Schur–Weyl Duality for Unipotent Upper Triangular Matrices

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Schur–Weyl duality for unipotent upper triangular matrices

Megan Danielle Ly

B.A., Loyola Marymount University, 2012

M.S., University of Colorado Boulder, 2015

A thesis submitted to the

Faculty of the Graduate School of the

University of Colorado in partial fulﬁllment

of the requirements for the degree of

Doctor of Philosophy

Department of Mathematics

2018 This thesis entitled: Schur–Weyl duality for unipotent upper triangular matrices written by Megan Danielle Ly has been approved for the Department of Mathematics

Nathaniel Thiem

Richard M. Green

Date

The ﬁnal copy of this thesis has been examined by the signatories, and we ﬁnd that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. iii

Ly, Megan Danielle (Ph.D., Mathematics)

Schur–Weyl duality for unipotent upper triangular matrices

Thesis directed by Nathaniel Thiem

Schur–Weyl duality is a fundamental framework in combinatorial representation theory. It intimately relates the irreducible representations of a group to the irreducible representations of its centralizer algebra. We investigate the analogue of Schur–Weyl duality for the group of unipotent upper triangular matrices over a ﬁnite ﬁeld. In this case, the character theory of these upper triangular matrices is “wild” or unattainable. Thus we employ a generalization, known as supercharacter theory, that creates a striking variation on the character theory of the symmetric group with combinatorics built from set partitions. In this thesis, we present a combinatorial formula for calculating a restriction and induction of supercharacters based on statistics of set partitions and seashell inspired diagrams. We use these formulas to create a graph that encodes the decomposition of a tensor space, and develop an analogue of Young tableaux, known as shell tableaux, to index paths in this graph. These paths also help determine a basis for the maps that centralize the action of the group of unipotent upper triangular matrices. We construct a part of this basis by determining copies of certain modules inside a tensor space to construct projection maps onto supermodules that act on a standard basis. Dedication

To the mathematician that has inspired me the most throughout my life: my mother. If I turn out to be half as smart, half as patient, and half as kind, I would consider myself successful. v

Acknowledgements

I would like to express my gratitude and appreciation to my advisor Nathaniel Thiem for his guidance and dedication to helping me succeed. His support has been invaluable, especially his generosity with his time and adamant attempts to make research exciting. I would like to thank the other members of my committee for their encouragement. In particular, Richard Green’s feedback over the years has helped me grow as a mathematician and writer.

A special thank you goes to all the skaters on the ﬁgure skating team at CU Boulder for pushing me to be better both on and oﬀ the ice. I also thank the women in my ballet class and the janitors at the ice rink for their kindness in reminding me of my daily accomplishments.

Lastly, I owe many thanks to my wonderful family and friends for supporting me throughout my graduate career. I express my love and thanks to my partner, Caleb, who has stuck by my side during this challenging time. Contents

Chapter

1 Introduction 1

2 Preliminaries 5

2.1 Character Theory ...... 5

2.2 Supercharacter Theories ...... 11

2.2.1 A supercharacter theory for Un ...... 12

2.3 Set Partition Combinatorics ...... 14

2.3.1 An uncolored supercharacter theory ...... 16

2.3.2 A colored supercharacter theory ...... 18

3 Branching Rules 22

3.1 Restriction ...... 22

3.2 Induction and Superinduction ...... 32

4 Shell Tableaux 37

5 Schur–Weyl dualities 47

5.1 Classical Schur–Weyl duality ...... 47

5.2 The partition algebra ...... 48

5.3 A unipotent analogue of Schur–Weyl duality ...... 49

5.3.1 Dimensions of centralizer subalgebras ...... 51 vii

5.3.2 The path basis ...... 55

5.3.3 Decomposing V ⊗k ...... 56

5.3.4 Projections for pairs of paths ...... 67

⊗k 5.3.5 The centralizer algebra EndU2 (V )...... 68

Bibliography 73

Appendix

A Sage Code 75 Chapter 1

Introduction

Schur–Weyl duality forms an archetypal situation in combinatorial representation theory involving two actions that complement each other. In the basic setup, a G-module M of a ﬁnite group G is tensored together k times to form the tensor space

M ⊗k = M ⊗ · · · ⊗ M . | {z } k factors

⊗k ⊗k The commuting actions of G and its centralizer algebra Zk = EndG(M ) on M produce a decomposition

⊗k ∼ M λ λ M = G ⊗ Zk as a (G, Zk)-bimodule λ λ λ where the G are irreducible G-modules and the Zk are irreducible Zk-modules. This bimodule decomposition intimately relates the irreducible representations of G with the irreducible representations of Zk.

In the classical situation, the general linear group GLn(C) of n × n matrices over the ﬁeld

⊗k C of complex numbers acts on the tensor space V of an n dimensional vector space V , and its centralizer algebra is the symmetric group Sk on the k tensor factors. More recently, the study of new versions of Schur–Weyl duality has led to many remarkable discoveries about algebras of operators on tensor space that are full centralizers of each other. For example,

(1) the Brauer algebra is the centralizer of the symplectic and orthogonal groups acting on

n ⊗k tensor space (C ) [10]; 2

(2) the partition algebra is the centralizer of the symmetric group acting on the tensor space

V ⊗k of its permutation representation V [14].

My research focuses on a unipotent analogue of Schur–Weyl duality.

For a positive integer n and a power of a prime q = pr, consider the finite group of unipotent n × n upper-triangular matrices    1 ∗ · · · ∗     .  . 0 1 . Un =   . ..  . . ∗      0 ··· 0 1  with ones on the diagonal and entries ∗ in the finite field Fq with q elements. Since Un is a

Sylow p-subgroup of GLn(Fq), then every p-group of GLn(Fq) is conjugate to a subgroup of Un.

Embedding every ﬁnite p-group in Sn ⊆ GLn(Fq) as permutation matrices, it follows that every p-group is isomorphic to a subgroup of Un. This is akin to how every ﬁnite group is isomorphic to a subgroup of Sn, so it is not unreasonable to hope that the representation theories of Un and Sn have comparable structures.

Unlike the combinatorially rich representation theory of Sn [18], the representation theory of Un is well-known to be intractable or “wild” [13]. Nevertheless, Andr´e[2, 3, 4, 5] and Yan [21] constructed a workable approximation that has been useful in studying Fourier analysis [12], ran- dom walks [7], and Hopf algebras [1]. In [12] Diaconis and Isaacs generalize this idea to arbitrary

ﬁnite groups to develop the notion of supercharacter theory. Supercharacter theory approximates the character theory of a ﬁnite group by replacing conjugacy classes with certain unions of conjugacy classes called “superclasses” and irreducible characters with certain linear combinations of irreducible characters called “supercharacters”.

We study a coarsening of Andre and Yan’s traditional super-representation theory on Un [9] 3 where there is a one-to-one correspondence between      supercharacters   Set partitions of  ! .  of Un   {1, 2, . . . , n} 

It is becoming ever more apparent that the set partition combinatorics of this super-representation theory is analogous to the classical partition combinatorics of the representation theory of the symmetric group, but with some important diﬀerences.

⊗k 1 In Chapter 3, we study the decomposition of V where V = CUn ⊗CUn−1 as a Un- supermodule. Much like the partition algebra, we have

V ⊗k =∼ (IndUn ResUn )k(1) Un−1 Un−1 | {z } k times where the trivial supercharacter is restricted and induced k times. We provide a combinatorial formula calculating a restriction of supercharacters from Un to Un−1 where the coeﬃcients of the supercharacters of Un−1 are a product of powers of q and q − 1 based on statistics of set partitions and seashell inspired diagrams. For example, a shell formed by two set partitions is shown below

Using Frobenius reciprocity, we obtain a corresponding formula for inducing supercharacters. To- gether these formulas are known as branching rules. As opposed to the representation theory of the symmetric group, they depend on the embedding of Un−1 in Un.

In Chapter 4, we use the branching rules to create a graph that encodes the decomposition of V ⊗k known as the Bratteli diagram. For the symmetric group, paths in the Bratteli diagram are indexed by a set of combinatorial objects called Young tableaux. We create an analogue of Young tableaux, known as shell tableaux, that is built from the combinatorics of the previous chapter using a generalization of shells. Next, we construct a bijection between shell tableaux and paths in the Bratteli diagram. When q = 2, we remove a condition on shell tableaux to produce a bijection 4 with weighted paths in the Bratteli diagram. In contrast with the symmetric group, these weights account for the multiplicities in our Bratteli diagram.

In Chapter 5, we apply our results to the theory of Schur–Weyl duality. Since we are ap- proximating by supercharacters, we consider a subalgebra of the centralizer algebra that treats

⊗k supermodules as irreducibles. We show that the decomposition of V as a Un-supermodule given by the Bratteli diagram produces a decomposition of this subalgebra. We also prove that the dimension of this subalgebra is the product of the weights of pairs in paths in the Bratteli diagram, which is a polynomial in q. Moreover, these paths also help index a basis of the centralizer subalgebra. We construct a part of this basis by determining copies of certain modules inside of

V ⊗k to compute projection maps onto supermodules that act on a standard basis of V ⊗k. While determining these projections is generally unattainable, it seems to be tractable in this case. We illustrate this when n = 2 and q = 2 to produce a basis for the full centralizer algebra.

Our construction of a basis is one of the first attempts to carve out a framework for studying a well-defined piece of a centralizer algebra, leading to surprising findings about the super- representation theory of Un. In particular, we have shown the standard local branching rules of

Un can largely be realized explicitly as modules, which is fairly unexpected in general. The shell combinatorics developed from this may help compute in other algebraic structures related to the supercharacter theory of Un, such as the Hopf algebra of symmetric functions in noncommuting variables. It also turns out that when the supercharacters coincide with irreducible characters, the dimensions of the centralizer algebras are interesting sequences found in the On-Line Encyclopedia of Integer Sequences (OEIS). However, the dimensions of centralizer subalgebras lead to numerous new combinatorial sequences yet to be explored. Chapter 2

Preliminaries

We develop background material on character theory, supercharacter theories, and set partition combinatorics with a focus on a supercharacter theory for the group of unipotent upper triangular matrices.

2.1 Character Theory

Let G be a finite group and V be a finite dimensional vector space over the field of complex numbers C.A representation over C is a homomorphism ρ : G ! GL(V ) where GL(V ) is the group of invertible linear transformations of V . We say V is a left G-module with the action

gv = ρ(g)v g ∈ G, v ∈ V.

All groups have a trivial representation that sends each g ∈ G to the identity transformation which maps every vector in V to itself. Furthermore, when G acts on a basis B for V the permutation representation of V is deﬁned by ρ(g)v = gv for all v ∈ B. In particular, we can form the vector space CG over C whose elements are formal linear combinations

X agg, ag ∈ C. g∈G The permutation representation arising from the action of G on itself is known as the regular representation.

In a sense, understanding representations of a group G reduces to studying invariant subspaces of the regular representation. If V is a G-module and W is a subspace such that ρ(g)(W ) = W 6 for all g ∈ G, then W is a submodule of V . The corresponding representation is known as a subrepresentation. We say a G-module is irreducible it contains no proper nonzero submodules.

We also refer to the representation ρ as irreducible.

