside the varied monomial crystal ℳ푐(Φ) rather than the monomial crystal ℳ(Φ). 8.5.15 Theorem Q Let Φ be a root datum of symmetric, bipartite, acyclic Cartan type (퐼, ⋅). Then the image of the map 휙 ∶ 휋0(픐(푊 ) ) → ℳ푐(Φ) is the varied product monomial crystal ℳ푐(휆, Q), where

휆0 휆푖,푐 R[푖,푐] ℳ푐(휆, Q) = 푒 ⋅ ∏ ℳ(푒 ⋅ 푦푖,푐 ), (8.5.16) (푖,푐)∈Supp Q

∨ and 휆 = 휆0 + ∑(푖,푐)∈R 휆푖,푐 is any decomposition of 휆 such that ⟨훼푖 , 휆푖,푐⟩ = R[푖, 푐] for all (푖, 푐) ∈ R. The proof follows the approach outlined in Proposition 7.7 of [Kam+19a]. 8.5.17 Proof Firstly, we reason that this works for a monomial concentrated in a single point. When the multiset Q is 푛 푝 concentrated, i.e. it is of the form Q = {(푘, 푝) }, then the vector space 푊 is concentrated in the term 푊푘 , so it is concentrated over a single vertex with a ℂ×-action by a single weight. In this case a result2of Nakajima’s (references in Section 8 of [Nak01b]) gives that the graded quiver variety (or rather, the set attracting to it)

1 This is almost exactly what appears in [Nak01b], but we have switched the min and max appearing in the definitions of 푛푒,푘 and 푛푓 ,푘 . We believe that this is a typo in Nakajima’s original paper, since the 푒푖 and 푓푖 crystal operators are not obviously partially inverse without this change.

83 8 Nakajima quiver varieties

is the usual Lagrangian variety 픏(휆) ⊆ 픐(휆), and hence the crystal afforded by the quiver variety isthe Q 휆 highest weight crystal ℬ(휆). Therefore in this case we have 휋0(픐(푊 ) ) ≅ ℬ(휆) ≅ ℳ(푒 ⋅ 푦Q), and the claim follows since 휙 is a nonzero crystal morphism between two connected seminormal crystals, and hence must be an isomorphism. Next, we consider the general case. Fix a factorisation of (휆, R) into dominant pairs

푛 푛 (휆1, (푘1, 푝1) 1 ), … , (휆푁 , (푘푁 , 푝푁 ) 푁 ). (8.5.18)

An arbitrary monomial 푝 of the product monomial crystal ℳ푐(휆, Q) has an according factorisation 푝 = 푝1 ⋯ 푝푁 into monomials coming from the fundamental subcrystals. 푛 Let Q푟 = {(푘푟 , 푝푟 ) 푟 }, so that each monomial 푝푟 is an element of the fundamental subcrystal ℳ푐(휆푟 , Q푟 ), and let 푊 = 푊 [1] ⊕ ⋯ ⊕ 푊 [푟] be a factorisation of 푊 such that 푊 [푟] is concentrated in the (퐼, ℤ)-degree (푘푟 , 푝푟 ) with dimension 푛푟 . Since the claim holds on fundamental subcrystals, for each 푟 there is an isomorphism Q ∼ 휙[푟]∶ 휋0(픐(푊 [푟]) 푟 ) −→ℳ푐(휆푟 , Q푟 ) of crystals, and hence for each monomial 푝푟 there exists some 퐼-graded Q vector space 푉 [푟] and fixed point (퐵[푟], 푖[푟], 푗[푟]) ∈ ℳ(푊 [푟], 푉 [푟]) 푟 mapping to 푝푟 under 휙[푟]. It is then straightforward to check that

(퐵[1] ⊕ ⋯ ⊕ 퐵[푟], 푖[1] ⊕ ⋯ ⊕ 푖[푟], 푗[1] ⊕ ⋯ ⊕ 푗[푟]) ∈ ℳ(푊 , 푉 [1] ⊕ ⋯ ⊕ 푉 [푟]) (8.5.19)

is a ⋄Q-fixed point mapping to 푝 under 휙. This shows that ℳ푐(휆, Q) ⊆ im 휙. −1 To get the opposite inclusion, consider a T such that 퐹Q (T) is nonempty (and hence is a connected com- Q −1 ponent of 픐(푊 ) ). By Proposition 4.1.2 of [Nak01a] the variety 퐹Q (T) is homotopic to its projective sub- −1 variety 퐹Q (T) ∩ 픏(푊 ). The maximal torus 푇푊 ⊆ GL퐼 (푊 ) acts on this subvariety, and hence there exists a fixed point by Borel’s theorem. Lemma 3.2 of[Nak01b] implies that such a fixed point can be decomposed Q as a sum of (퐵[푟], 푖[푟], 푗[푟]) as above, where each (퐵[푟], 푖[푟], 푗[푟]) is an element of 휋0(픐(푊 [푟]) 푟 ), and hence −1 휙(퐹Q (T)) ∈ ℳ푐(휆, Q), showing that im 휙 ⊆ ℳ푐(휆, Q).

Having shown that 휙 is a crystal isomorphism, it is immediate that the subset ℳ푐(휆, Q) is a seminormal abstract int ∼ crystal of an 풪푞 (Φ)-module, and together with the isomorphism ℳ푐(Φ) −→ℳ(Φ) given in Lemma 6.2.5 implies that the product monomial crystal ℳ(휆, R) is indeed a crystal: Theorem 6.3.5 is proven. Q Finally, we also note that Theorem 8.5.15 together with the fact that 휋0(픐(푊 ) ) is connected when Q is concentrated in a single point imply that the product monomial crystal ℳ(휆, R) is well-defined, no matter how the multiset R is broken up into fundamental pairs, since the identity Eq. (6.3.4) is proven.

2The acyclic assumption on (퐼, ⋅) is necessary for this result to hold.

84 9 Generalised Schur modules

The irreducible polynomial representations of GL푛(ℂ) are parametrised by partitions 휆 with at most 푛 rows, each simple module 퐿(휆) having an explicit construction by applying a certain Schur functor 풮휆 ∶ Vectℂ → Vectℂ to the defining representation of GL푛. For this reason, the representation 퐿(휆) is sometimes called a Schur module: the image of the defining representation under a Schur functor. There is a natural generalisation of the functor 풮휆 to any diagram 퐷 of boxes in the plane, recovering 풮퐷 = 풮휆 in the case that 퐷 is a Young diagram of the partition 휆. The image of the defining representation under such a functor 풮퐷 is accordingly called a generalised Schur module.

Our eventual aim for these final two chapters is to show a correspondence between diagrams 퐷 and multisets R 푛 such that the product monomial crystal ℳ(휆, R) ⊆ ℳ(GL푛) is the crystal of a generalised Schur module 풮퐷(ℂ ). In order to do this we put two character formulae to work, the first being our own formula Theorem 7.2.3 on the product monomial crystal side, and the second formula due Magyar, Reiner, and Shimozono on the generalised Schur module side. Unfortunately, this only gets us as far as the statement ‘they match for large enough 푛’, and so we spend Chapter 10 showing some stability properties of the monomial crystal (which are interesting in their own right) to deduce the result in general.

In this chapter we will first define generalised Specht modules, the symmetric group analogues of the generalised Schur modules, which can be used to give a quick definition of the Schur modules from which their stability properties are evident. We then introduce the generalised Schur functors 풮퐷 which give a second definition of a generalised Schur module as the image of the defining representation under 풮퐷, and show that (in characteristic zero) these two different definitions of generalised Schur modules coincide. The Schur functors arespecial cases of flagged Schur functors, in a similar way to how the highest-weight modules 퐿(휆) are special cases of De- mazure modules 퐿푤 (휆). The flagged Schur functors are more amenable to inductive analysis, and we give results due to Magyar, Reiner, and Shimozono about the characters of flagged Schur modules. Finally, we give a direct correspondence between diagrams 퐷 and multisets R which we use to show that the characters of the product 푛 monomial crystal ℳ(휆, R) and the generalised Schur module 풮퐷(ℂ ) coincide for column-convex diagrams 퐷, provided that 푛 is large enough.

9.1 Generalised Specht modules

Perhaps the quickest way of defining generalised Schur modules is via their analogue for representations of symmetric groups, the generalised Specht modules. We first recall some necessary combinatorics. 9.1.1 Definition (Partitions) A partition is a weakly decreasing finitely supported sequence 휆 ∶ ℙ → ℕ. A partition 휆 can be regarded as a finite list 휆 = (휆1, … , 휆푙) where 휆1 ≥ ⋯ ≥ 휆푙 > 0 and 휆푟 = 0 for 푟 > 푙. Each element 휆푖 ≥ 1 is called a part of 휆, the sum of the parts |휆| = ∑푖≥1 휆푖 is called the size of 휆, and the size of the support ℓ(휆) = 푙 is called the length of 휆. When 휆 has size 푛 we say that 휆 is a partition of 푛. The empty partition is the unique partition of 0 and is denoted by ∅. We define the sets 1. Part of all partitions, 2. Part(≤ 푙) of all partitions of length at most 푙, 3. Part푛 of all partitions of size 푛, 4. Part푛(≤ 푙) = Part푛 ∩ Part(≤ 푙) of all partitions of 푛 with length at most 푙.

85 9 Generalised Schur modules

The first two sets are always infinite (besides the special case Part(≤ 0) = {∅}), while the last two sets are always finite. It is common to represent a partition 휆 pictorially via its Young diagram 퐷(휆) ⊆ ℙ × ℙ, where 퐷(휆) is the subset consisting of those (푖, 푗) such that 푗 ≤ 휆푖. The coordinates (푖, 푗) are read similarly to matrices, with 푖 increasing down the page and 푗 increasing to the right. The Young diagrams of the partitions 휆 ∈ Part5 are pictured as follows:

(5) (4, 1) (3, 2) (3, 1, 1) (2, 2, 1) (2, 1, 1, 1) (1, 1, 1, 1, 1)

The partitions Part푑 of size 푑 index the conjugacy classes in the symmetric group 픖푑 , classifying a permutation by its sorted list of cycle lengths. By the general theory of representations of finite groups, the partitions of size 푑 also index the set {Σ휆 ∣ 휆 ∈ Part푑 } of pairwise nonisomorphic irreducible representations of ℂ[픖푑 ]. The constructions we will give of these modules Σ휆 are called Specht modules, and the reader can find a full account of this from the traditional perspective (which we will briefly go over here) in Chapter 7of[Ful96]). We also mention that there is an alternative beautiful derivation of these modules, including a canonical decomposition into lines, due to Okounkov and Vershik [VO05] and explained well in Chapter I.2 of [Kle05].