It is important to determine when two irreducible representations are isomorphic. A G- module homomorphism between two G-modules V and W is a linear transformation ϕ : V ! W such that

ϕ(gv) = gϕ(v) g ∈ G, v ∈ V.

A G-module homomorphism from V to itself is known as a G-module endomorphism, and a G- module isomorphism is a bijective G-module homomorphism. Schur’s lemma characterizes G- module homomorphisms.

Proposition 2.1.1 (Schur’s Lemma [16, Lemma 1.5]). Let V and W be irreducible G-modules.

(1) If ϕ : V ! W is a G-module homomorphism then either ϕ is the zero map or an isomor-

phism.

(2) If ϕ : V ! V is a G-module endomorphism then ϕ is a scalar multiple of the identity

endomorphism.

Given a representation ρ of G, the character of G aﬀorded by ρ (and its corresponding

G-module V ) is the function

χ : G −! C

g 7! tr(ρ(g)) where tr denotes the trace of a linear transformation with respect to a basis. If χ is a character of an irreducible representation, we say χ is an irreducible character. The set of irreducible characters of G is denoted as Irr(G).

We refer to the character aﬀorded by the trivial representation as the trivial character 1.

Similarly, the character χreg corresponding to the regular representation is called the regular character. The degree of a character χ is the value χ(1). Characters of degree 1 are called linear

× characters, and are equivalent to homomorphisms from G to the multiplicative group C of C. 7

A class function of G is a function from G to C that takes a constant value on each conjugacy class. Let cl(G) denote the set of conjugacy classes of G and deﬁne

−1 cf(G) = C-span{ϕ : G ! C | ϕ(hgh ) = ϕ(g) for all g, h ∈ G} to be the space of class functions. Note that characters are class functions, since if χ is a character of ρ,

χ(hgh−1) = tr(ρ(hgh−1)) = tr(ρ(h)ρ(g)ρ(h−1)) = tr(ρ(g)) = χ(g) for all g, h ∈ G.

The space of class functions has an inner product

1 X hϕ, θi = ϕ(g)θ(g), |G| g∈G where θ(g) is the complex conjugate of θ(g). The next proposition shows that with respect to this inner product, the irreducible characters form an orthonormal basis for the space of class functions.

Proposition 2.1.2 ([16, Corollary 2.14, Theorem 2.8]). For irreducible characters χ and ψ of G,

hχ, ψi = δχ,ψ, so that every class function ϕ of G can be uniquely expressed in the form

X ϕ = hϕ, χiχ. χ∈Irr(G)

We say an irreducible character χ in the decomposition of a character ψ with hψ, χi 6= 0 is a constituent of ψ. In addition, the inner product can also be used to determine if a character is irreducible.

Proposition 2.1.3 ([16, Corollary 2.17]). A character χ of G is irreducible if and only if

hχ, χi = 1. 8

We now look at ways of building new representations of a group G from existing representations. A G-module V is the direct sum V = U ⊕ W of two submodules U and W if every element of v ∈ V can be written uniquely as a sum

v = u + w u ∈ U, w ∈ W.

If U aﬀords the character χ and W aﬀords the character ψ, then the character of V is χ + ψ.

Proposition 2.1.4 (Maschke’s Theorem [16, Theorem 1.9]). Let G be a ﬁnite group and V be a non-zero G-module. Then V can be decomposed as a direct sum

V = W1 ⊕ W2 ⊕ · · · ⊕ Wk of irreducible submodules Wi of V .

Thus irreducible modules are the building blocks of all ﬁnite dimensional G-modules.

In addition, we introduce a product operation of two G-modules V and W . Choose bases

{v1, . . . , vn} for V and {w1, . . . , wm} for W . The tensor product V ⊗C W is the set of formal sums of the form X aij(vi ⊗ wj), aij ∈ C.

When the ﬁeld is clear we simply write V ⊗ W . We deﬁne an action of G on V ⊗ W by setting

g(vi ⊗ wj) = gvi ⊗ gwj, g ∈ G, and extending linearly to all of V ⊗ W . If V and W aﬀord characters χ and ψ respectively, then

V ⊗ W aﬀords the character χ ψ given for g ∈ G by

(χ ψ)(g) = χ(g)ψ(g). (2.1)

We can also relate representations of a group G to representations of its subgroups. If H is

G a subgroup of G and V is a G-module, then V is also an H-module denoted by ResH (V ). The aﬀorded character, given by restricting the character χ of the G-module V to H, is called the

G restriction ResH (χ) of χ to H. For h ∈ H and ϕ ∈ cf(G), this produces a map

G ResH : cf(G) ! cf(H) 9 deﬁned by

G ResH (ϕ)(h) = ϕ(h).

Conversely, we can build up a representation of a group G from a representation of its subgroup H to obtain a map from cf(H) to cf(G). The right action of G on itself makes CG into a right G-module. Given an H-module W , the induced module of W is the G-module

G ∼ IndH (W ) = CG ⊗CH W with the action

g · (g0 ⊗ w) = (gg0) ⊗ w g, g0 ∈ G, w ∈ W.

If ψ is the character of W , the character aﬀorded by CG ⊗CH W is known as the induced character

G IndH (ψ) of ψ to G, and a map

G IndH : cf(H) ! cf(G) is deﬁned by   1 X  χ(g) if g ∈ H, IndG (θ)(g) = χ˙(xgx−1) whereχ ˙(g) = H |H| x∈G  0 if g 6∈ H, for θ ∈ ch(H) and g ∈ G.

Proposition 2.1.5 (Frobenius reciprocity [16, Lemma 5.2]). Let H be a subgroup of G. Suppose

ϕ is a class function of G and θ is a class function of H. Then

G G hIndH (θ), ϕi = hθ, ResH (ϕ)i.

When H =∼ G/N for a normal subgroup N of G we have an alternative way to use a represen- ∼ tation of H to construct a representation of G. In this case G = N o H. Let W be an H-module and 1 X z = n ∈ Z( G), N |N| C n∈N 10 the center of the group algebra CG. Then CzN H is a right H-module under right multiplication. The inﬂation module

G ∼ InfH (W ) = CzN H ⊗CH W = CzN ⊗CH W is a G-module with the action for g = n0h0 ∈ G given by

0 0 0 g · (zN h ⊗ w) = (n h zN h) ⊗ w = zN h h ⊗ w, h ∈ H, w ∈ W.

Its character is

G InfH (ψ) = ψ ◦ π where ψ is the character aﬀorded by W and π is the projection π : G ! G/N. In this construction irreducible representations of H correspond to irreducible representations of G.

Proposition 2.1.6 ([16, Lemma 2.22]). Let H =∼ G/N for a normal subgroup N of G. Then

G InfH (ψ) ∈ Irr(G) if and only if ψ ∈ Irr(H).

Note we can generalize representations to other algebraic structures. Let F be a ﬁeld. An

F-algebra is an F-vector space that is also a ring with 1 such that for all c ∈ F, x, y ∈ A, we have

(cx)y = c(xy) = x(cy).

For example, Mn(F), the set of n × n matrices over F, is an F-algebra. The vector space FG with addition and multiplication given by

X X X agg + bgg = (ag + bg)g g∈G g∈G g∈G and X X X agg × bgg = (agbh)gh g∈G g∈G g,h∈G forms the group algebra. In addition End(V ), the set of linear transformations of an F-vector space

V , and EndA(V ), the set of A-module linear transformations of V , are F-algebras. The algebra

EndA(V ) is known as the centralizer algebra of V . 11

If A and B are F-algebras, then a linear transformation ϕ : A ! B satisfying ϕ(1) = 1 and

ϕ(xy) = ϕ(x)ϕ(y) for all x, y ∈ A is an algebra homomorphism.A representation of an F-algebra

A on a ﬁnite dimensional F-vector space V is an algebra homomorphism ρ : A ! End(V ). We say V together with the action

av = ρ(a)v a ∈ A, v ∈ V, is a left A-module. If V is a left A-module and a right B-module with compatible actions where

(av)b = a(vb) a ∈ A, b ∈ B, v ∈ V then we say V is an (A, B)-bimodule.

There is a natural (A, EndA(V ))-bimodule structure on a left A-module V that allows us to study the centralizer algebra EndA(V ).

Theorem 2.1.7 (Double Centralizer [14, Theorem 5.4]). Let F be an algebraically closed ﬁeld and

A be a ﬁnite dimensional F-algebra. Suppose V is an A-module where

∼ M λ V = mλA λ∈Vˆ

λ for an index set Vˆ of irreducible A-modules A and positive integers mλ. If Z = EndA(V ), then

∼ M (a) Z = Mmλ (F); λ∈Vˆ (b) as an (A, Z)-bimodule, we have M V =∼ Aλ ⊗ Zλ λ∈Vˆ where the Zλ for λ ∈ Vˆ are irreducible Z-modules.

2.2 Supercharacter Theories

Supercharacter theory arises as a natural generalization of character theory. Since determining the conjugacy classes and irreducible characters of Un is impossible or “wild” [13], Andr´e

[2, 3, 4, 5] and Yan [21] constructed a workable resemblance of character theory by “clumping” 12 together some irreducible characters and some conjugacy classes. This clumping method can be thought of as a way to coarsen the partition of a group into its conjugacy classes that is compatible with certain sums of irreducible characters. In [12] Diaconis and Isaacs generalize this idea to arbitrary ﬁnite groups.

A supercharacter theory of a group G consists of a set of superclasses K and a set of supercharacters X such that

(a) the set K is a partition of G into unions of conjugacy classes,

(b) the set X is a set of characters such that each irreducible character of G is a constituent of

exactly one supercharacter,

(d) the supercharacters are constant on superclasses.

Every group G has two “trivial” supercharacter theories: the usual character theory with K = cl(G) and X = Irr(G), and the supercharacter theory with K = {{1},G−{1}} and X = {1, χreg−1} where 1 is the trivial character of G and χreg is the regular character. While many ﬁnite groups have several supercharacter theories [12], preference is given to supercharacter theories that strike a balance between computability and producing better approximations of the usual character theory.

2.2.1 A supercharacter theory for Un

We focus on the supercharacter theory on Un given in [20] that is a slight coarsening of the traditional supercharacter theory of Andr´eand Yan.

Let Un be the subgroup of unipotent upper-triangular matrices of the general linear group

GLn(Fq) over the ﬁnite ﬁeld Fq with q elements, Bn be the normalizer of Un in GLn(Fq) consisting of upper triangular matrices, and

un = Un − 1 13 be the nilpotent Fq-algebra of strictly upper triangular matrices. The subgroup Bn acts by left and right multiplication on un, and the superclasses are given by the two-sided orbits

BnunBn ! K

BnxBn 7! 1 + BnxBn.

+ × Following the construction in [9], ﬁx a nontrivial homomorphism ϑ : Fq ! C . The Fq-vector space of n × n matrices gln(Fq) decomposes in terms of upper triangular matrices bn and strictly lower triangular matrices ln as

gln = bn ⊕ ln.

Identifying ln with gln/bn makes ln a canonical set of coset representatives in gln/bn. For v ∈ gln deﬁne

v¯ = (v + bn) ∩ ln.

Then for v ∈ ln,

C-span{av | a ∈ Bn} is Un-supermodule with left action

uw = ϑ(tr((u − 1)w)(uw) for u ∈ Un, w ∈ ln and right action

−1 −1 wu = ϑ(tr(w(u − 1))(wu ) for u ∈ Un, w ∈ ln.

The two-sided orbits from extending these actions on ln to the normalizer subgroup Bn yields corresponding supercharacters given by,

BnlnBn ! X

|Bnv| X BnvBn 7! g 7! ϑ(tr((g − 1)w)). |BnvBn| w∈BnvBn

In constructing the supercharacters of Un it is more common to construct a module structure

∗ on the dual un, where un = Un − 1 as in [12]. However, the actions of Bn on ln are a translation of

∗ the actions on un that make studying modules more straightforward [9]. 14

By elementary row and column operations we may choose orbit representatives for the two- sided action of Bn on un and ln so that there is a one to one correspondence between    

 superclasses   u − 1 has at most one 1 

! u ∈ Un  of Un   in every row and column 

   

 supercharacters   v has at most one 1 

! v ∈ ln .  of Un   in every row and column  These representatives are indexed by set partitions.

2.3 Set Partition Combinatorics

Deﬁne [n] = {1, 2, . . . , n}.A set partition λ of [n] is a subset {(i, j) ∈ [n] × [n] | i < j} such that if (i, k), (j, l) ∈ λ, then i = j if and only if k = l. We represent each set partition λ ` [n] diagrammatically as a set of arcs on a row of n nodes so that if (i, j) ∈ λ, then there is an arc connecting the ith node to the jth node. For example,

1 2 3 4 5 6 {1 _ 3, 3 _ 5, 2 _ 6} ! or . 1 2 3 4 5 6

In these diagrams it is natural to draw the arcs above or below the nodes. We will use both orientations to compare set partitions. We typically refer to the pair (i, j) as an arc in λ and write

(i, j) = i _ j or (i, j) = i ^ j to specify the arc. For each arc (i, j) ∈ λ we call i the left endpoint and j the right endpoint. The sets of left and right endpoints of λ are given by

le(λ) = {i ∈ [n] | (i, j) ∈ λ, for some j ∈ [n]}

re(λ) = {j ∈ [n] | (i, j) ∈ λ, for some i ∈ [n]}.

We say two arcs conﬂict if they have the same the same left or right endpoints. Thus no arcs conﬂict in a set partition. 15

We obtain the more traditional deﬁnition of set partitions by taking part(λ) for λ ` [n] to be the set of equivalence classes on [n] given by the reﬂexive transitive closure of i ∼ j if (i, j) ∈ λ.

For instance, ! part = {{1, 3, 5}, {2, 6}, {4}}. 1 2 3 4 5 6

Note the connected components of the diagram are the parts of the set partition and the arcs are the adjacent pairs of elements in each part.

There are some natural statistics on set partitions [11]. For a set partition λ ` [n] the dimension is X dim(λ) = j − i − 1. i_j∈λ For a pair of set partitions λ, µ ` [n] deﬁne

CRS(λ, µ) = {((i, k), (j, l)) ∈ λ × µ | i < j < k < l}, crs(λ, µ) = |CRS(λ, µ)|,

λ λ λ NSTµ = {((i, l), (j, k)) ∈ λ × µ | i < j < k < l}, nstµ = |NSTµ| as the crossing set, crossing number, nesting set, and nesting number respectively. To illustrate, if

λ = and µ = , then we have

λ µ dim(λ) = 3, crs(λ, λ) = 1, nstλ = 0, dim(µ) = 4, crs(µ, µ) = 0, nstµ = 1.

Superimposing λ and µ, where the arcs of λ are dashed

λ ∪ µ = yields

λ CRS(λ, µ) = {(1 _ 4, 2 _ 6), (1 _ 4, 3 _ 5)}, NSTµ = ∅ 16 but,

µ CRS(µ, λ) = ∅, NSTλ = {(2 _ 6, 3 _ 5)}.

While it is not generally true that CRS(λ, µ) = CRS(µ, λ), it follows from the deﬁnition of a crossing number that for all set partitions λ, µ, ν ` [n],

CRS(λ, µ ∪ ν) = CRS(λ, µ) + CRS(λ, ν) (2.2)

CRS(λ ∪ µ, ν) = CRS(λ, ν) + CRS(µ, ν). (2.3)

It will also be of interest to consider set partitions where the arcs are labeled or colored by

× × an element of Fq . An Fq -colored set partition of [n] is a pair (λ, φ), where λ is a set partition of

× × [n] and φ : λ ! Fq is a coloring of the arcs by elements of Fq . By convention, if φ((i, j)) = a and the orientation of the arc is speciﬁed, we write a labeled arc as i _a j or i ^ j. For example, a

1 1 2

× is an F3 -colored set partition of [6].

2.3.1 An uncolored supercharacter theory

We describe the correspondence between set partitions and the superclasses and supercharacters of Un. Given a set partition λ ` [n], we construct a representative uλ of a superclass of Un by   1 if i _ j ∈ λ or i = j (uλ)i,j =  0 otherwise. 17

For instance, the correspondence between λ and uλ is given as follows   0 0 1 0 0 0        0 0 0 0 0 1        λ =  0 0 0 0 1 0  ! uλ − 1 =   .    0 0 0 0 0 0         0 0 0 0 0 0      0 0 0 0 0 0

The corresponding superclass Kλ is

Kλ = 1 + Bn(uλ − 1)Bn.

Similarly, a representative vλ for the two-sided action of Bn on ln is   1 if j _ k ∈ λ, (vλ)k,j =  0 otherwise so that

λ ∼ V = C-span{avλ | a ∈ Bn}

λ and for g ∈ Un, the corresponding supercharacter χ is deﬁned as

|Bnvλ| X χλ(g) = ϑ(tr((g − 1) v)). |BnvλBn| v∈BnvλBn Amazingly, many properties of these supercharacters can be determined using statistics of set partitions.

Proposition 2.3.1 ([9, Bragg, Thiem, Proposition 2.1]). For λ, µ ` [n], we have   |λ∩µ| dim(λ) |λ−µ| if i < j < k, i _ k ∈ λ  (−1) q (q − 1)  λ λ  nstµ χ (uµ) = q then i _ j, j _ k∈ / µ,    0 otherwise. 18

In particular the trivial supercharacter 1 is the supercharacter χ∅ corresponding to the empty set partition of [n], and the degree of each supercharacter is

χλ(1) = qdim(λ)(q − 1)|λ|.

It also follows from the formula that supercharacters factor as tensor products of arcs

K χλ = χi_j where (χ ψ)(g) = χ(g)ψ(g). (2.4) i_j∈λ

With respect to the inner product the supercharacters form an orthogonal set.

Proposition 2.3.2. For λ, µ ` [n], we have

λ µ |λ| crs(λ,λ) hχ , χ i = δλµ(q − 1) q .

Proposition 2.3.2 can be proved from [20, Thiem, (2.3)]. In light of Proposition 2.1.3, the crossing number crs(λ, λ) helps measure how close a supercharacter is to being irreducible.

2.3.2 A colored supercharacter theory

If instead of considering the orbits of the full subgroup Bn, we consider the Un orbits on

∗ the group un and its dual un, then we obtain the traditional supercharacter theory of Andr´eand

× Yan. In this case the combinatorics depends on the ﬁnite ﬁeld Fq and is based on Fq -colored set partitions.

× In this traditional supercharacter theory, the superclass Kλ,φ corresponding to an Fq -colored set partition (λ, φ) of [n] is   φ(i _ j) if i _ j ∈ λ,   Kλ,φ = 1 + Un(uλ,φ − 1)Un, where (uλ,φ)i,j = 1 if i = j,    0 otherwise.

The supercharacters are given by the following proposition. 19

+ × × Proposition 2.3.3. Let ϑ : Fq ! C be a nontrivial homorphism. For Fq -colored set partitions (λ, φ) and (µ, ψ) of [n], we have   dim(λ) if i < j < k, i _ k ∈ λ  q Y  λ ϑ(φ(i _ l)ψ(i _ l)) λ,φ  nstµ χ (uµ,ψ) = q i_l∈λ then i _ j, j _ k∈ / µ,    0 otherwise, where ψ(i _ l) = 0 if i _ l∈ / µ.

There is a nice relationship between this colored supercharacter theory and the uncolored supercharacter theory we are considering.

Proposition 2.3.4. For λ ` [n], we have

[ λ X λ,φ Kλ = Kλ,φ and χ = χ , φ∈Col Fq(λ) φ∈ColFq (λ)

× where ColFq (λ) denotes the set of all Fq -colorings of λ.

Proof. Let λ ` [n]. For each s, t ∈ Tn, the group of diagonal matrices, there exists a unique

φ ∈ ColFq (λ) such that

s(uλ − 1)t = uλ,φ − 1.

Seeing that Bn = Un o Tn, we obtain

Kλ = {1 + xs(uλ − 1)ty | x, y ∈ Un, s, t ∈ Tn}

= {1 + x(uλ,φ − 1)y | x, y ∈ Un, φ ∈ ColFq (λ)} [ = Kλ,φ. φ∈Col Fq(λ)

λ λ Since χ is constant on the superclasses Kλ, it follows that χ takes a constant value each superclass Kλ,φ for φ ∈ ColFq (λ). Thus it suﬃces to show that

λ X λ,φ χ (uµ,ψ) = χ (uµ,ψ)

φ∈ColFq (λ) 20

× for an Fq -colored set partition (µ, ψ). By Propositions 2.3.3 and 2.3.1 if i _ l ∈ λ, and i _ j ∈ µ or j _ k ∈ µ for i < j < k then

X λ,φ λ χ (uµ,ψ) = 0 = χ (uµ,ψ). φ∈Col Fq(λ) Otherwise, we obtain

dim(λ) X λ,φ q X Y χ (uµ,ψ) = λ ϑ(φ(i _ l)ψ(i _ l)). qnstµ φ∈Col Fq(λ) φ∈Col Fq(λ) i_l∈λ If

λ − µ = {i1 _ l1, . . . , it _ lt} and λ ∩ µ = {j1 _ k1, . . . , js _ ks},

× then we can identify any Fq -coloring φ with a tuple

× t+s (a1, . . . , at, b1, . . . , bs) ∈ (Fq ) .

As a result, we obtain

X Y ϑ(φ(i _ l)ψ(i _ l))

φ∈Col Fq(λ) i_l∈λ t s X Y X Y = ϑ(φ(ir _ lr)ψ(ir _ lr)) · ϑ(φ(jr _ kr)ψ(jr _ kr)). (2.5) × r=1 × r=1 a1,...,at∈Fq b1,...,bs∈Fq

Since ir _ lr ∈/ µ, it follows that ψ(ir _ lr) = 0, which implies

t t X Y X Y X t ϑ(φ(ir _ lr)ψ(ir _ lr)) = ϑ(0) = 1 = (q − 1) . (2.6) × r=1 × r=1 × a1,...,at∈Fq a1,...,at∈Fq a1,...,at∈Fq

+ Because ϑ is a nontrivial linear character of Fq , we have

s s X Y X Y ϑ(φ(jr _ kr)ψ(jr _ kr) = ϑ(brψ(jr _ kr)) × r=1 × r=1 b1,...,bs∈Fq b1,...,bs∈Fq s X Y = ϑ(br) × r=1 b1,...,bs∈Fq !s X = ϑ(b) × b∈Fq

= (−1)s. (2.7) 21

Substituting (2.6) and (2.7) into Equation (2.5) yields

X Y ϑ(φ(i _ l)ψ(i _ l)) = (−1)|λ∩µ|(q − 1)|λ−µ|

φ∈Col Fq(λ) i_l∈λ as t = |λ − µ| and s = |λ ∩ µ|. Therefore, we obtain

|λ∩µ| dim(λ) |λ−µ| X λ,φ (−1) q (q − 1) λ χ (uµ,ψ) = λ = χ (uµ,ψ) qnstµ φ∈Col Fq(λ) by Proposition 2.3.3.