In order to construct the Specht module Σ휆, we first fix a bijective tableau 푇 of shape 휆, which is a bijective map 푇 ∶ 퐷(휆) → [푑]. The symmetric group 픖푑 has a free transitive action on the set of bijective tableaux by postcomposition, 휎 ⋅ 푇 = 휎 ∘ 푇 . The tableau 푇 determines a row stabilising subgroup 푅푇 ⊆ 픖푑 of elements permuting the entries of 푇 within their rows, and a column stabilising subgroup 퐶푇 of elements permuting the entries of 푇 within their columns. For example, shown below is a bijective tableau 푇 of 휆 = (3, 2) together with its row and column stabilising subgroups:

푇

2 5 4 푅푇 = 픖({2, 4, 5}) × 픖({1, 3}) 퐶푇 = 픖({1, 2}) × 픖({3, 5}) × 픖({4}) 1 3

Define the following elements of the group algebra ℂ[픖푑 ]:

휋 푐푇 = ∑ (−1) 휋, 푟푇 = ∑ 휎, 푦푇 = 푐푇 푟푇 . (9.1.2) 휋∈퐶푇 휎∈푅푇

The element 푦푇 is called the Young symmetriser associated to 푇 and is a pseudo-idempotent of ℂ[픖푑 ], meaning that 2 × 푦푇 = 푧푦푇 for some nonzero scalar 푧 ∈ ℂ . The left submodule Σ푇 = ℂ[픖푑 ]푦푇 is called a Specht module associated −1 −1 −1 to 휆. Since 푅휎푇 = 휎푅푇 휎 and 퐶휎푇 = 휎퐶푇 휎 , we have 푦휎푇 = 휎푦푇 휎 and hence a different choice of bijective tableau 휎푇 will yield an isomorphic Specht module, with the map

−1 Σ푇 → Σ휎푇 , 푥 ↦ 푥휎 (9.1.3) giving an isomorphism of representations. Hence we can define the Specht module Σ휆 up to isomorphism to be any one of the Σ푇 where 푇 is a bijective tableau of shape 휆. 9.1.4 Theorem Fix a 푑 ≥ 1. The Specht modules {Σ휆 ∣ 휆 ∈ Part푑 } give a complete list of pairwise nonisomorphic irreducible representations of the symmetric group 픖푑 over the complex numbers ℂ. Everything we have done so far is completely classic. We will generalise this construction in one of the most naïve ways possible, by replacing the Young diagram 퐷(휆) ⊆ ℙ × ℙ with an arbitrary subset 퐷 ⊆ ℙ × ℙ of size 푑,

86 9.1 Generalised Specht modules called a diagram. We can define bijective tableaux 푇 ∶ 퐷 → [푑] and the row and column stabilising subgroups 푅푇 and 퐶푇 in the same way as before, and we end up with a generalised Young symmetriser 푦푇 . This symmetriser is still a pseudo-idempotent of the group algebra, but the associated generalised Specht module Σ퐷 ≅ Σ푇 = ℂ[픖푑 ]푦푇 is no longer irreducible in general. As the representation theory of ℂ[픖푑 ] is semisimple, the generalised Specht module Σ퐷 decomposes as a direct sum of Specht modules. 9.1.5 Definition (Generalised Littlewood-Richardson coefficients) 휆 The multiplicity 푐퐷 ∶= [Σ휆 ∶ Σ퐷] of the irreducible module Σ휆 inside the generalised Specht module Σ퐷 is called a generalised Littlewood-Richardson coefficient. As the isomorphism class of Σ퐷 is invariant under row 휆 or column permutations of 퐷, so are the coefficients 푐퐷. We will justify the name generalised Littlewood-Richardson coefficients. A diagram is skew if it is equal to the difference 퐷(휈) ⧵ 퐷(휆) of two Young diagrams. By a theorem in Section 3 of [JP79], the multiplicity of the Specht module Σ휇 in the generalised Specht module Σ퐷(휈)⧵퐷(휆) is equal to the inner product ⟨푠휈/휆, 푠휇⟩ of symmetric 휈 functions, which is the Littlewood-Richardson coefficient 푐휆휇 counting the number of Littlewood-Richardson tableaux of shape 퐷(휈)⧵퐷(휆) and weight 휇 (Section I.9 of [Mac95]). Hence the generalised Littlewood-Richardson 휇 휈 coefficient 푐퐷(휈)⧵퐷(휆) is equal to the Littlewood-Richardson coefficient 푐휆휇. 9.1.6 Example Let 퐷 be the 5-box diagram 퐷 = {(1, 1), (2, 2), (2, 3), (3, 2), (4, 3)} ⊆ ℙ × ℙ. By applying the row permutation (234) and the column permutation (132), 퐷 may be rearranged into a skew diagram 퐷′ = 퐷(3, 2, 2, 1)⧵퐷(2, 1). We show this rearrangement pictorially, using colours to mark the original and final positions of boxes: 퐷 퐷′

We can then state the multiplicities of Specht modules inside the generalised (skew) Specht module by

휆 푐퐷(3,2,2,1)⧵퐷(2,1) = [Σ휆 ∶ Σ퐷(3,2,2,1)⧵퐷(2,1)] = ⟨푠(3,2,2,1)/(2,1), 푠휆⟩, (9.1.7)

which are easily calculated by any software package1dealing with symmetric functions or Littlewood- Richardson coefficients. In this case, we get

⊕2 Σ퐷 ≅ Σ퐷′ ≅ Σ(2,1,1,1) ⊕ Σ(3,1,1) ⊕ Σ(2,2,1) ⊕ Σ(3,2). (9.1.8)

(2,2,1) Hence the generalised Littlewood-Richardson coefficient 푐퐷 = 2. The generalised Littlewood-Richardson coefficients are really a strict generalisation of Littlewood-Richardson coefficients, since not all diagrams can be made via row and column permutations into a skew shape. For example the following diagram

cannot be rearranged by column and row permutatinos to be skew. In addition to [JP79], the modules Σ퐷 have been studied in [Liu10], [Liu15], and various works of Reiner and Shimozono which we will cite in what follows.

1Or by hand, if it’s a rainy day.

87 9 Generalised Schur modules

9.2 Generalised Schur modules

Let 푉 be a finite-dimensional complex vector space, and GL(푉 ) the associated general linear group. The tensor ⊗푑 power 푉 of the defining representation is naturally a (GL(푉 ), 픖푑 )-bimodule, where GL(푉 ) acts along the diagonal 푔 ⋅ (푣1 ⊗ ⋯ 푣푑 ) = 푔푣1 ⊗ ⋯ ⊗ 푔푣푑 and 픖푑 acts on the right by permuting tensor factors. For each diagram ⊗푑 퐷 we get a left GL(푉 )-module 풮퐷(푉 ) = 푉 ⊗ℂ[픖푑 ] Σ퐷 called a generalised Schur module, where 푑 = |퐷| is the number of boxes in the diagram 퐷. When 퐷 = 퐷(휆) is the Young diagram of a partition, the generalised Schur module 풮휆(푉 ) is irreducible, and we call it a Schur module.

As a consequence of Schur-Weyl duality, the generalised Schur module 풮퐷(푉 ) decomposes in terms of the ir- 휆 reducibles 풮휆(푉 ) with the same decomposition multiplicites 푐퐷 given by the generalised Littlewood-Richardson coefficients, where the sum is taken over the restricted set Part푑 (≤ dim 푉 ) of partitions with length at most (dim 푉 ): 휆 풮퐷(푉 ) ≅ ⨁ 푐퐷풮휆(푉 ). (9.2.1) ℓ(휆)≤dim 푉

Note that if ℓ(휆) > dim 푉 then 풮휆(푉 ) = 0, so the above sum would still be valid without this restriction, but must be read more carefully. Eq. (9.2.1) has an important consequence for what we will call the stability of the generalised Littlewood-Richardson coefficients: although the decomposition of the generalised Schur module 풮퐷(푉 ) depends 휆 on dim 푉 , the coefficients 푐퐷 do not. Via an alternative definition of the generalised Schur module (Definition 9.2.5 휆 and Lemma 9.2.7) and the Pieri rule, we can see that 푐퐷 = 0 whenever the length of 휆 is larger than the number of rows of 퐷. Eq. (9.2.1) gives the following interpretation of generalised Littlewood-Richardson coefficients, which should be 휆 read in two parts: firstly a stability part about the ideal coefficients 푐퐷 being determined for dim 푉 large enough, and secondly a restriction part about how to apply those coefficients to unstable 푉 . 9.2.2 Corollary (Generalised Littlewood-Richardson coefficients via Schur modules) Let 퐷 be a diagram of 푑 boxes, and 푉 a vector space such that dim 푉 is at least the number of rows of 퐷. Then 휆 휆 the coefficients 푐퐷 appearing in the decomposition 풮퐷(푉 ) ≅ ⨁휆 푐퐷풮휆(푉 ) are the generalised Littlewood- Richardson coefficients of Definition 9.1.5. Furthermore, if 푈 is any vector space, then the decomposition of 휆 풮퐷(푈 ) into irreducible modules is given by 풮퐷(푈 ) ≅ ⨁ℓ(휆)≤dim 푈 푐퐷풮휆(푈 ). Although the definition we have given above of generalised Schur modules is concise and naturally givesthe stability result Corollary 9.2.2, it is is quite difficult to use for more ‘hands-on’ work and does not lend itself toa straightforward filtration by smaller modules. We turn instead to the notionofa Schur functor. The symmetric algebra 푆•(푉 ) and exterior algebra Λ•(푉 ) are graded-commutative Hopf algebras associated to 푉 . Fixing a degree 푑 ∈ ℕ, iterated comultiplication followed by taking the (1, … , 1)-graded piece gives maps into the tensor power 푉 ⊗푑 : 푑 Δ ⊗푑 푆 (푉 ) −→푉 , 푣1 ⋯ 푣푑 ↦ ∑ 푣휎(1) ⊗ ⋯ ⊗ 푣휎(푛), 휎∈픖푑 (9.2.3) 푑 Δ ⊗푑 휎 Λ (푉 ) −→푉 , 푣1 ⋯ 푣푑 ↦ ∑ (−1) 푣휎(1) ⊗ ⋯ ⊗ 푣휎(푛). 휎∈픖푑 Similarly, we have maps from the tensor power 푉 ⊗푑 to the degree-푑 part of the symmetric and exterior algebras by taking iterated multiplication:

⊗푑 푚 푑 푉 −→푆 (푉 ), 푣1 ⊗ ⋯ ⊗ 푣푑 ↦ 푣1 ⋯ 푣푑 , (9.2.4) ⊗푑 푚 푑 푉 −→Λ (푉 ), 푣1 ⊗ ⋯ ⊗ 푣푑 ↦ 푣1 ⋯ 푣푑 . For a diagram 퐷, let cols(퐷)∶ ℙ → ℕ and rows(퐷)∶ ℙ → ℕ be the finitely supported functions counting the number of boxes in each column or row. For a finitely supported function 훼 ∶ ℙ → ℕ, let 푆훼 (푉 ) be the tensor product of symmetric powers 푆훼 (푉 ) = 푆훼(1)(푉 ) ⊗ 푆훼(2)(푉 ) ⊗ 푆훼(3)(푉 ) ⊗ ⋯ ,

88 9.2 Generalised Schur modules noting that 푆0(푉 ) ≅ ℂ and hence this tensor product is isomorphic to a finite tensor product. We similarly define Λ훼 (푉 ), hence we get the spaces Λcols(퐷)(푉 ) and 푆rows(퐷)(푉 ). There are two distinguished bijective tableaux associated to 퐷, the column ordered tableau 퐶 corresponding to ordering columns from left-to-right, and from top-to-bottom within a column, andthe row ordered tableau 푅 corresponding to ordering rows from top-to-bottom, and left-to-right within a row. There is a unique permutation 휋퐷 such that 휋퐷 ∘ 퐶 = 푅.

퐷 퐶 푅

4 6 휋퐷 = (13624) 1 2 1 3 2 5 4 5 3 6

We may now define the Schur functor associated to 퐷. 9.2.5 Definition (Generalised Schur functor)

Let 퐷 ⊆ ℙ × ℙ be a diagram of 푑 boxes in the plane, with cols(퐷), rows(퐷), and 휋퐷 as defined above. Given a vector space 푉 , define the map 휓퐷,푉 to be the composition

cols(퐷) Δ ⊗푑 휋퐷 ⊗푑 푚 rows(퐷) 휓퐷,푉 ∶ Λ (푉 ) −→푉 −−→푉 −→푆 (푉 ). (9.2.6)

Define the Schur functor on a vector space by 풮퐷(푉 ) = im 휓퐷,푉 . For a map 푓 ∶ 푉 → 푊 , let 풮퐷(푓 )∶ 풮퐷(푉 ) → rows(퐷) rows(퐷) rows(퐷) 풮퐷(푊 ) be the map obtained by restricting the natural map 푓 ∶ 푆 (푉 ) → 푆 (푊 ) to im 휓퐷,푉 .

It is straightforward to check that the image of 풮퐷(푓 ) does indeed land in the subspaces im 휓퐷,푉 , and hence 풮퐷 defines an endofunctor in the category of complex vector spaces. Endofunctors of vector spaces naturally produce new representations from old representations: if 휌 ∶ 퐺 → Endℂ(푉 ) is a representation of a group 퐺 on the vector space 푉 , then we obtain a new representation of 퐺 on 풮퐷(푉 ) by letting the group element 푔 ∈ 퐺 act by 풮퐷(휌(푔)). Our second definition of a generalised Schur module of GL(푉 ) is the image 풮퐷(푉 ) of the defining representation under the Schur module. We note that column permutations of 퐷 do not affect the functor 풮퐷 at all, while row permutations of 퐷 give functors isomorphic to 풮퐷. We can also interpret Definition 9.2.5 in terms of a ‘wiring diagram’ defined by 퐷. The data 퐷 is equivalent to a bipartite graph on ℙ×ℙ (and the isomorphism class of 풮퐷 depends only on the isomorphism class of this bipartite graph). Drawing all of the edges in this graph in an ordered fashion, we can then group edges together and treat the whole thing as a string diagram.

rows 푆2(푉 ) 푆1(푉 ) 푆2(푉 ) 푆1(푉 ) 푆rows(퐷)(푉 ) 1 2 3 4 1 2 3 푚 1 2 ̃휋 휓퐷,푉 3 퐷 4 1 2 3 Δ columns Λ3(푉 ) Λ2(푉 ) Λ1(푉 ) Λcols(퐷)(푉 )

Above we have shown a diagram 퐷, its associated bipartite graph embedded into the plane, and a string diagram (read from bottom-to-top). The junctions in the bottom orange part of the string diagram should be understood as iterated comultiplication in the exterior algebra followed by projection onto the (1, … , 1)-graded piece, as in Eq. (9.2.3). The junctions in the top green part of the diagram are the iterated multiplications 푉 ⊗푘 ↠ 푆푘(푉 ) into ⊗푑 ⊗푑 the symmetric power, as in Eq. (9.2.4). The middle blue part of the diagram is a map ̃휋퐷 ∶ 푉 → 푉 permuting tensor factors according to the permutation 휋퐷 = (13624) defined before.

89 9 Generalised Schur modules

Now that we have given two different definitions of generalised Schur modules, we should show that theyare equivalent. In this proof we will have to make use of the characteristic-zero assumption on the base field, a reflection of how semisimplicity of both categories is partly responsible for this equivalence. 9.2.7 Lemma (Equivalence of definitions of generalised Schur modules)

Let 퐷 be a diagram of size 푑, with 퐶 its column ordered tableau, 푅 its row ordered tableau, and 휋퐷 ∈ 픖푑 the unique permutation such that 휋퐷 ∘ 퐶 = 푅. Recall that to the column-stabilising subgroup of 퐶 we associate 휎 the alternating sum over its elements 푐퐶 = ∑휎 (−1) 휎, and to the row-stabilising subgroup of 푅 we associate 푘 ⊗푑 the sum over its elements 푟푅 = ∑휎 휎. Let Γ (푉 ) ⊆ 푉 denote the subspace of tensors which are symmetric under the action of 픖푘. Consider the following diagram:

휋 Λcols(퐷)(푉 ) Δ 푉 ⊗푑 퐷 푉 ⊗푑 푚 푆rows(퐷)(푉 ) ⋅푟푅 Δ (9.2.8) ⊗푑 rows(퐷) 푉 푐퐶 Γ (푉 )

It is easy to check that the triangle on the right commutes, just by checking that the operation Δ ∘ 푚 is equal ⊗푑 to the action of 푟푅. The two maps on the left have the same image inside 푉 , and since the vertical map Δ is an isomorphism (a strictly characteristic-zero phenomenon) we have that the image im 휓퐷,푉 of the top ⊗푑 horizontal map is isomorphic to the image of the bottom composition 푉 푐퐶 휋퐷푟푅. −1 −1 As subgroups of 픖 we have 휋 RowStab 휋 = RowStab −1 = RowStab , hence 휋 푟 휋 = 푟 , showing 푑 퐷 푅 퐷 휋퐷 ∘푅 퐶 퐷 푅 퐷 퐶 that ⊗푑 ⊗푑 ⊗푑 푉 푐퐶 휋퐷푟푅 = 푉 푐퐶 푟퐶 휋퐷 ≅ 푉 푐퐶 푟퐶 , (9.2.9) where the last isomorphism is because right multiplication by a permutation is an isomorphism 푉 ⊗푑 → 푉 ⊗푑 . ⊗푑 ⊗푑 Therefore we have that the image of 휓퐷,푉 is isomorphic to 푉 푐퐶 푟퐶 = 푉 ⊗ℂ[픖푑 ] Σ퐶 where Σ퐶 is the Specht module associated to the tableau 퐶 of the diagram 퐷. We briefly remark on how this setup can be extended to the positive characteristic case. 9.2.10 Remark There is a dual notion of Weyl functor 풲퐷, defined as the image of the composition

휋−1 rows(퐷) Δ |퐷| 퐷 |퐷| 푚 cols(퐷) 휑퐷,푉 ∶ Γ (푉 ) −→푉 −−−→푉 −→Λ (푉 ), (9.2.11)

where Γ푑 (푉 ) denotes the 푑th divided power algebra (see Appendix 2.4 of [Eis95] for a definition of the divided power algebra). On a morphism 푓 ∶ 푉 → 푊 , the Weyl functor is the restriction of the natural map cols(퐷) cols(퐷) cols(퐷) 푓 ∶ Λ (푉 ) → Λ (푊 ) to the image of 휑퐷,푉 . The Schur and Weyl functors 풮퐷 and 풲퐷 make sense for modules over any commutative unital ring 푘, and behave well under base change (in fact, when 퐷 is a Young diagram or skew diagram, both functors are universally free [ABW82]).

For a partition 휆, the Schur module 풮휆(푉 ) is what we would call the induced module of GL(푉 ) corresponding to the weight 휆, isomorphic to the space of sections Γ(퐺/퐵, ℒ휆), while the Weyl module 풲휆(푉 ) is its dual. When we say dual, we mean the composition of the normal dual with the anti-involution of 퐺 switching positive and negative roots — in the case of a module 푀 over GL푛, this is the vector space Hom푘(푀, 푘) with the action (푔 ⋅ 푓 )(푣) = 푓 (푔푇 푣) where 푔푇 is the transpose matrix.