This proposition allows us to translate many results on the colored supercharacter theory to the uncolored one. Chapter 3

Branching Rules

An important property of the supercharacters of Un is that their restriction to any subgroup is a linear combination of supercharacters with nonnegative integer coefficients [12]. However, the coefficients in the restriction decompositions are not well understood [20]. We provide a combinatorial formula for calculating the restriction of supercharacters of Un to Un−1. Using Frobenius reciprocity, we obtain a corresponding formula for inducing supercharacters. Since these formulas depend on the number of nonzero elements in the field Fq, fix

t = q − 1 for this chapter.

3.1 Restriction

We consider the restriction of supercharacters from Un to Un−1 by embedding Un−1 ⊆ Un as

Un−1 = {u ∈ Un | (u − 1)ij 6= 0 implies i < j < n}.

Since supercharacters decompose into tensor products of arcs (2.4), for λ ` [n], we have

K K χλ = χi_l and ResUn (χλ) = ResUn (χi_l). Un−1 Un−1 i_l∈λ i_l∈λ

Consequently we compute restrictions for each χi_l and use the tensor product to glue together the resulting restrictions. 23

The restriction of the supercharacter χi_l is given using the formulas in [20] for computing restrictions in the colored supercharacter theory [cf. Section 2.3.2].

Proposition 3.1.1. For 1 ≤ i < l ≤ n, the restriction ResUn (χi_l) is given by Un−1   χi_l if l 6= n,  ResUn (χi_l) = Un−1 X  t 1 + χi_k if l = n.  i

X a X a ResUn (χi_l) = ResUn (χi_l) = χi_l = χi_l, Un−1 Un−1 × × a∈Fq a∈Fq and for l = n, we have X a X X b X ResUn (χi_l) = ResUn (χi_l) = 1 + χi_k = t 1 + χi_k . Un−1 Un−1 a∈ × a∈ × i

Intuitively, restricting an arc corresponds to removing the last node and reattaching the arc in all possible ways.

We now use the tensor product to glue together the resulting restrictions. For 1 ≤ i < l, deﬁne X X χi_×l = t 1 + χi_k and χi×_l = t 1 + χj_l . i

Proposition 3.1.2. For 1 ≤ i < l ≤ n and 1 ≤ j < k ≤ n such that (i, l) 6= (j, k),   {i_l,j_k}  χ if k 6= l, i 6= j,  i_l j_k  χ χ = χi_l χj_×k if i < j < k = l,    χi_l χj×_k if i = j < k < l. Proof. Let 1 ≤ i < l ≤ n and 1 ≤ j < k ≤ n such that (i, l) 6= (j, k). For k 6= l and i 6= j, the tensor product χi_l χj_k is given by

X X a b X X a b χi_l χj_k = χi_l χj_k = χ{i_l,j_k} = χ{i_l,j_k}, × × × × a∈Fq b∈Fq a∈Fq b∈Fq 24 for i < j < k = l, we have

X X a b X X a X c χi_l χj_l = χi_l χj_l = χi_l 1 + χj_k = χi_l χj_×l, × × × × a∈ q b∈ q a∈ q b∈ q j

X X a b X X a X c χi_l χi_k = χi_l χi_k = χi_l 1 + χj_k = χi_l χi×_k × × × × a∈ q b∈ q a∈ q b∈ q i

Thus the tensor product provides a rule for resolving conﬂicting arcs that have the same right endpoint by removing the smaller arc and reattaching it in all possible ways.

Next we work toward providing a combinatorial description of the coeﬃcients in the tensor product based on statistics of set partitions and seashell inspired diagrams.

0 Deﬁnition 3.1.3. Let s ∈ {s, s + 1} for s ∈ Z≥1 and 1 ≤ i ≤ l ≤ n.A shell of size n and width l − i is a set of arcs on n nodes of the form

s s0−1 [ [ {ir _ lr} ∪ {ir ^ lr+1} r=1 r=1 where i = i1 < ··· < is ≤ ls0 < ··· < l1 = l.

For example, some shells of size 6 and width 6 − 2 are

{2 _ 6}, {2 _ 6} ∪ {2 ^ 5}

{2 _ 6, 3 _ 5} ∪ {2 ^ 5}, {2 _ 6, 3 _ 5} ∪ {2 ^ 5, 3 ^ 4}. 25

A whorl is pair of consecutive arcs (i _ l, i ^ j) in a shell corresponding to a 360◦ rotation in the spiral conﬁguration. Following the notation of Deﬁnition 3.1.3, the number of whorls of a shell is ls + s0 − 1m 2 as each arc is half a whorl. We use the convention that whorls are counted from the right endpoint l spiraling inward. For instance, in the shell below we count the two whorls (1 _ 5, 1 ^ 4) and

(2 _ 4, 2 ^ 3) as follows 1 4

1 . 2 2 1

3 4

If the whorls of a shell are given by (i1 _ l1, i1 ^ l2),..., (is _ ls, is ^ ls+1) we say the pair

(i1 _ l1, i1 ^ l2) is the outer whorl and the other whorls are inner whorls.

We can use shells to determine the partitions that appear in the restriction of a supercharacter. More precisely, drawing the arcs of a partition µ ` [n − 1] below the nodes, and identifying the nodes with the leftmost n − 1 nodes of a partition λ ` [n] allows us to characterize the partitions with nonzero coeﬃcients in the restriction of λ as the partitions µ ` [n−1] such that the symmetric diﬀerence between λ and µ form a shell.

Deﬁnition 3.1.4. For λ ` [n] and 1 ≤ i < l ≤ n with i∈ / le(λ), the shell set Cλ,i_l of λ ∪ {i _ l} is

Cλ,i_l = µ ` [n − 1] | (λ ∪ {i _ l}) − µ ∪ µ − (λ ∪ {i _ l}) is a shell of width l − i .

This corresponds to all the ways to reattach the arc i _ l and “straighten” the resulting diagram by resolving all the conﬂicting arcs that share the same right endpoint. 26

Example 3.1.5. Suppose λ = {1 _ 4, 3 _ 5} ` [6]. Consequently, we have

λ ∪ {2 _ 6} = and     Cλ,2_6 = , , .   The seashells created by the symmetric diﬀerences between λ ∪ {2 _ 6} and µ ∈ Cλ,2_6 are shown as solid lines

while the arcs in λ ∩ µ are dashed.

It will be of interest to examine the shell sets Cλ,i_l by considering the right endpoints re(λ).

Deﬁnition 3.1.6. For each j _ k ∈ λ with i < j < k < l deﬁne λ|j7!i as the set partition obtained by replacing j _ k with i _ k and leaving everything else in λ the same. That is,

λ|j7!i = λ ∪ {i _ k} − {j _ k}.

With this notation we can describe the shell set Cλ,i_l as a union of shells with half a whorl, shells with one whorl, and shells with greater than one whorl.

Lemma 3.1.7. For λ ` [n], and i∈ / le(λ), the shell set is given by

Cλ,i_l = {λ} ∪ {λ ∪ {i _ k} | i < k < l, k∈ / re(λ)} ∪ {µ ∈ Cλ|j7!i,j_k | i

Proof. By deﬁnition {λ, λ ∪ {i _ k} | i < k < l, k∈ / re(λ)} ⊆ Cλ,i_l, so it suﬃces to show that

Cλ,i_l\{λ, λ ∪ {i _ k} | i < k < l, k∈ / re(λ)} = {µ ∈ Cλ|j7!i,j_k | i

0 There exist j = j1 < ··· < js < ks0 < ··· < k1 = k with s ∈ {s, s + 1} such that

(λ|j7!i ∪ {j _ k}) − µ = {j1 _ k1, j2 _ k2, . . . , js _ ks}, and

µ − (λ|j7!i ∪ {j _ k}) = {j1 _ k2, j2 _ k3, . . . , js0−1 _ ks0 } 27 if and only if there exist i < j = j1 < ··· < js < ks0 < ··· < k1 = k < l such that

(λ ∪ {i _ l}) − µ = {i _ l, j1 _ k1, . . . , js _ ks}, and

µ − (λ ∪ {i _ l}) = {i _ k, j1 _ k2, . . . , js0−1 _ ks0 }.

Thus µ ∈ Cλ|j7!i,j_k for some i

Definition 3.1.8. For each µ ∈ Cλ,i_l define the shell coefficient of λ ∪ {i _ l} and µ as

t|(λ∪{i_l})−µ|qcrs((λ∪{i_l})∩µ,(λ∪{i_l})−µ) cλ,i_l = µ qcrs((λ∪{i_l})∩µ,µ−(λ∪{i_l})) where t = q − 1 and crs(·, ·) is the crossing number of two set partitions given in Section 2.3.

λ,i_l We can associate each shell coeﬃcient cµ to the shell created by the symmetric diﬀerence of

λ ∪ {i _ l} and µ. The next lemma shows the shell coefficient is the product of the shell coefficient of the outer whorl with the shell coefficient of the inner whorls.

Lemma 3.1.9. Let λ ` [n], i∈ / le(λ), and h _ l, j _ k ∈ λ with 1 ≤ h < i < j < k < l ≤ n. If

µ ∈ Cλ|j7!i,j_k then

λ,i_l λ,i_l λ|j7!i,j_k cµ = cλ ∪{i_k}cµ .

λ|j i,j_k Proof. Let µ ∈ C 7! . By construction i∈ / le(λ|j7!i), so i _ l∈ / µ. Thus we have

(λ ∪ {i _ l}) − µ = {i _ l} ∪ (λ − µ), hence

(λ ∪ {i _ l}) ∩ µ = λ ∩ µ.

Substituting this and applying the crossing number equation (2.2), it follows that

t|(λ∪{i_l})−µ|qcrs((λ∪{i_l})∩µ,(λ∪{i_l})−µ) cλ,i_l = µ qcrs((λ∪{i_l})∩µ,µ−(λ∪{i_l})) t|{i_l}∪(λ−µ)|qcrs(λ∩µ,{i_l}∪(λ−µ)) = qcrs(λ∩µ,µ−(λ ∪{i_l})) t|i_l|qcrs(λ∩µ,i_l)t|λ−µ|qcrs(λ∩µ,λ−µ) = . qcrs(λ∩µ,µ−(λ ∪{i_l})) 28

Similarly since j _ k ∈ λ and i _ k ∈ µ − λ, we have

µ − (λ ∪ {i _ l}) = {i _ k} ∪ (µ − (λ|j7!i ∪ {j _ k})) and thus

λ − µ = λ|j7!i ∪ {j _ k} − µ.

By the crossing number equation (2.2),

|i_l| crs(λ∩µ,i_l) |(λ|j7!i ∪ j_k)−µ| crs(λ∩µ,(λ|j7!i ∪ j_k)−µ) λ,i_l t q t q cµ = qcrs(λ∩µ,{i_k}∪(µ−(λ|j7!i ∪ j_k))) t|i_l|qcrs(λ∩µ,i_l) t|(λ|j7!i ∪ j_k)−µ|qcrs(λ∩µ,(λ|j7!i ∪ j_k)−µ) = · . qcrs(λ∩µ,i_k) qcrs(λ∩µ,µ−(λ|j7!i ∪ j_k))

Moreover any arc in λ that crosses with i _ k or i _ l must be in µ, implying

λ,i_l λ|j7!i,j_k = cλ ∪{i_k}cµ .

Theorem 3.1.10. For λ ` [n], i∈ / le(λ), and 1 ≤ i < l ≤ n, we have

λ i_×l X λ,i_l µ χ χ = cµ χ µ∈Cλ,i_l

λ,i_l λ,i_l where C is the shell set of λ ∪ {i _ l} and cµ is the shell coeﬃcient of λ ∪ {i _ l} and µ.

Before proving the theorem we state a lemma about the q-analogue of a crossing number. In general, the q-analogue of a nonnegative integer n is

qn − 1 [n] = . q q − 1

Lemma 3.1.11. For λ ` [n], and 1 ≤ j < l ≤ n where j∈ / le(λ), we have

X qcrs(λ,j_l) − 1 qcrs(λ,j_k) = = [crs(λ, j _ k)] , q − 1 q i_k∈λ i

Proof. Let λ ` [n], 1 ≤ j < l ≤ n, and j∈ / le(λ). If the set of arcs in λ that cross with j _ k is given by

{i _ k ∈ λ | i < j < k < l} = {i1 _ k1, i2 _ k2, . . . , ir _ kr}, then for 1 ≤ s ≤ r

{i _ k ∈ λ | i < j < k < ks} = {i1 _ k1, i2 _ k2, . . . , is−1 _ ks−1}.

By the deﬁnition of the crossing number

r r X X X qr − 1 qcrs(λ,j_l) − 1 qcrs(λ,j_k) = q#{i_k∈λ|i

We are now ready to prove Theorem 3.1.10.

Proof. We induct on l − i. For the base case assume l − i = 1. Then Cλ,i_l = {λ}, and we obtain

λ i_×l λ 1 λ λ,i_l λ χ χ = χ t = tχ = cλ χ as desired.

Assume the formula holds for all k − j < l − i. Then, this yields

X χλ χi_×l = χλ t 1 + χi_k i

X X X = tχλ + t χλ∪{i_k} + t χλ χi_×k + t χλ∪{i_k}−{j_k} χj_×k. i

Recall from Deﬁnition 3.1.6 that λ|j7!i = λ ∪ {i _ k} − {j _ k} for each j _ k ∈ λ such that i < j < k < l. By the induction hypothesis the tensor product χλ χi_×l is

λ X λ∪{i_k} X X λ,i_k µ X X λ|j7!i,j_k µ = tχ + t χ + t cµ χ + t cµ χ i

By Lemma 3.1.11, the coeﬃcient of χλ will be

X X qcrs(λ,i_l) − 1 t + t cλ,i_k = t 1 + tqcrs(λ,i_k) = t 1 + t · = tqcrs(λ,i_l) = cλ,i_l. λ t λ h

Similarly for λ ∪ {i _ k0} where i < k0 < l and k0 ∈/ re(λ), the coeﬃcient of χλ ∪{i_k0} is

X X qcrs(λ,i_k) X t + t cλ,i_k = t 1 + t = t 1 + tqcrs(λ−ν,i_k) , µ qcrs(λ,i_k0) h

qcrs(λ−ν,i_l) − 1 qcrs(λ,i_l)−crs(λ,i_k0) − 1 tqcrs(λ,i_l) t 1 + t · = t 1 + t · = = cλ,i_l t t qcrs(λ,i_k0) λ ∪{i_k}

0 0 by Lemma 3.1.11. If j _ k ∈ λ is such that i < j < k < l then we have λ|j7!i = λ ∪ {i _ k0} − {j _ k0}. Let ν = {h _ k ∈ λ | (h _ k, i _ k0) ∈ CRS(λ, i _ k0)}. Using Lemma 3.1.9, the

0 coeﬃcient of χµ for each µ ∈ Cλ|j7!i,j_k is

0 0 0 λ|j7!i,j_k X λ,i_k λ|j7!i,j_k X λ,i_k λ|j7!i,j_k tcµ + t cµ = tcµ + t cλ∪i_k0 cµ h

0 crs(λ,i_k) λ|j7!i,j_k X q = tcµ 1 + t qcrs(λ,i_k0) h

Applying Lemmas 3.1.11 and 3.1.9 yields

crs(λ−ν,i_l) crs(λ,i_l)−crs(λ,i_k0) 0 q − 1 0 q − 1 tcλ|j7!i,j_k 1 + t · = tcλ|j7!i,j_k 1 + t · t t

crs(λ,i_l) 0 tq λ|j7!i,j_k = cµ qcrs(λ,i_k0) 0 λ,i_l λ|j7!i,j_k = cλ ∪{i_k}0 cµ

λ,i_l = cµ .

Substituting this into the equation for χλ χi_×l and applying Lemma 3.1.7 we obtain

λ i_×l λ,i_l λ X λ,i_l λ ∪{i_k} X X λ,i_l µ χ χ = cλ χ + cλ ∪{i_k}χ + cµ χ i

This combinatorial description of the coeﬃcients in the tensor product leads to a combinatorial description of the coeﬃcients in the restriction to Un−1.

Corollary 3.1.12. For λ ` [n], the restriction ResUn (χλ) is given by Un−1

X ResUn (χλ) = cλχµ Un−1 µ µ `[n−1] where   δλµ if n∈ / re(λ),   t|λ−µ|qcrs(λ∩µ,λ−µ) cλ = if µ ∈ Cλ−{i_n},i_n, µ qcrs(λ∩µ,µ−λ)    0 otherwise.

Proof. Applying Propositions 3.1.1, 3.1.2, and Theorem 3.1.10 respectively,

K K ResUn (χλ) = ResUn (χi_l) = χj_l χi_×n Un−1 Un−1 i_l∈λ j_l∈λ l6=n

λ−{i_n} i_×n X λ µ = χ χ = cµχ µ∈Cλ−{i_n},i_n 32 where t|λ−µ|qcrs(λ∩µ,λ−µ) cλ = cλ−{i_n},i_n = . µ µ qcrs(λ∩µ,µ−λ)

Example 3.1.13. Similar to Example 3.1.5, let

λ = so that     Cλ−{2_6},2_6 = , , .   Drawing the arcs of µ = {1 _ 4, 2 _ 3 _ 5} below the nodes of λ as shown below

illustrates that t1 · q1 cλ = = tq µ q0 since

λ−µ = {2 _ 6}, CRS(λ∩µ, λ−µ) = {(1 _ 4, 2 _ 6)}, crs(λ∩µ, λ−µ) = 1, crs(λ∩µ, µ−λ) = 0.

We can calculate the other coeﬃcients in the same manner to obtain

ResU6 (χλ) = tqχ{1_4,3_5} + tqχ{2_3_5,1_4} + t2qχ{1_4,2_5}. U5

3.2 Induction and Superinduction

While the restriction of a supercharacter of Un is a nonnegative integer linear combination of supercharacters, an induced supercharacter may not be a sum of supercharacters. In fact, the induced character may not even be a superclass function; for an example see [12, Section

6]. If instead we generalize to superinduction by averaging over superclasses in the same way that 33 induction averages over conjugacy classes, then the constructed function will be a linear combination of supercharacters with rational coeﬃcients [12, Lemma 6.7].

Suppose H ⊆ G and χ is a superclass function of H. If Kg is the superclass containing g ∈ G,

G then the superinduction SIndH (χ) is   χ(x) if x ∈ H G 1 X  SIndH (χ)(g) = |G : H| χ˙(x) whereχ ˙(x) = |Kg| x∈Kg  0 if x 6∈ H.

A nice property of superinduction is that the analog of Frobenius reciprocity holds.

Proposition 3.2.1 (Frobenius Reciprocity [15, Lemma 5.2]). Let H be a subgroup of G. Suppose

ϕ is a superclass function of G and θ is a class function of H. Then

G G hSIndH (θ), ϕiG = hθ, ResH (ϕ)iH .

However, superinduced characters are not necessarily characters so it is useful to know when superinduction is equivalent to induction.

[19, Section 3.2] examines some cases when this occurs for a larger class of p-groups known as algebra groups. If J is a ﬁnite dimensional nilpotent associative algebra over Fq, then the algebra group based on J is G = {1 + x | x ∈ J} under the multiplication (1 + x)(1 + y) = 1 + x + y + xy.

In particular, Marberg and Thiem show if we embed Un−1 into Un by

Un−1 = {u ∈ Un | (u − 1)ij 6= 0 implies i < j < n}

then for any superclass function χ of Un−1,

SIndUn (χ) = IndUn (χ). Un−1 Un−1

They also provide some conditions when superinduction is the same as induction.

Proposition 3.2.2 ([19, Theorem 3.1]). Let H be a subalgebra group of an algebra group G, and suppose

(1) no two superclasses of H are in the same superclass of G, and 34

(2) x(h − 1) + 1 ∈ H for all x ∈ G, h ∈ H.

Then the superinduction of any superclass function χ of H is

G G SIndH (χ) = IndH (χ).

If we embed Un−1 into Un by

Un−1 = {u ∈ Un | un−1,n = 0 and ui,n−1 = 0 for i < n − 1} then we have the following corollary.

Corollary 3.2.3. Let Un−1 = {u ∈ Un | un−1,n = 0 and ui,n−1 = 0 for i < n − 1}. Then the superinduction any superclass function χ of Un−1 is

SIndUn (χ) = IndUn (χ). Un−1 Un−1

Proof. It suﬃces to show the hypotheses of the previous theorem hold. Because there is an injective function from superclasses of Un−1 to Un then no two superclasses of Un−1 are in the same superclass of Un.

Let x ∈ Un, h ∈ Un−1 and u = x(h − 1) + 1. Since hi,n−1 − 1 = 0 we have ui,n−1 = 0 for i < n − 1. Similarly, xn−1,j = 0 for j < n − 1 and hn−1,j − 1 = 0 for j ≥ n − 1 implies un−1,n = 0.

This shows u ∈ U . Therefore, SIndUn (χ) = IndUn (χ) for any superclass function χ of U n−1 Un−1 Un−1 n−1 by Proposition 3.2.2.