In the characteristic-zero case the functors 풮퐷 and 풲퐷 are isomorphic, and furthermore both are isomorphic

to the functor (−) ⊗픖|퐷| Σ퐷 of tensoring with a generalised Specht module, however this is no longer true in the case of positive characteristic. A modern perpective for the study of the functors 풮퐷 and 풲퐷 is notion of polynomial functors.

90 9.3 Flagged Schur modules

9.3 Flagged Schur modules

The generalised Schur functors 풮퐷 are not straightforward to study. The results of [JP79; ABW82], which give a basis of 풮퐷(푉 ) when 퐷 is a Young diagram or skew shape, depend in an essential way on a partial ordering on semistandard tableaux. Such a partial ordering simply does not exist when 퐷 is an arbitrary diagram, so a different approach is required. The approach we use here will follow that of Magyar, Reiner, Shimozonoand others, studying the Schur module 풮퐷(푉 ) by certain 퐵-stable quotients defined by the rows of the diagram 퐷.

푛 We now need to be clear about what we mean by a Borel subgroup 퐵. For each 푛 ≥ 0, let ℂ• be the full flag 푛 푛 푛−1 푖 푖−1 of quotient spaces ℂ• = (ℂ ↠ ℂ ↠ ⋯ ↠ ℂ), where the map ℂ ↠ ℂ sends the coordinate vector 푒푖 to 푛 푛 zero. We say that a group element 푔 ∈ GL(푉 ) preserves a flag 퐹• if 푔(퐹푖) ⊆ 퐹푖 for all 푖. Let 퐵(ℂ• ) ⊆ GL(ℂ ) be 푛 the subgroup preserving the coordinate flag ℂ• , which is precisely the subgroup of upper-triangular matrices. 푛 푛 푛 푛 푛 Defining 푇 (ℂ ) ⊆ GL(ℂ ) to be the subgroup of diagonal matrices, we get a pinning (푇 (ℂ ) ⊆ 퐵(ℂ• ) ⊆ GL(ℂ )) of the reductive algebraic group GL(푉 ), a realisation of the root datum GL푛 of type A푛−1. 9.3.1 Definition (Flagged Schur module) Let 퐷 be a diagram with 푑 boxes fitting within the first 푟 rows, meaning that 퐷 ⊆ [푟] × ℙ, and fix a full 푟 flag ℂ• of quotient spaces. When row 푖 of the diagram has row푖(퐷)-many boxes, we can apply the symmetric row (퐷) 푟 푖 푟 row (퐷) 푟 power functor 푆 푖 to the surjection ℂ ↠ ℂ determined by the flag ℂ• to get a surjection 푆 푖 (ℂ ) ↠ row (퐷) 푖 푆 푖 (ℂ ). Taking the tensor product of these and precomposing with the map 휓퐷,ℂ푟 defining the Schur module gives a map

휓퐷,ℂ푟 휙퐷 Λcols(퐷)(ℂ푟 ) −−−−→푆rows(퐷)(ℂ푟 ) −−→→ ⨂ 푆row푖(퐷)(ℂ푖). (9.3.2) 푖≥1 푟 푟 The image of this map is the flagged Schur module ℱ퐷(ℂ ). It is naturally a 퐵(ℂ•)-module, but is no longer in general a GL(ℂ푟 )-module. 푛 푟 Note that we have ℱ퐷(ℂ ) = ℱ퐷(ℂ ) for all 푛 ≥ 푟, and furthermore that the definition does not make sense 푟 for 푛 < 푟. Therefore we can use the notation ℱ퐷(ℂ ) = ℱ (퐷) and leave it implied we are working inside some flag of length at least 푟.

This quotient is quite straightforward to reason about in terms of a basis of the target space 푆rows(퐷)(ℂ푟 ) of the rows(퐷) 푟 Schur functor map 휓퐷,ℂ푟 . As a tensor product of symmetric powers, 푆 (ℂ ) has a basis indexed by tableaux 푇 ∶ 퐷 → [푟] which are row-semistandard, meaning that the entries along each row weakly increase. In terms rows(퐷) 푟 row푖(퐷) 푖 of this basis, the quotient 푆 (ℂ ) ↠ ⨂푖≥1 푆 (ℂ ) simply kills any tableaux having an entry in row 푖 which is larger than 푖. 9.3.3 Example

Below are shown two row-semistandard tableaux 푇1, 푇2 for a diagram 퐷, and another row-semistandard ′ ′ tableau 푇3 for a different diagram 퐷 . ′ 푇1 푇2 푇3 1 1 1 1 1 1 1 2 2 2 3 2 1 1 3 2 3 3 1 2 3 3 4 1 2 푒11 ⊗ 푒2 ⊗ 푒23 푒11 ⊗ 푒3 ⊗ 푒12 푒11 ⊗ 푒3 ⊗ 푒12

The tableau 푇1 survives the quotient 휙퐷 because every entry of row 푖 is at most 푖. The tableau 푇2 gets killed ′ in the quotient 휙퐷 because there is a 3 in row 2. The third tableau 푇3 survives the quotient 휙퐷′ , because by ′ shifting 푇2 down a row we have removed the only problem. However, 푇3 is a tableau for a different diagram to 푇2, and they do not represent elements in the same flagged Schur module. 푟 The example above shows that the isomorphism class of the flagged Schur module ℱ퐷(ℂ ) is not invariant under 푟 row permutations of the diagram 퐷, in contrast to the Schur module 풮퐷(ℂ ). However, it is clear from the defini-

91 9 Generalised Schur modules tions that a permutation of the columns of 퐷 leaves the flagged Schur module unchanged. For further properties of these modules (and their dual equivalents, the flagged Weyl modules), the reader can consult Sections 2 and 5 of [RS99]. Many of the results known about generalised Schur modules are due to geometric constructions of this module as sections of a line bundle over a (generally singular) variety, first explored in [Mag98a; Mag98b]. In this setting, the flagged Schur module (or the dual construction, the flagged Weyl module) naturally arise, andin[RS95; RS98] a Demazure-type character formula is given for the characters of the flagged Schur modules of percentage-avoiding diagrams 퐷. The only diagrams we will encounter are northwest (or can be made northwest via a column permutation) which are automatically percentage-avoiding, and hence these results apply. (A diagram 퐷 is northwest if whenever (푗, 푘), (푖, 푙) ∈ 퐷 with (푖 < 푗) and (푘 < 푙), then (푖, 푘) ∈ 퐷). From now on, we will restrict ourselves to diagrams which are column-convex, meaning that the columns have no gaps. A column-convex diagram satisfies the northwest property after a column permutation has been applied, and hence the results in the above paper will apply to our situation. In order to use the results of Reiner and Shimozono to write down a character formula for the flagged Schur module ℱ퐷 for a column-convex diagram 퐷, it will help to have a convenient way of encoding the data of a column-convex diagram 퐷. We will encode a column-convex diagram 퐷 fitting within 푟 rows as a sequence of 푟 partitions. 9.3.4 Definition (Partition sequences and diagrams) A partition sequence of length 푟 ≥ 0 is a sequence 휆 = (휆(1), … , 휆(푟)) of partitions, such that ℓ(휆(푖)) ≤ 푖 (the 푖th partition has at most 푖 rows). For each 0 ≤ 푖 ≤ 푟, let 휆푖 = (휆(1), … , 휆(푖)) denote the prefix of 휆 of length 푖. When 휆 is a partition sequence of length 푟, we define the associated diagram 퐷(휆) inductively by:

1. For 푖 = 0, 퐷(휆0) = ∅, the empty diagram, 2. For 푖 > 0, 퐷(휆푖) is obtained by shifting the contents of the previous diagram 퐷(휆푖−1) down one row, and placing the Young diagram of the partition 휆(푖) to the right of the previous diagram, with the first row of 휆(푖) in row 1. The diagram 퐷(휆푖) is always column-convex and contained within the rows {1, … , 푖}. 9.3.5 Example Consider the partition sequence 휆 = (∅, (1, 1), (2, 1), (1, 1, 1, 1), (2, 1, 1)), which we could picture as the following sequence of Young diagrams:

∅

휆(1) 휆(2) 휆(3) 휆(4) 휆(5)

The associated sequence of diagrams 퐷(휆푖) is shown below.

row 1 ∅ ∅ 2 3 4 5

퐷(휆0) 퐷(휆1) 퐷(휆2) 퐷(휆3) 퐷(휆4) 퐷(휆5)

92 9.4 Polynomial characters of GL푛

The above example makes it clear that the diagram 퐷(휆) is always column-convex. Conversely, given any column- convex diagram 퐷 its columns may be sorted in such a way that there exists a partition sequence 휆 with 퐷(휆) = 퐷. This sorting can be done by placing columns with entries appearing higher to the right, and breaking ties by putting longer columns on the left. 푛 In order to read the next lemma, we remind the reader that any integer vector 훼 = (훼1, … , 훼푛) ∈ ℤ can be treated 훼 푛 × 훼1 훼푛 as a weight of GL푛, the function 푒 ∶ 푇 (ℂ ) → ℂ taking the torus element diag(푡1, … , 푡푛) to 푡1 ⋯ 푡푛 . (This is explained further in Section 9.4). Each integer partition 휆 = (휆1, …) of length at most 푛 defines a dominant weight 휆 푒 of GL푛. 9.3.6 Lemma (Character of a flagged Schur module) Let 휆 be a partition sequence of length 푟. The characters of the flagged Schur modules ℱ (퐷(휆푖)) satisfy the following recurrence: 1. For 푖 = 0, ch ℱ (퐷(휆0)) = ch ℱ (∅) = 1, 푖 휆(푖) 푖−1 2. For 푖 > 0, ch ℱ (퐷(휆 )) = 푒 ⋅ 휋1 ⋯ 휋푖−1 (ch ℱ (퐷(휆 ))). 9.3.7 Proof The case for 푖 = 0 is clear. The inductive case follows from Theorem 23 of [RS98], noting that moving the diagram 퐷(휆푖−1) down one row can be done by applying the successive row permutations (푖 − 1, 푖), … , (1, 2), which corresponds to the application of Demazure operators 휋푖−1, … , 휋1, and adding a Young diagram 휆 in the top row corresponds to multiplication by 푒휆. 9.3.8 Example Consider Example 9.3.5 above. By the recursion rule given in Lemma 9.3.6, the character of the flagged Demazure module ℱ (퐷(휆)) is

(2,1,1) (1,1,1,1) (2,1) 0 0 ch ℱ (퐷(휆)) = 푒 휋1휋2휋3휋4(푒 휋1휋2휋3(푒 휋1휋2(푒 휋1푒 ))). (9.3.9)

9.4 Polynomial characters of GL푛

In this section we will explain how the characters of flagged Schur modules fit into the more well-known frame- work of characters of Schur modules. We begin by fixing a choice of root datum of GL푛. 9.4.1 Definition Fix an 푛 ≥ 1. The root datum Φ = GL푛 is the type (퐼, ⋅) = A푛−1 root datum with the following presentation:

푋(GL푛) = ℤ {휖1, … , 휖푛} , 훼푖 = 휖푖 − 휖푖+1 for 1 ≤ 푖 ≤ 푛 − 1, ∨ ∨ ∨ ∨ ∨ ∨ 푋 (GL푛) = ℤ {휖1 , … , 휖푛 } , 훼푖 = 휖푖 − 휖푖+1 for 1 ≤ 푖 ≤ 푛 − 1, (9.4.2) ∨ ⟨휖푖, 휖푗 ⟩ = 훿푖푗.

A weight 휆 = 휆1휖1 + ⋯ + 휆푛휖푛 is dominant iff 휆1 ≥ ⋯ ≥ 휆푛, and polynomial if 휆푖 ≥ 0 for all 1 ≤ 푖 ≤ 푛. There is an alternative basis of weights which we call fundamental weights by abuse of notation (as GL푛 is not semisimple, it does not really have fundamental weights). These fundamental weights are 휛푖 = 휖1 + ⋯ + 휖푖 ∨ for 1 ≤ 푖 ≤ 푛, and we have ⟨휛푖, 훼푗 ⟩ = 훿푖푗 for 1 ≤ 푖, 푗 ≤ 푛 − 1, with 휛푛 = 휖1 + ⋯ + 휖푛 generating the null space ∨ 푋(GL푛)0 = ⋂푖∈퐼 ker⟨−, 훼푖 ⟩. A weight is dominant polynomial if and only if it expands positively in the basis of the 휛푖. 푛 The polynomial weights of GL푛 are indexed by compositions of 푛, meaning finite sequences 휆 = (휆1, … , 휆푛) ∈ ℕ . A polynomial weight is dominant if and only if it is a partition. The category of representations with polynomial weights is closed under taking direct sums and tensor products, and furthermore to reach any non-polynomial representation one only needs to tensor with an appropriate negative power of the one-dimensional determinant representation 퐿(휛1). From here on, we restrict our attention only to polynomial weights, which we will denote 푝 by 푋(GL푛) .

93 9 Generalised Schur modules

푝 There is an isomorphism of algebras ℤ[푋(GL푛) ] ≅ ℤ[푥1, … , 푥푛] between the monoid algebra of the polynomial 휆 휆 휆1 휆푛 weights and the polynomial ring in 푛 variables, taking 푒 to 푥 ∶= 푥1 ⋯ 푥푛 . Furthermore, the Weyl group action on 푋(GL푛) preserves the polynomial weights, and hence descends to the usual action of the symmetric group 픖푛 on the polynomial ring ℤ[푥1, … , 푥푛] by permuting coordinates. Since we have realised the polynomial ring and 픖 the ring of symmetric polynomials ℤ[푥1, … , 푥푛] 푛 as the character ring and Weyl-invariant character ring of a root datum, we will get interesting new bases of these rings by considering characters of Demazure modules and highest-weight modules respectively. 푛 When 휆 is a partition (a dominant polynomial weight), the Schur module 풮휆(ℂ ) gives an explicit construction 픖 of the highest-weight module 퐿(휆), and when the resulting characters are viewed inside ℤ[푥1, … , 푥푛] 푛 they are called Schur polynomials and usually denoted ch 퐿(휆) = 푠휆(푥1, … , 푥푛). The Schur polynomials form a basis for 픖 the ring of invariants ℤ[푥1, … , 푥푛] 푛 , by a slight modification of the argument in Lemma 5.1.2 for polynomial weights. The Schur polynomials are extremely well-studied, and we could point the reader to several references [Ful96; Mac95; Sta97]. When 훼 is a composition (an arbitrary polynomial weight), the character of the Demazure module 퐷(훼) with Demazure lowest weight 훼 is called a key polynomial, and usually denoted ch 퐷(훼) = 휅훼 . Computing these key polynomials is quite straightforward. If 훼 is a partition then 퐷(훼) = 퐿푒(훼) = 퐿(훼)훼 is the one-dimensional weight 훼 휆 space of weight 훼 and hence 휅훼 = 푥 . In terms of Demazure operators, we have that 휅훼 = 휋푤 (푥 ), where 휆 is dominant and 푤 ∈ 푊 is the shortest permutation such that 푤휆 = 훼. This together with the zero-Hecke property (Remark 5.2.10) gives that

휅푠푖훼 if 훼푖 > 훼푖+1, 휋푖(휅훼 ) = { (9.4.3) 휅훼 otherwise.

훼 푥푖 Finally, we can rewrite the Demazure operator 휋푖 ∶ ℤ[푥1, … , 푥푛] → ℤ[푥1, … , 푥푛] using the fact that 푒 푖 = : 푥푖+1

휆 푠푖휆 휆 푥푖푥 − 푥푖+1푥 휋푖(푥 ) = . (9.4.4) 푥푖 − 푥푖+1 Some example of key polynomial computations are below:

(1,1,0) 휅(1,1,0) = 푥 (1,1,0) (1,0,1) 휅(1,0,1) = 휋2(휅(1,1,0)) = 푥1 ⋅ (푥2 + 푥3) = 푥 + 푥 (9.4.5) (1,1,0) (1,0,1) (0,1,1) 휅(0,1,1) = 휋1(휅(1,0,1)) = 푥 + 푥 + 푥 .

The last character 휅(0,1,1) is symmetric, and equal to the Schur polynomial 푠(1,1,0). This is a general fact: the Schur polynomial 푠(휆1,…,휆푛) in 푛 variables is equal to the key polynomial 휅(휆푛,…,휆1). It was shown in [RS95] that, similarly to how the highest-weight module 퐿(휆) can be constructed as a Schur 푛 module 풮휆(ℂ ) using the Young diagram associated to 휆, the flagged Schur module ℱ (훼) has character 휅훼 , where 훼 is treated as a left-justified diagram having 훼푖 boxes in row 푖. The next image shows three diagrams 퐷, with the character of the flagged Schur module ℱ (퐷) written below.

1 1 1 2 2 2 3 3 3 4 4 4

(5,3,2,0) 휅(5,3,2,0) = 푥 휅(2,0,5,3) 휅(0,2,3,5) = 푠(5,3,2,0)

We again see how Demazure modules interpolate between a single highest-weight space and the full space of a representation, and here they have a very concrete interpretation in terms of boxes in the plane. To the author’s knowledge, it is still unknown whether ℱ (훼) is isomorphic to the Demazure module 퐷(훼), despite them both having the same character 휅훼 (the character of a 퐵-module does not determine its isomorphism class). There are

94 9.5 Type A truncations combinatorial interpretations of these diagrams as well, for example a Demazure crystal for 퐷(훼) can be built using key tableau, which are certain fillings of the diagrams above by the letters 1, … 푛. For a good survey, one can read [AG19].

9.5 Type A truncations

Throughout this section, we work in type (퐼, ⋅) = A푛 for 푛 large enough. When considering a partition sequence of length 푟, we will need to be working in A푛 for 푛 ≥ 푟. Since we will only be working with product monomial crystals whose data is supported over the first 푟 nodes of the diagram, the particular choice of 푛 does not matter for any of the statements we make, provided 푛 ≥ 푟. 9.5.1 Definition (Multisets and truncations associated to partition sequences) Let 휆 be a partition sequence of length 푟. Define the associated multisets R(휆푖) and the associated upward- closed set 퐽푖 inductively as follows: 0 1. For 푖 = 0, the multiset R(휆 ) = ∅ is empty, and 퐽0 is the complement of the downward-closed set generated by (1, −1). 2. For 푖 > 0, let 퐽푖 be the union of 퐽푖−1 with the upward-closed set generated by (1, 3 − 2푖), and let the difference R(휆푖) − R(휆푖−1) be supported on the elements along the north-east diagonal beginning at (1, 3 − 2푖) and have weight 휆(푖).

푟 When 휆 has length 푟, we set R(휆) = R(휆 ) and 퐽(휆) = 퐽푟 . The definition above is more easily interpreted as a picture in terms of monomial diagrams. Rather thansaying ‘A푛 for 푛 large enough’, we will draw the half-infinite path A+∞ and work in this setting until we need to compute the character of a whole crystal ℳ(휆, R). While we are working with the truncations ℳ(휆, R, 퐽) it will not matter which A푛 we are working in, so long as all monomial in question are supported over A푛 ×̇ ℤ. 9.5.2 Example Taking the same partition sequence 휆 = (∅, (1, 1), (2, 1), (1, 1, 1, 1), (2, 1, 1)) as in Example 9.3.5, a schematic 푖 picture of what the data R(휆 ) and 퐽푖 looks like for every 0 ≤ 푖 ≤ 5 is shown below.