Unlike in the representation theory of the symmetric group, the decomposition of induced characters depends on the embedding of Un−1 into Un. If we instead consider right modules then superinduction is equivalent to induction for the following embeddings

Un−1 = {u ∈ Un | (u − 1)ij 6= 0 implies 1 < i < j} and

Un−1 = {u ∈ Un | u1,2 = 0 and u2,j = 0 for 2 < j} 35

[19, Section 3.1]. However, it is not known if superinduction is the same as induction for other embeddings. In our case we use the embedding of Un−1 ⊆ Un obtained by removing the last column so that superinduction is in fact induction.

We now derive a corresponding formula for induction from restriction.

Corollary 3.2.4. For µ ` [n − 1], the induction IndUn (χµ) is given by Un−1

X IndUn (χµ) = dλχλ, Un−1 µ λ `[n] where   δλµ if n∈ / re(λ)   t|µ−λ|qcrs(µ−λ,λ∩µ) dλ = if µ ∈ Cλ−{i_n},i_n µ qcrs(λ−µ,λ∩µ)    0 otherwise.

Proof. Let λ ` [n] and µ ` [n − 1]. Frobenius reciprocity, Proposition 3.2.1, shows

hχλ, SIndUn (χµ)i = hResUn (χλ), χµi . Un−1 Un Un−1 Un−1

Thus if X X IndUn (χµ) = dγ χγ and ResUn (χλ) = cλχν Un−1 µ Un−1 ν γ ν then the inner product, Proposition 2.3.2, yields

crs(λ,λ) |λ| λ crs(µ,µ) |µ| λ q t dµ = q t cµ.

λ Therefore, the coeﬃcient dµ is

t|µ|−|λ|qcrs(µ,µ) dλ = cλ µ qcrs(λ,λ) µ t|µ|−|λ|qcrs(µ,µ) t|λ−µ|qcrs(λ∩µ,λ−µ) = · qcrs(λ,λ) qcrs(λ∩µ,µ−λ) t|µ−λ|qcrs(µ,µ)−crs(λ∩µ,µ−λ) = qcrs(λ,λ)−crs(λ∩µ,λ−µ) since |µ − λ| = |µ| − |λ| + |λ − µ|. From the crossing number equation (2.2) we obtain 36

t|µ−λ|qcrs(µ−(λ∩µ),µ−(µ−λ)) dλ = µ qcrs(λ−(λ∩µ),λ−(λ−µ)) t|µ−λ|qcrs(µ−λ,λ∩µ) = . qcrs(λ−µ,λ∩µ)

Together Corollaries 3.1.12 and 3.2.4 for decomposing restricted and induced supercharacters are known as branching rules, which we restate due to their importance.

Theorem 3.2.5 (Branching Rules). For λ ` [n], the restriction ResUn (χλ) is given by Un−1

For µ ` [n − 1], the induction IndUn (χµ) is given by Un−1

Since the branching rules are simple and easily computable, we include code in Appendix A for a program in SAGE that takes a supercharacter as input and outputs these restriction and induction decompositions. This enables us to quickly compute meaningful examples of restricting and inducing a supercharacter multiple times. While these formulas allow us to better understand restriction and induction, they are also useful for Schur–Weyl duality. Chapter 4

Shell Tableaux

We use the branching rules to create a graph known as the Bratteli diagram. For the symmetric group, paths in the Bratteli diagram are indexed by a set of combinatorial objects called Young tableaux [18]. Building from the combinatorics of the previous chapter, we create an analogue of

Young tableaux known as shell tableaux and construct a bijection between shell tableaux and paths in the Bratteli diagram.

For k ∈ Z≥1, consider

V k = (IndUn ResUn )k(1) Un−1 Un−1 | {z } k times where 1 is the trivial supercharacter of Un that is restricted and induced k times. Let

k Zˆk = λ ` [n] corresponding to supercharacters of V

Un k Zˆ 1 = µ ` [n − 1] corresponding to supercharacters of Res (V ) . k+ 2 Un−1

The Bratteli diagram Λ(n) is the graph with

ˆ 1 ˆ (a) vertices {(λ, k) | k ∈ Z≥0, λ ∈ Zk} ∪ {(µ, k + ) | k ∈ Z≥0, µ ∈ Z 1 }, 2 k+ 2

(b) an edge (λ, k) (µ, k + 1 ) if hResUn (χλ), χµi= 6 0, ! 2 Un−1

(d) an edge labeling m : E ! Z≥1 on the set of edges E deﬁned by (q − 1)|λ−µ|qcrs(λ∩µ,λ−µ) m((λ, k) ! (µ, k + 1 )) = 2 qcrs(λ∩µ,µ−λ)

(q − 1)|µ−λ|qcrs(µ−λ,λ∩µ) m((µ, k + 1 ) ! (λ, k + 1)) = . 2 qcrs(λ−µ,λ∩µ)

1 Recall from the branching rules, Theorem 3.2.5, that the edge labeling m((λ, k) ! (µ, k + 2 )) is the restriction coefficient which specifies the multiplicity that χµ appears in ResUn (χλ). Similarly, the Un−1 1 edge labeling m((λ, k + 2 ) ! (µ, k + 1)) is the induction coefficient which specifies the multiplicity that χλ appears in IndUn (χµ). Un−1 When drawing the Bratteli diagram, we place all the vertices (λ, l) in the lth row and simply write λ. For example, the Bratteli diagram for Λ(3) up to row 3 is

k = 0

1 k = 2

k = 1 t t t 1 k = 1 2 t k = 2 t t t t 1 k = 2 2 t k = 3 where t = q − 1.

0 1 k− 1 k A path P in the Bratteli diagram Λ(n) to λ ∈ Zˆk is a sequence P = (λ , λ 2 , . . . , λ 2 , λ = λ) such that for 0 ≤ r ≤ k − 1,

1 r r+ 2 1 (a)( λ , r) and (λ , r + 2 ) are vertices in Λ(n)

1 1 r r+ 2 1 r+ 2 1 r+1 (b)( λ , r) ! (λ , r + 2 ) and (λ , r + 2 ) ! (λ , r + 1) are edges in Λ(n).

For instance, P = , , , , , , 39 is a path in Λ(3).

Taking the edge labeling into account, we say the weight wt(P ) of a path P is the product

k−1 1 Y r r+ r+ 1 r+1 2 1 2 1 m((λ , r) ! (λ , r + 2 ))m((λ , r + 2 ) ! (λ , r + 1)) r=1

λ of its edge labels. The sum of the weights of the paths to λ ∈ Zˆk is the multiplicity that χ appears

1 k 2 1 1 2 1 2 in V . The path given above has weight t since m((λ , 1) ! (λ , 2 )) = t and m((λ , 2) !

1 2 2 1 (λ , 2 )) = t.

Let Pk(λ) be the set of paths in Λ(n) to λ ∈ Zˆk. There is a combinatorial way to encode paths in Pk(λ) using a generalization of shells.

0 Deﬁnition 4.1.1. Let s ∈ {s, s + 1} for s ∈ Z≥1 and 1 ≤ i ≤ l ≤ n.A generalized shell of width l − i is a set of arcs on n nodes of the form

s s0−1 [ [ {j _ min Lr | j ∈ Ir} ∪ {max Ir ^ m | m ∈ Lr+1} r=1 r=1 where Ir,Lr ⊆ [n] with {i} = I1 < ··· < Is ≤ Ls0 < ··· < L1 = {l}.

For subsets I,L ⊆ [n] we say I < L if i < l for each i ∈ I and l ∈ L. If max I = min L, we say

I ≤ L. It follows that a generalized shell with |Ir| = 1 and |Lr| = 1 for all r is simply a shell in the sense of Deﬁnition 3.1.3. Some generalized shells of size 6 and width 6 − 2 are

{2 _ 6} ∪ {2 ^ m | m ∈ {3, 4}}, {2 _ 6} ∪ {j _ 5 | j ∈ {3, 4}} ∪ {2 ^ 5}.

A labeled shell is a pair (ς, τ) for a generalized shell ς and a map τ : ς ! Z≥0. We say the labeling τ is strict if every pair of arcs (i, l), (j, m) ∈ ς with dim(i, j) > dim(j, m) satisﬁes

τ(i, j) < τ(j, m), and τ(j, m) 6= τ(i, l) + 1 if i = j or l = m. If τ(i, j) = a, we write the labeled arc as (i, j; a). When the orientation of the arc is speciﬁed we write (i _ j; a) or (i ^ j; a). For example, in the case of the shell 40

4 {(2 _ 6; 3), (3 _ 5; 6), (2 ^ 5; 4)}.

From strictly labeled shells, we deﬁne the key notion shell tableaux.

Deﬁnition 4.1.2. A shell tableau T = (ς1, . . . , ςk) of length k is a sequence of strictly labeled shells

r ς of size n and width n − ir such that

r r k (1) for 1 ≤ r < k, ς = {(n _ n; a)} or |ς | ≥ 2, and ς = {(ik _ n; a)};

Pk r (2) each arc has a distinct label in {1, 2,..., r=1 |ς |};

(3) the two smallest labels of each labeled shell ςr are less than the smallest label in ςr+1;

(4) for l 6= m and i < j ≤ min{l, m}, if (i, l; a) ∈ ςrl then there exists a minimal b > a such

that (i, m; b) ∈ ςrm if and only if (j, min{l, m}; b + 1) ∈ ςrmin{l,m} ;

(5) for i 6= j and max{i, j} ≤ l < m, if (j, m; a) ∈ ςrj then there exists a minimal b > a such

that (i, m; b) ∈ ςri if and only if (max{i, j}, l; b + 1) ∈ ςrmax{i,j} .

Conditions 1–3 provide the basic set up of the shells and labeling that generalizes the condition of increasing entires along the rows and columns in standard Young tableaux. Intuitively conditions

4 and 5 say a strictly labeled shell in a shell tableau has inner whorls if and only if its outer whorl conﬂicts with the outer whorl of another shell. As an example consider the tableau

1 3 5 8 ! 6 T = , , , 7 2 4 of length 4. By condition 4, the inner half whorl (3 _ 5; 6) lies in ς2 since (2 ^ 5; 4) ∈ ς2 conﬂicts with (2 _ 6; 5) ∈ ς3.

Let ST k denote the set of shell tableaux of length k. 41

Deﬁnition 4.1.3. Deﬁne the map

sh : (Z≥0, ST k) −! Set of Arcs  

k  i 6= l and τ((i, l)) is maximal  [ (a, T ) 7−! (i, l) ,  r  r=1  among all labels b ∈ ς with b ≤ a  and sh(T ) = sh(|T |,T ) be the shape of a shell tableau T .

For T in the example above, we have

sh(T ) = because τ(1, 4) = 2 is the maximal label ς1, τ(3, 5) = 6 is the maximal label in ς2 and τ(2, 3) is the

3 maximal label in ς . For λ ` [n], let ST k(λ) denote the set of shell tableaux of shape λ.

Theorem 4.1.4. Let λ ∈ Zˆk. There is a bijection between Pk(λ) and ST k(λ).