(1, 1) 퐽0 = 퐽1

퐽2 ⧵ 퐽1

퐽3 ⧵ 퐽2

퐽4 ⧵ 퐽3

퐽5 ⧵ 퐽4

A+∞

We start with the upward-closed set 퐽0 (note that 퐽0 is not finitely-generated, since A+∞ is infinite). The set 퐽1 adds a single point (1, 1), the set 퐽2 adds two more points, then 퐽3 three more, and so on. We end up with

95 9 Generalised Schur modules

퐽5, which is the entire green shaded region.

In terms of the ‘fundamental weights’ of A+∞, our partition sequence 휆 is the sequence of weights (0, 휛2, 휛1 + 푖 휛2, 휛4, 휛1 + 휛3). The construction of the R(휆 ) multiset from the previous one is simply to include elements (푖) (3) from 퐽푖 ⧵ 퐽푖−1 with multiplicities according to the weights given by 휆 . For example, here the weight of 휆 is 휛1 + 휛2, and therefore we include into R the two points (1, −3) and (2, −2) of 퐽3 ⧵ 퐽2 which will give that weight.

푖 Our construction of the R(휆 ) and 퐽푖 is designed to be compatible with our inductive character formula Theo- rem 7.2.3. From the picture above it is clear that we can go from the set 퐽푖−1 to 퐽푖 by adding (푖 − 1) points to the truncating set, and each one added can be taken to be minimal by adding the rightmost one first. After this is done, 푖 we place the new elements of R(휆 ) over the new points 퐽푖, noting that (퐽푖 ⧵ 퐽푖−1) ⊆ 휕퐽푖 and so every new element 푖 푖 푖 of R(휆 ) is added along the boundary. We get a recurrence of characters for the truncations ℳ(wt 휆 , R(휆 ), 퐽푖) which matches the recurrence for the flagged Schur modules Lemma 9.3.6. 9.5.3 Lemma 푖 푖 Let 휆 be a partition sequence of length 푟. The characters of the truncations ℳ(wt 휆 , R(휆 ), 퐽푖) satisfy the recurrence 1. For 푖 = 0, ch ℳ(0, ∅, 퐽1) = 1, 2. For 푖 > 0, 푖 푖 휆(푖) 푖−1 푖−1 ch ℳ(wt 휆 , R(휆 ), 퐽푖) = 푒 ⋅ ch ℳ(wt 휆 , R(휆 ), 퐽푖) (9.5.4) 휆(푖) 푖−1 푖−1 = 푒 ⋅ 휋1 ⋯ 휋푖−1 ch ℳ(wt 휆 , R(휆 ), 퐽푖−1). Therefore, the character of the flagged Schur module ℱ (퐷(휆)) matches the character of the truncated crystal ℳ(wt 휆, R(휆), 퐽(휆)). We can apply two different stabilisation results to see that the characters of the corresponding product monomial crystal and Schur module match. 9.5.5 Theorem Let 휆 be a partition sequence of length 푟, and work in type 퐼 = A푛 for 푛 ≥ 푟. Then the characters of the Schur 푟 module 풮퐷(휆)(ℂ ) and the product monomial crystal ℳ(wt 휆, R(휆)) are equal. 9.5.6 Proof By Corollary 7.3.9, the character of the whole product monomial crystal can be obtained from any of its

truncations by applying the Demazure operator 휋푤퐼 corresponding to the longest element of the Weyl group 푊퐼 ≅ 픖푛+1. Hence 푟 푟 ch ℳ(wt 휆, R(휆)) = 휋푤퐼 ch ℳ(wt 휆 , R(휆 , 퐽푟 )). (9.5.7) 푛 By Theorem 21 of [RS98], the character of the Schur module 풮퐷(ℂ ) can be obtained from the character of 푛 the flagged Schur module ℱ퐷(ℂ ) by applying the Demazure operator 휋푤퐼 , hence 푛 푛 ch 풮퐷(ℂ ) = 휋푤퐼 ch ℱ퐷(ℂ ). (9.5.8) Since both the characters appearing on the right-hand sides of the above expressions are equal, the result follows. It is worth pointing out that we really did need to use both results Corollary 7.3.9 and (Theorem 21, [RS98]) in the statement above. Despite the fact that we know that the character of the flagged Schur module matches the character of the truncated product monomial crystal, and that both are positive sums of Demazure characters even, we cannot apply the stabilising operator 휋푤 and expect to get the character of the containing representation 퐼 푛 (resp. crystal) unless we know for sure that the representation ℱ퐷(ℂ ) has a filtration by Demazure modules (resp. the crystal is a sub-Demazure crystal). In this case due to our result that the truncations of the product monomial crystals are Demazure (Theorem 7.3.7) we know that each truncation is a sub-Demazure crystal. Interestingly it is still unknown whether the flagged Schur modules admit a filtration by Demazure modules (Reiner, personal communication). Theorem 21 of [RS98] is proven in an entirely different manner, by a construction due to Magyar 푛 [Mag98b] realising 풮퐷(ℂ ) as the space of sections of a line bundle over a ‘configuration variety’ depending on

96 9.5 Type A truncations the diagram 퐷. The result Theorem 9.5.5 is good, but is not as strong as we want since it only holds for 푛 large enough compared to the height of the diagram 퐷. If the partition sequence 휆 is of length 푟, but each partition has length at most 5 (for example), then it is easy to see from Example 9.5.2 that the corresponding multiset R(휆) is defined in type A5. However our above result Theorem 9.5.5 is not applicable in type A5 if 푟 > 5. In the following chapter, we will develop some stability properties of the product monomial crystals defined when R is held fixed and the type A푛 varies, so that we can strengthen the above result to all 푛.

10 Stability of decomposition

휆 In Chapter 9 we discussed the generalised Littlewood-Richardson coefficients 푐퐷 associated to a diagram 퐷, and remarked on their stability properties in terms of Schur functors (Corollary 9.2.2). Namely, for any vector space 휆 whose dimension is large enough, every coefficient 푐퐷 can be read off from the decomposition of the generalised Schur module 풮퐷(푉 ), and furthermore these stable coefficients can be used to deduce the decomposition of 풮퐷(푈 ) for any dim 푈 ≤ dim 푉 .

In this chapter we will see that the product monomial crystal ℳ(휆, R) ⊆ ℳ(GL푛) has similar behaviour: there ex- 휆 ist stable coefficients 푐R giving the decomposition of ℳ(휆, R) into irreducible crystals whenever 푛 is large enough, and furthermore these stable coefficients can be used to deduce the decomposition of ℳ(휆, R) ⊆ ℳ(GL푚) whenever 푚 ≤ 푛. Once we have this result, we can apply Theorem 9.5.5 to show that whenever 퐷 and R correspond, 휆 휆 we have 푐퐷 = 푐R, and hence the product monomial crystal ℳ(휆, R) ⊆ ℳ(GL푛) is the crystal of the generalised 푛 Schur module 풮퐷(ℂ ) for all 푛.

10.1 Stability of restriction

Let Φ be a root datum of Cartan type (퐼, ⋅). Recall from Definitions 3.1.1 and 3.4.1 that each subset 퐽 ⊆ 퐼 defines a Cartan datum (퐽, ⋅) by restriction, and a restricted root datum Φ퐽 of type (퐽, ⋅) by keeping the weight and coweight lattices the same, but only remembering the simple roots and coroots indexed by 퐽. For each Φ-crystal 퐵, we have a Φ퐽 -crystal 퐵퐽 by restriction: keeping the set 퐵 the same, but remembering only the crystal operators indexed by 퐽. As an example, we show the GL4-crystal ℬ(2, 0, 0, 0), and the restrictions ℬ(2, 0, 0, 0){1,2} and ℬ(2, 0, 0, 0){1,3}.

ℬ(2, 0, 0, 0) ℬ(2, 0, 0, 0){1,2} ℬ(2, 0, 0, 0){1,3}

A3 1 2 3

As the example shows, we expect a restricted crystal 퐵퐽 to often have many connected components, for example with 퐽 = {1, 2} in the example we see that ℬ(2, 0, 0, 0)퐽 is a disjoint union of three connected crystals. When

99 10 Stability of decomposition

퐽 is a single element 퐽 = {푖}, rather than writing Φ{푖} and 퐵{푖}, we will simply write Φ푖 and 퐵푖. The connected components of the restricted crystal 퐵푖 are exactly the 푖-strings inside 퐵. Suppose that (휆, R) is a dominant pair defining a product monomial crystal ℳ(휆, R) ⊆ ℳ(Φ). Whenever 퐽 ⊆ 퐼 is a subset such that the multiset R is supported over 퐽 ×̇ ℤ, this data also defines a product monomial crystal ℳ(휆, R) ⊆ ℳ(Φ퐽 ), a Φ퐽 -crystal rather than a Φ-crystal. In this case we will say that the multiset R lives over 퐽 ⊆ 퐼. We write ℳ(Φ, 휆, R) for the Φ-crystal, and ℳ(Φ퐽 , 휆, R) for the Φ퐽 -crystal to keep notation as clear as possible. We get a different Φ퐽 -crystal by the restriction ℳ(Φ, 휆, R)퐽 , and the arguments following will have to do with the interplay between the Φ-crystal ℳ(Φ, 휆, R), its restricted Φ퐽 -subcrystal ℳ(Φ, 휆, R)퐽 , and the Φ퐽 - crystal ℳ(Φ퐽 , 휆, R). 10.1.1 Example

Let Φ = GL3, a root datum of Cartan type (퐼, ⋅) = A2, with vertices 퐼 = {1, 2}. The data 휆 = 2휖1, R = {(1, 3), (1, 1)} defines a Φ-product monomial crystal ℳ(Φ, 휆, R), and also a Φ{1}-product monomial crystal ℳ(Φ{1}, 휆, R). These two crystals are pictured below using monomial diagrams, where the point (1, 3) is 푘 circled in green, and we write a 푦 next to the point (푖, 푐) to indicate that the multiplicity of 푦푖,푐 in that monomial is 푘. ℳ(Φ, 휆, R) ℳ(Φ{1}, 휆, R) 푦 푦 푦 푦 푦

푦 푦 푦 푦 푦 푦−1 푦−1 푦−1 푦−1

푦 푦 푦−1 푦−1 푦 푦−1 푦−1 푦−1 푦−1 푦−1 푦−1 푦−1

Both components of the Φ{1}-crystal ℳ(Φ{1}, 휆, R) appear inside the restricted Φ{1}-crystal ℳ(Φ, 휆, R){1}. This is a general fact, which we will make precise in what follows.