0 1 1 k− 1 k Proof. Let λ ∈ Zˆk. Given a path P = (λ , λ 2 , λ , . . . , λ 2 , λ ) ∈ Pk(λ) we will recursively deﬁne a sequence

T0,T 1 ,T1,...,T 1 ,Tk 2 k− 2

j where Tj is a shell tableau of length j and shape λ , and T 1 is a shell tableau of length j + 1 and j+ 2 j+ 1 shape λ 2 . Let T0 be the empty shell tableau of length 0.

j+ 1 j (1) If λ 2 = λ , we deﬁne

1 2 j+1 Tj+ 1 = (ςj+ 1 , ςj+ 1 , . . . , ς 1 ), 2 2 2 j+ 2

where   ςr if r < j + 1, r  j ς 1 = j+ 2  (n, n; |Tj| + 1) if r = j + 1.

j+ 1 j (2) If λ 2 6= λ , suppose t t [ [ {is _ ls} ∪ {is ^ ls+1} s=1 s=1 42

r r+ 1 j is the shell created by the symmetric diﬀerence of λ and λ 2 . Since sh(Tj) = λ , for

rs 1 ≤ s ≤ t,(is, ls; as) is an arc with maximal label as ≤ |Tj| in a diagram ςj of Tj. Let

1 2 j+1 Tj+ 1 = (ςj+ 1 , ςj+ 1 , . . . , ς 1 ), 2 2 2 j+ 2

where   ςr if r 6= r for any s,  j s  r r ς 1 = ς + (is, ls+1; |Tj| + s) if r = rs for some s, j+ 2 j    (n, n; |Tj| + t + 1) if r = j + 1.

j+1 j+ 1 (3) If λ = λ 2 , deﬁne Tj+1 = T 1 . j+ 2

j+1 j+ 1 (4) If λ 6= λ 2 , suppose t t−1 [ [ {is _ ls} ∪ {is ^ ls+1} s=1 s=1

r r+ 1 j+ 1 is the shell created by the symmetric diﬀerence of λ and λ 2 . Since sh(T 1 ) = λ 2 j+ 2

then for 1 ≤ s ≤ t − 1, (is, ls+1; as) is an arc with maximal label as ≤ |T 1 | in a distinct j+ 2

rs diagram ς 1 in Tj+ 1 . We deﬁne j+ 2 2

1 2 j+1 Tj+1 = (ςj+1, ςj+1, . . . , ςj+1 ),

where   ςr if r 6= r for any s,  j+ 1 s  2  r r ςj+1 = ς + (i , l ; |T 1 | + s) if r = r for some s, j+ 1 s+1 s+1 j+ s  2 2   (i1, l1, |T 1 |) if r = j + 1. j+ 2

1 1 In the above construction, we have T 1 = (ς 1 ) where ς 1 = {(n _ n; 1)} is a shell tableau of 2 2 2 j length 1 and shape ∅. If Tj is a shell tableau of length j and shape λ , then T 1 has length j + 1 j+ 2 and

j j+ 1 j+ 1 j j+ 1 sh(T 1 ) = (λ ∩ λ 2 ) ∪ (λ 2 − λ ) = λ 2 . j+ 2

It is straightforward to check that T 1 satisﬁes conditions 1–4. Since Tj is a shell tableau, it suﬃces j+ 2

rs to prove condition 5 for the arcs (is, ls+1; |Tj| + s). For s > 1, consider (is, ls; as) ∈ ς 1 . Then j+ 2 43

rs−1 rs (is−1, ls; |Tj| + s − 1) lies in ς 1 where |Tj| + s − 1 > as is minimal, and (is, ls+1; |Tj| + s) ∈ ς 1 . j+ 2 j+ 2

Thus condition 5 holds, so T 1 is in fact a shell tableau. A similar argument can be used to verify j+ 2 j each Tj is a shell tableau of length j and shape λ .

For λ ∈ Zˆk, deﬁne

ϕ : Pk(λ) −! ST k(λ)

P 7! Tk. The map ϕ is bijective since the construction of the sequence of shell tableaux can be reversed

1 2 k as follows. Given a shell tableau T = (ς , ς , . . . , ς ) of shape λ, let Tk = T .

j (1) If ς = {(n _ n; a)}, deﬁne T 1 = Tj. j− 2

j (2) If (i _ n; a) ∈ ς for i < n, let (i1, l1; a1), (i2, l2; a2),..., (it, lt; at) be the arcs in Tj with

rs is _ ls ∈ ςj and as ≥ a. We deﬁne

1 2 j Tj− 1 = (ςj− 1 , ςj− 1 , . . . , ς 1 ), 2 2 2 j− 2

where   ςr if r 6= r for any s,  j s  r r ς 1 = ς − (is, ls; as) if r = rs for some s, j− 2 j    (n, n, |Tj| − t + 1) if r = j.

j 1 2 j−1 r r (3) If ς = {(n _ n; a)}, deﬁne Tj = (ςj , ςj , . . . , ςj ) where ςj = ς 1 . j+ 2

j (4) If (i _ n; a) ∈ ς for i < n, let (i1, l1; a1), (i2, l2; a2),..., (it, lt; at) be the arcs in Tj with

rs is _ ls ∈ ςj and as > a. We deﬁne

1 2 j−1 Tj = (ςj , ςj , . . . , ςj ),

where  r  ς 1 if r 6= rs for any s, r j+ 2 ςj =  r  ς 1 − (is, ls; as) if r = rs for some s. j+ 2 44

Therefore the inverse of ϕ is

−1 ϕ : ST k(λ) −! Pk(λ)

T 7! P = (sh(T1), sh(T2),..., sh(Tk)).

Example 4.1.5. For the path

! P = ∅, ∅, , , , , , ,

the sequence of shell tableaux is

T0 = ( )

1 ! T 1 = 2

1 ! T1 =

1 3 ! T 1 = , 1 2 2

1 3 ! , T2 = 2

1 3 5 ! T 1 = , , 2 2 2 4

1 3 5 ! , 6 T3 = , 2 4

1 3 5 8 ! 6 T 1 = , , , 3 2 7 2 4

1 3 5 8! , 6 T4 = , , . 7 2 4 45

Note that each shell ςr keeps track of the arc introduced at the rth row of the Bratteli diagram from inducing ResUn (V ⊗r−1). Un−1 When q = 2, then q − 1 = 1 so that many of the edges in the Bratteli diagram have weight

1. In this case, we can account for the weights of paths in the Bratteli diagram by removing the second condition in the deﬁnition of a strict labeling. A semi-strict shell tableau is a shell tableau where we allow τ(j, m) = τ(i, l)+1 for every pair of arcs (i, l; τ(i, l)) and (j, m; τ(j, m)) in a labeled shell with dim(i, l) > dim(j, m) and i = j or k = l. This is reminiscent of semi-standard Young tableaux where we allow the entries along the rows to be weakly increasing.

Suppose SST k(λ) is the set of semi-strict shell tableaux of length k and shape λ. Recall the sum of the weights of paths to λ is the multiplicity of χλ in V k. When q = 2, this is the number of semi-strict shell tableaux.

Proposition 4.1.6. Let q = 2 and λ ∈ Zˆk. Then

X w(P ) = |SST k(λ)|.

P ∈Pk(λ)

1 k Proof. Let q = 2 and λ ∈ Zˆk. Let T = (ς , . . . , ς ) ∈ ST k(λ) be the shell tableau corresponding

0 1 k− 1 k to the path P = (λ , λ 2 , . . . , λ 2 , λ ) via the bijection in the proof of Theorem 4.1.4. Since the weight of P is the product of its edge labels, it suﬃces to consider the label of a single edge. Recall

1 r r+ 2 1 the label of an edge (λ , r) ! (λ , r + 2 ) in P is

1 1 crs(λr∩λr+ 2 ,λr−λr+ 2 ) 1 2 r r+ 2 1 m((λ , r) ! (λ , r + 2 )) = 1 1 . 2crs(λr∩λr+ 2 ,λr+ 2 −λr)

Suppose t t [ [ {is _ ls} ∪ {is ^ ls+1} s=1 s=1 1 r r+ rs is the shell created by the symmetric diﬀerence of λ and λ 2 where (is, ls+1; as) ∈ ς . For

1 ≤ s ≤ t, deﬁne the set   r r+ 1 r r+ 1  (j _ m, is _ ls) ∈ CRS(λ ∩ λ 2 , λ − λ 2 ), 

Ys = m . r r+ 1 r+ 1 r  (j ^ m, is ^ ls+1) ∈/ CRS(λ ∩ λ 2 , λ 2 − λ )  46

Note that t X r r+ 1 r r+ 1 r r+ 1 r+ 1 r |Ys| = crs(λ ∩ λ 2 , λ − λ 2 ) − crs(λ ∩ λ 2 , λ 2 − λ ). s=1 rs For each subset Xs ⊆ Ys, add the arcs (is, l; bl) ∈ ς for l ∈ Xs. There is a unique relabeling

Pk r Pt of the arcs with a distinct label in {1, 2,..., r=1 |ς |+ s=1 |Xs|} so that the order of the labels of the original arcs in T is preserved, and every pair of arcs (i, l; τ(i, l)) and (j, m; τ(j, m)) in a labeled shell with dim(i, l) > dim(j, m) satisﬁes τ(i, l) < τ(j, m). Then each (X1,...,Xt) determines one

Pt |Y | of the 2 s=1 s semi-strict shell tableaux.

Example 4.1.7. Consider the path P from Example 4.1.5, ! P = ∅, ∅, , , , , , , ,

and corresponding tableaux

1 3 5 8! 6 T = , , , . 7 2 4

1 3 3 2 1 The path P has weight 2 since m((λ , 3) ! (λ , 3 2 )) = 2. The shell created by the symmetric

3 3 1 diﬀerence between λ and λ 2 is

3 3 1 3 3 1 and the set Y1 = {4} as (1 _ 4, 2 _ 6) ∈ CRS(λ ∩ λ 2 , λ − λ 2 ), but (1 ^ 4, 2 ^ 3) ∈/

3 3 1 3 1 3 CRS(λ ∩ λ 2 , λ 2 − λ ). The two semi-strict tableaux corresponding to ∅ and Y1 are

1 3 5 8 ! 6 T = , , , 7 2 4 and 1 3 5 9 ! ˜ , 6 T = , 8 , 7 2 4 respectively. Chapter 5

Schur–Weyl dualities

Schur–Weyl duality is a fundamental framework in combinatorial representation theory [14].

Classically it relates the irreducible representations of the symmetric group Sn to the irreducible representations of the general linear group GLn(C) via their commuting actions. More recently, the study of new versions of Schur–Weyl duality has led to many remarkable discoveries about algebras of operators on tensor space that are full centralizers of each other [10, 14, 17].

5.1 Classical Schur–Weyl duality

First, we review the classical Schur–Weyl duality of GLn(C) and Sk. Let V be an n dimensional complex vector space. Consider the tensor space

V ⊗k = V ⊗ · · · ⊗ V . | {z } k factors

⊗k ⊗k The general linear group GLn(C) acts on V diagonally, and the symmetric group Sk acts on V by permuting factors. Schur–Weyl duality gives rise to the following two statements.

(a) These actions commute and each action generates the full centralizer of the other. That is,

if the representations arising from each action are

ρ ⊗k π CGLn(C) −! End(V ) − CSk

then the images of ρ and π are

⊗k ⊗k ρ(CGLn(C)) = EndSk (V ) and π(CSk) = EndGLn(C)(V ). 48

(b) This double-centralizer relationship in Theorem 2.1.7 produces

⊗k ∼ M λ λ V = G ⊗ Sk as a (GLn(C),Sk)-bimodule, λ ` k

λ λ where the G are irreducible GLn(C)-modules and the Sk are irreducible Sk-modules.