Let 풵(Φ) ⊆ ℳ(Φ) be the subgroup generated by the 푧푖,푘 for all (푖, 푘) ∈ 퐼 ×ℤ̇ , so that the product monomial crystal 휆 ℳ(Φ, 휆, R) is contained inside the coset 푒 ⋅ 푦R ⋅ 풵(Φ). It is clear from the definition of the monomial crystal (Definition 6.1.1) that each coset of 풵(Φ) is a subcrystal of ℳ(Φ). For any subset 퐽 ⊆ 퐼, let 훾퐽 ∶ 풵(Φ퐽 ) ↪ 풵(Φ) be the linear inclusion map taking 푧푖,푘 ∈ 풵(Φ퐽 ) to 푧푖,푘 ∈ 풵(Φ) (note that the definition of 푧푖,푘 depends on the Cartan matrix, so this map is less obvious than it seems). If (휆, R) lives over 퐽 then the product monomial crystal ℳ(Φ퐽 , 휆, R) makes sense. We define an inclusion of cosets 휆 휆 휆 휆 휓휆,R,퐽 ∶ 푒 푦R풵(Φ퐽 ) ↪ 푒 푦R풵(Φ), 푒 푦R푧 ↦ 푒 푦R훾퐽 (푧), (10.1.2) which is in fact a map of Φ퐽 -crystals, where the crystal on the right is equipped with the Φ퐽 -crystal structure through restriction. The image of 휓휆,R,퐽 consists of all monomials whose 푧푖,푘-exponent is zero for 푖 ∉ 퐽. Note that the map 휓휆,R,퐽 is an affine, rather than linear, map of abelian groups.

100 10.1 Stability of restriction

10.1.3 Lemma The inclusion 휓휆,R,퐽 maps the product monomial crystal ℳ(Φ퐽 , 휆, R) into the set ℳ(Φ, 휆, R). Furthermore, the image of this map consists of all monomials 푝 ∈ ℳ(Φ, 휆, R) such that SuppR(푝) ⊆ 퐽 ×̇ ℤ. 10.1.4 Proof First consider the case where (휆, R) is a fundamental pair concentrated at some (푖, 푐). The restricted crystal ℳ(Φ, 휆, R)퐽 decomposes into a disjoint union of Φ퐽 -crystals. Let us write ℳ(Φ, 휆, R)퐽 = 퐵 ⊔ 퐶 where 퐵 휆 is the connected component containing the highest-weight element 푒 ⋅ 푦R and 퐶 are the other connected 휆 components. Since the inclusion 휓휆,R,퐽 maps the 퐽-highest-weight element 푒 ⋅ 푦R ∈ ℳ(Φ퐽 , 휆, R) to the 휆 퐼-highest-weight element 푒 ⋅ 푦R of 퐵, the Φ퐽 -crystal map 휓휆,R,퐽 gives an isomorphism ℳ(Φ퐽 , 휆, R) → 퐵.

It remains to be seen that SuppR(푝) ⊈ 퐽 ×̇ ℤ for all 푝 ∈ 퐶. As ℳ(Φ, 휆, R) is connected as a Φ-crystal, every 푟 monomial 푝 ∈ ℳ(Φ, 휆, R) may be written 푝 = 푓 푟1 ⋯ 푓 푙 (푒휆 ⋅푦 ) for some 푙 ≥ 0, 푟 ≥ 1 and 푖 ∈ 퐼. The monomial 푖1 푖푙 R 푖 푘 푝 belongs to the top component 퐵 if and only if {푖1, … , 푖푙} ⊆ 퐽 because 퐵 is a highest-weight Φ퐽 -crystal with 휆 highest-weight element 푒 ⋅ 푦R. Hence every monomial 푝 ∈ 퐶 has SuppR(푝) ⊈ 퐽 ×̇ ℤ, completing the proof of the claim in the case where (휆, R) is a fundamental pair.

For the general case where (휆, R) is a dominant pair, we use a factorisation into fundamental pairs (휆1, R1), 휆 …, (휆푟 , R푟 ), and note that for each factorised monomial 푝 = 푒0 푝1 ⋯ 푝푟 ∈ ℳ(Φ, 휆, R) we have

휓퐽,휆,R(푝1 ⋯ 푝푟 ) = 휓퐽,휆1,R1 (푝1) ⋯ 휓퐽,휆푟 ,R푟 (푝푟 ). (10.1.5) The result then follows from the case of a fundamental pair.

In order to really understand what the affine map 휓휆,R,퐽 is doing in Lemma 10.1.3, we return to our previous 휆 −1 example, this time written out in terms of the 푒 ⋅ 푦R ⋅ 푧S labelling. 10.1.6 Example

The crystals ℳ(Φ, 휆, R) and ℳ(Φ{1}, 휆, R) of Example 10.1.1 are shown as monomial diagrams below, this −1 time always using the factorisation 푦R ⋅ 푧S . ℳ(Φ, 휆, R) ℳ(Φ{1}, 휆, R) 푦 푦 푦 푦 푦 푦푧−1 푦 푦푧−1

푦 푦 푦 푦 푦 푦 푦푧−1 푦 푧−1 푧−1 푧−1 푧−1 푧−1

푦 푦 푦 푦 푦 푦푧−1 푦푧−1 푦푧−1 푦푧−1 푦푧−1 푧−1 푧−1 푧−1 푧−1 푧−1 푧−1 푧−1 푧−1 푧−1

The claims of Lemma 10.1.3 should be clear in the picture above.

101 10 Stability of decomposition

A highest-weight element 푝 ∈ ℳ(Φ퐽 , 휆, R) of the smaller Φ퐽 -crystal gets mapped to a highest-weight element 휓휆,R,퐽 (푝) of the larger Φ-crystal: we can see that 푒푗(휓휆,R,퐽 (푝)) = ⊥ for all 푗 ∈ 퐽 because 휓 is a morphism of Φ퐽 - crystals, and if 푖 ∈ 퐼 ⧵ 퐽 then all exponents 휓휆,R,퐽 (푝)[푖, 푘] are zero or negative by the definition of the auxiliary monomial 푧푖,푘. Therefore we get the following: 10.1.7 Lemma The map 휓휆,R,퐽 restricted to highest-weight elements is an injective map of sets

h.w. h.w. Ψ휆,R,퐽 ∶ ℳ(Φ퐽 , 휆, R) ↪ ℳ(Φ, 휆, R) . (10.1.8)

Furthermore, the image of this map consists of all highest-weight monomials 푝 ∈ ℳ(Φ, 휆, R)h.w. such that SuppR(푝) ⊆ 퐽 ×̇ ℤ. 10.1.9 Proof The map is injective since 휓휆,R,퐽 is, and its image consists of highest-weight elements as reasoned in the previous discussion. By Lemma 10.1.3, all we need to prove about the claim describing the image is one h.w. inclusion: that a highest-weight monomial 푝 ∈ ℳ(Φ, 휆, R) satisfying SuppR(푝) ⊆ 퐽 ×̇ ℤ is in the image of Ψ휆,R,퐽 . By Lemma 10.1.3 it has a preimage 푞 ∈ ℳ(Φ, 휆, R), so we just need to show that 푞 is highest-weight, but this is clear because 휓휆,R,퐽 is a morphism of Φ퐽 -crystals.

The inclusions Ψ휆,R,퐽 can be viewed as a directed system consisting of those subsets 퐽 ⊆ 퐼 such that R lives over 퐽, in the sense that the inclusions ℳ(Φ퐽 , 휆, R) ↪ ℳ(Φ퐾 , 휆, R) ↪ ℳ(Φ, 휆, R) are compatible for 퐽 ⊆ 퐾 ⊆ 퐼. One can ask: is there a smaller subset 퐽 ⊆ 퐼 such that the product monomial crystal ℳ(Φ퐽 , 휆, R) captures all necessary information to determine the isomorphism class of ℳ(Φ, 휆, R)? 10.1.10 Lemma Let (휆, R) be a dominant pair for Φ. Consider the set 푋 ⊆ 퐼 ×̇ ℤ defined by

푋 = up(R) ∩ (down({(푖, 푐 − 2)} ∣ (푖, 푐) ∈ R) ∪ Supp R) . (10.1.11)

h.w. h.w. Then if 푋 is contained in 퐽 ×̇ ℤ for some 퐽 ⊆ 퐼, the map Ψ휆,R,퐽 ∶ ℳ(Φ퐽 , 휆, R) ↪ ℳ(Φ, 휆, R) is a bijection. 10.1.12 Proof h.w. By Lemma 6.5.4 and Corollary 6.6.5, every highest-weight element 푞 ∈ ℳ(Φ, 휆, R) satisfies SuppR(푞) ⊆ 푋 ⊆ 퐽 ×̇ ℤ, and hence by the description of the image in Lemma 10.1.7 the claim follows.

10.2 Application: decomposing a product

The results of the previous section can be applied to decompose the product monomial crystal into a product of crystals, in some cases. Suppose we are working with the root datum Φ = SL9 of Cartan type (퐼, ⋅) = A8, and we have 휆 = 2휛1 + 2휛8 with the multiset R = {(1, 1), (1, 3), (8, 2), (8, 4)} as pictured here:

In this special case, the set 푋 appearing in Lemma 10.1.10 is precisely 푋 = Supp R, which lives over 퐽 = {1, 8}. Therefore to determine the highest-weight elements of the crystal ℳ(Φ, 휆, R) it suffices to determine the highest- weight elements of the (much smaller) crystal ℳ(Φ퐽 , 휆, R). Since the Cartan datum (퐽, ⋅) is disconnected, we have

102 10.3 Stability of the product monomial crystal for GL푛 an isomorphism

∼ ℳ(Φ1, 2휛1, R1) = {(1, 1), (1, 3)}) ⊗ ℳ(Φ8, 2휛8, R8) = {(8, 2), (8, 4)}) −→ℳ(Φ퐽 , 휆, R), (10.2.1) with the isomorphism being multiplication of monomials. We can determine these crystals using our previous results about characters:

ℳ(Φ1, 2휛1, R1) ≅ ℬ(Φ1, 2휛1) ⊕ ℬ(Φ1, 휛2), ℳ(Φ2, 2휛8, R8) ≅ ℬ(Φ8, 2휛8) ⊕ ℬ(Φ8, 휛7). (10.2.2)

Notice that both ℬ(Φ1, 휛2) and ℬ(Φ8, 휛7) are single-element crystals which we are tempted to say are trivial, but with our definition the restricted root data Φ1 and Φ8 retain the original weight lattice, so these single-element crystals have weight 휛2 and 휛7. This makes everything ‘fit back together’ in the nicest possible way. We canthen take the monomial-wise product of these to get the isomorphism of Φ퐽 -crystals

ℳ(Φ퐽 , 휆, R) ≅ ℬ(2휛1 + 2휛8) ⊕ ℬ(2휛1 + 휛7) ⊕ ℬ(휛2 + 2휛8) ⊕ ℬ(휛2 + 휛7). (10.2.3) By Lemma 10.1.10, these are precisely the highest-weights of the large crystal ℳ(Φ, 휆, R), and hence we have the isomorphism ℳ(Φ, 휆, R) ≅ ℬ(2휛1 + 2휛8) ⊕ ℬ(2휛1 + 휛7) ⊕ ℬ(휛2 + 2휛8) ⊕ ℬ(휛2 + 휛7), (10.2.4) this time of Φ-crystals.

10.3 Stability of the product monomial crystal for GL푛

Recall the discussion of polynomial weights and Schur functions from Section 9.4. In that discussion we worked with the polynomial weights of GL푛 for a particular 푛, here we want to switch to considering some ‘limit’ where we can work with characters for any 푛. A neat way to package this limit up is to consider a particular root datum for the ‘infinite Cartan type’ A+∞.

The infinite Cartan datum A+∞ has index set 퐼 = ℙ, with the bilinear form given by 푖⋅푖 = 2, and 푖⋅푗 = −1 whenever |푖 − 푗| = 1. The Dynkin diagram is a half-infinite path:

A+∞

Let Φ be the ‘root datum’ whose weight lattice 푋(Φ) is the free ℤ-module with basis 휖1, 휖2,…, coweight lattice is ∨ ∨ ∨ the free ℤ-module with basis 휖1 , 휖2 ,…, the pairing ⟨휖푖, 휖푗 ⟩ = 훿푖푗, and simple roots 훼푖 = 휖푖 − 휖푖+1 and simple coroots ∨ ∨ ∨ 훼푖 = 휖푖 − 휖푖+1. This is not really a root datum, since the weight and coweight lattices have infinite rank, and the pairing is not perfect. However, there exists a unique basis 휛푖 = 휖1 + ⋯ + 휖푖 of 푋(Φ) dual to the simple coroots.

A weight 휆 = ∑푖≥0 휆푖휖푖 ∈ 푋(Φ) is polynomial if 휆푖 ≥ 0 for all 푖. Any dominant weight is automatically polynomial, since a weight is dominant if and only if 휆1 ≥ 휆2 ≥ ⋯, and since 휆 ∈ 푋(Φ) is finitely supported this sequence must end in zeros. Therefore we get a bijection 푋(Φ)+ ≅ Part between dominant weights and partitions, where the weight 휆 = ∑푖≥0 휆푖휖푖 is identified with the partition (휆1, 휆2, …).

Now, suppose that R is a finite multiset based in A+∞×ℤ̇ . It determines a unique dominant weight 휆 = ∑(푖,푐)∈R 휛푖 ∈ 푋(Φ). Whenever 푛 ≥ 1 is such that R lives over [푛 − 1] ×̇ ℤ, the data of R and 휆 determine a product monomial crystal ℳ(Φ[푛−1], 휆, R), a Φ[푛−1]-crystal. All of the weights of ℳ(Φ[푛−1], 휆, R) belong to the submodule ∑푖∈[푛] ℤ휖푖, and hence we may consider ℳ(Φ[푛−1], 휆, R) as a GL푛-crystal. 10.3.1 Lemma Let R be a finite multiset based in A+∞ ×̇ ℤ, with associated dominant weight 휆 = ∑(푖,푐)∈R 휛푖 ∈ 푋(Φ). Then 휇 there exists some 푛 ≥ 1 and coefficients 푐휆,R such that

⊕푐휇 ℳ(Φ[푛−1], 휆, R) ≅ ⨁ ℬ(GL푛, 휇) 휆,R as GL푛 -crystals, (10.3.2) 휇

103 10 Stability of decomposition

and whenever R lives over [푚 − 1] ×̇ ℤ for some 푚 ≥ 1 we have

⊕푐휇 ℳ(Φ[푚−1], 휆, R) ≅ ⨁ ℬ(GL푚, 휇) 휆,R as GL푚 -crystals. (10.3.3) ℓ(휇)≤푚

Lemma 10.3.1 should be compared with Corollary 9.2.2, as they are similar in two ways: firstly they assert the existence of certain stable coefficients, and secondly they give a restriction rule, and the restriction rules match. 10.3.4 Proof Since R is finite, the set 푋 ⊆ A+∞ ×̇ ℤ appearing in Lemma 10.1.10 is also finite, so there exists some 휇 푛 ≥ 1 such that 푋 ⊆ [푛 − 1] ×̇ ℤ which we use to define the coefficients 푐휆,R according to Eq. (10.3.2). Lemma 10.1.10 gives that Eq. (10.3.3) holds for all 푚 ≥ 푛. Now suppose that 푚 < 푛, and consider the map h.w. h.w. Ψ∶ ℳ(Φ[푚−1], 휆, R) ↪ ℳ(Φ[푛−1], 휆, R) appearing in Lemma 10.1.7 which has image consisting of those highest-weight monomials whose R-support is contained in [푚 − 1] ×̇ ℤ. A highest-weight monomial 푝 of ℳ(Φ[푛−1], 휆, R) has R-support contained in [푚 − 1] ×̇ ℤ if and only if wt(푝) is a partition with at most 푚 parts, giving that Eq. (10.3.3) holds when 푚 < 푛.

We remark that the statement of Lemma 10.3.1 would be a lot more ugly if we were to use SL푛 rather than GL푛, since the property of a monomial having R-support contained in [푚 − 1] ×̇ ℤ cannot be checked purely from its weight. In order to compute decompositions in the SL푛 case, one should do computations for GL푛, and restrict to SL푛 as a last step.

We now get our main result: for GL푚, the product monomial crystal is the crystal of a generalised Schur module, which follows directly from comparing Lemma 10.3.1 with Corollary 9.2.2, and checking that they both agree for some large 푛 as a result of Theorem 9.5.5. 10.3.5 Corollary Let 휆 be a partition sequence of length 푟, defining both a diagram 퐷(휆) by Definition 9.3.4 and a multiset R(휆) by Definition 9.5.1. Let 푚 ≥ 1 be an integer such that R(휆) lives over [푚 − 1] ×̇ ℤ. Then the product 푚 monomial crystal ℳ(GL푚, wt 휆, R(휆)) is the crystal of the generalised Schur module 풮퐷(휆)(ℂ ) associated to the column-convex diagram 퐷(휆). 10.3.6 Remark Corollary 10.3.5 applies to all product monomial crystals of some GL푚, since after applying a vertical shift R ↦ R′ of the form R′[푖, 푐] = R[푖, 푐 + 2푘] for some 푘 ∈ ℤ, R′ can be brought to the form where R′ = R(휆) for some partition sequence 휆. Similarly, Corollary 10.3.5 applies to all column-convex diagrams, which are all of the form 퐷(휆) after applying some column permutation. To finish this section, we give a worked example of using Lemma 10.3.1 to find stable coefficients and restrict them to a smaller GL푚. 10.3.7 Example

Let R = {(1, 5), (3, 1), (4, 6)}, with associated dominant weight 휆 = 휛1 + 휛3 + 휛4, which define a product monomial crystal ℳ(GL5, 휆, R) in type A4. The figure below shows the set Supp R as the circled points, and the set down((푖, 푐 − 2) ∣ (푖, 푐) ∈ R) ∩ up(R) in green: their union is the set 푋 appearing in Lemma 10.1.10.

A4 A+∞

104 10.3 Stability of the product monomial crystal for GL푛

The figure shows that R is not stable for GL5, but it is stable for GL6 and upwards, so we can determine the 휇 stable coefficients 푐휆,R by computing the decomposition of the product monomial crystal ℳ(GL푛, 휆, R) for any 푛 ≥ 6. Using a computer, we determine the decomposition of ℳ(GL6, 휆, R) to be

ℳ(Φ6, 휆, R) ≅ ℬ(2휛4)⊕ℬ(휛3+휛5)⊕ℬ(휛2+휛6)⊕ℬ(휛1+휛3+휛4)⊕ℬ(휛1+휛2+휛5)⊕ℬ(2휛1+휛6). (10.3.8)

We can then apply the restriction rule in Lemma 10.3.1 to deduce the decomposition of ℳ(GL5, 휆, R): we must discard all partitions with length greater than 5. As GL5-crystals, we get

ℳ(GL5, 휆, R) ≅ ℬ(2휛4) ⊕ ℬ(휛3 + 휛5) ⊕ ℬ(휛1 + 휛3 + 휛4) ⊕ ℬ(휛1 + 휛2 + 휛5). (10.3.9)

We could even further restrict to SL5-crystals along the morphism SL5 → GL5 of 퐴4 root data, which has the effect of quotienting the weight lattice by ℤ휛5:

ℳ(SL5, 휆, R) ≅ ℬ(2휛4) ⊕ ℬ(휛3) ⊕ ℬ(휛1 + 휛3 + 휛4) ⊕ ℬ(휛1 + 휛2). (10.3.10)

105

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