⊗k The bimodule decomposition of V makes studying the irreducible representations of GLn(C) and the irreducible representations of Sk two sides of the same coin [14].

5.2 The partition algebra

There are many groups besides GLn(C) and Sk that play analogous Schur–Weyl duality roles.

We turn to a version of Schur–Weyl duality obtained by restricting the action of GLn(C) to one of its subgroups.

⊗k Viewing Sn ⊆ GLn as the subgroup of permutation matrices, Sn acts on V diagonally

CAk(n) ⊆

⊗k GLn(C) −! V − CSk ⊆

⊗k and its corresponding centralizer algebra EndSn (V ) is an algebra described in terms of set partitions of {1, 2,..., 2k} known as the partition algebra CAk(n) [14].

⊗k The decomposition of V as a (Sn,Ak(n))-bimodule is

⊗k ∼ M λ λ V = Sn ⊗ Ak(n) ˆ λ ` Ak(n)

λ λ where the Sn are irreducible Sn-modules and the Ak(n) are irreducible Ak(n)-modules. As an

Sn-module, V is the permutation representation so that

V =∼ IndSn ResSn (1), Sn−1 Sn−1 49 where 1 is the trivial character of Sn [14, (3.16)]. In general, for any Sn-module M,

IndSn ResSn (M) =∼ S ⊗ ResSn (M) Sn−1 Sn−1 C n CSn−1 Sn−1 ∼ 1 = (CSn ⊗CSn−1 ) ⊗ M (5.1)

=∼ V ⊗ M by the deﬁnition of induction and the tensor identity (5.1) from [14, (3.18)]. Iterating this identity, it follows that

V ⊗k =∼ (IndSn ResSn )k(1) Sn−1 Sn−1 | {z } k times where the trivial character is restricted and induced k times.

5.3 A unipotent analogue of Schur–Weyl duality

We examine the analogue of Schur–Weyl duality for the group Un of unipotent upper- 1 1 triangular matrices. Let V = CUn ⊗CUn−1 where is the trivial supercharacter. By the deﬁnition of induction, V is given by

V = U ⊗ ResUn (1) = IndUn ResUn (1). C n CUn−1 Un−1 Un−1 Un−1

More broadly, we have the following generalization of the tensor identity (5.1) from [14, (3.18)].

Lemma 5.3.1. Let H be a subgroup of a group G. For a G-module M, the map

G 1 τ : CG ⊗CH ResH (M) −! (CG ⊗CH ) ⊗ M

g ⊗ m 7! (g ⊗ 1) ⊗ gm

g ⊗ g−1m (g ⊗ 1) ⊗ m [ is a G-module isomorphism.

Iterating this identity, we obtain

V ⊗k =∼ (IndUn ResUn )k(1), Un−1 Un−1 | {z } k times 50 where the trivial supercharacter is restricted and induced k times. This is reminiscent of the situation in the partition algebra from Section 5.2 where the permutation representation of the symmetric group is isomorphic to restricting and then inducing the trivial character.

⊗k The group Un acts diagonally on V and the goal of Schur–Weyl duality is to determine

⊗k the centralizer algebra EndUn (V ) generated by this action. By double centralizer theory, the decomposition of the centralizer algebra into irreducibles can be obtained by decomposing the

⊗k ⊗k tensor product V into irreducible Un-modules. However, we generally cannot decompose V into irreducible Un-modules as the representation theory of Un is well known to be “wild” [13].

Therefore, we approximate with supercharacters and consider the following centralizer subalgebras

˜ ⊗k λ ∼ λ λ Zk,n = {ϕ ∈ EndUn (V ) | for all (λ, k) ∈ Λ(n), ϕ(Un ) = Un or ϕ(Un ) = 0} ⊆

λ λ Z = {ϕ ∈ Z˜ | if ϕ(U ) = U , (λ, k) ∈ Λ(n), then ϕ| λ = aId, a ∈ }. k,n k,n n n Un C where Λ(n) is the Bratteli diagram.

Note that the ﬁrst subalgebra Z˜k,n distinguishes supermodules so that the supermodules satisfy the ﬁrst statement of Schur’s lemma 2.1.1. While this subalgebra may be interesting to study, we use both statements of Schur’s lemma in the next section to produce a bimodule decomposition

⊗k of V . Hence we focus on Schur–Weyl duality for the second subalgebra Zk,n, which distinguishes supermodules and treats them like irreducibles. That is, we seek to determine the centralizer

⊗k subalgebra Zk,n of the action of Un on V .

If all the supercharacters in the decomposition of V ⊗k are irreducible characters then we have

˜ ⊗k the equalities Zk,n = Zk,n = EndUn (V ) so that the centralizer subalgebras are in fact the full centralizer algebra. By Propositions 2.1.3 and 2.3.2, a supercharacter χλ is irreducible if and only if

hχλ, χλi = (q − 1)|λ|qcrs(λ,λ) = 1.

Thus the supercharacters coincide with irreducible characters when q = 2, and n ≤ 3 or k < 2.

However, the containment of subalgebras can be strict. To see this, suppose χλ is a supercharacter in the decomposition of V ⊗k that is not irreducible. Then χλ is a sum of at least two 51 irreducible characters. Projection onto one of these constituents is in the full centralizer algebra

⊗k ˜ λ EndUn (V ), but not the subalgebra Zk,n. Meanwhile, if χ has at least two isomorphic con-

λ stituents then permutation of the isomorphic constituents of χ is in the subalgebra Z˜k,n, but not

Zk,n.

5.3.1 Dimensions of centralizer subalgebras

⊗k We now show decomposing V as a Un-supermodule produces a decomposition of the centralizer subalgebra Zk,n using a supermodule analogue of double centralizer theory.

Theorem 5.3.2 (Double Centralizer Theorem). Suppose the decomposition of V ⊗k is given by

⊗k ∼ M λ V = mλUn . (λ,k)∈Λ(n) Then we obtain the decompositions

∼ M (a) Zk,n = Mmλ (C); (λ,k)∈Λ(n)

(b) as a (Un,Zk,n)-bimodule ⊗k ∼ M λ λ V = Un ⊗ Zk,n (λ,k)∈Λ(n) λ where the Z are irreducible Zk,n-modules.

Proof. We follow the proof of the Centralizer Theorem in [14]. Index the components of the

⊗k λ decomposition of V by dummy variables i so that

mλ ⊗k ∼ M M λ λ V = Un ⊗ i . (λ,k)∈Λ(n) i=1 This implies

⊗k ⊗k ∼ M M λ λ M M µ λ Zk,n ⊆ HomUn (V ,V ) = HomUn Un ⊗ j , Un ⊗ i . λ j µ i λ λ µ λ λ λ ∼ µ µ If ϕ ∈ Zk,n satisﬁes ϕ| λ λ ∈ HomUn (U ⊗ ,Un ⊗ ), then ϕ(U ⊗ ) = Un ⊗ , so λ = µ or Un ⊗j n j i n j i λ λ ϕ(Un ⊗ j ) = 0. Thus, we have the isomorphism

∼ M M λ λ µ µ Zk,n = δλµ HomUn (Un ⊗ j ,Un ⊗ i ). λ,µ i,j 52

λ λ λ For each (λ, k) ∈ Λ(n) and 1 ≤ i, j ≤ mλ, let πij : Un ⊗ j ! Un ⊗ i be the Un-supermodule isomorphism given by

λ λ λ λ πij(u ⊗ j ) = u ⊗ i , u ∈ Un .

λ λ λ λ λ Suppose π, ϕ ∈ Zk,n are such that π| λ λ = π and ϕ| λ λ : U ⊗ ! U ⊗ is a Un- Un ⊗j ij Un ⊗j n j n i −1 −1 λ λ λ λ supermodule isomorphism. Then ϕ ◦ π ∈ Zk,n with ϕ ◦ π (U ⊗ ) = U ⊗ . Thus ϕ| λ λ ◦ n i n i Un ⊗j λ −1 λ −1 λ λ (π ) is a multiple of the identity and ϕ| λ λ = ϕ| λ λ ◦ (π ) π is a multiple of π . Hence ij Un ⊗j Un ⊗j ij ij ij this yields mλ ∼ M M λ Zk,n = Cπij λ i,j=1 λ so the πij are a basis for the centralizer algebra. Therefore each element ϕ ∈ Zk,n can be written as mλ X X λ λ λ ϕ = zijπij zij ∈ C, (λ,k)∈Λ(n) i,j=1 and identiﬁed with an element of ⊕λMmλ (C) by mapping

mλ X X λ λ ϕ 7! zijeij, (λ,k)∈Λ(n) i,j=1

λ λ µ λ where eij is the matrix with a 1 in the (i, j) entry of the λth diagonal block. Since πijπkl = δλµδjkπil, then this mapping is a homomorphism so that we have

∼ M Zk,n = Mmλ (C). (λ,k)∈Λ(n)

µ µ L As a vector space, Zk,n = C-span{i | 1 ≤ i ≤ mµ} is isomorphic to the irreducible λ Mmλ (C)-

⊗k module of column vectors of length mµ. It follows that V is given by

⊗k ∼ M λ λ V = Un ⊗ Zk,n (λ,k)∈Λ(n) as a (Un,Zk,n)-bimodule with the action

λ µ µ µ (g ⊗ πij)(u ⊗ k ) = δλµδjk(gu ⊗ i ) u ∈ Un , g ∈ Un. 53

⊗k The Bratteli diagram Λ(n) encodes the decomposition of V as a Un-supermodule so it yields a decomposition of the centralizer subalgebra Zk,n. By construction, the sum of the weights

λ of paths to λ in row k is the multiplicity mλ of each Un-supermodule Un . Since

⊗k ∼ M λ V = mλUn as a Un-supermodule (λ,k)∈Λ(n) ∼ M λ λ = Un ⊗ Zk,n as a (Un,Zk,n)-bimodule (λ,k)∈ Λ(n) by the supermodule version of the Double Centralizer theorem 5.3.2, it follows that the dimension

λ λ of Zk,n is mλ. Therefore, the dimension of Zk,n is

λ X dim Zk,n = mλ = wt(P )

P ∈Pk(λ) where Pk(λ) is the set of paths in Λ(n) to λ in row k. By part (a) of the Double Centralizer ∼ L Theorem Zk,n = (λ,k)∈Λ(n) Mmλ (C), so the dimension of the centralizer subalgebra is

X λ 2 dim Zk,n = (dim Zk,n) (λ,k)∈Λ(n) X 2 = mλ (λ,k)∈Λ(n) X X 2 X X = wt(P ) = wt(P ) wt(Q).

(λ,k)∈Λ(n) P ∈Pk(λ) (λ,k)∈Λ(n) P,Q∈Pk(λ)

Thus we can calculate the dimension of the centralizer subalgebra from the Bratteli diagram.

Theorem 5.3.3. The dimension of the centralizer subalgebra Zk,n is a polynomial in q.

Proof. By deﬁnition, the weight of a path is the product of its edge labels, which are products of

a b powers of (q −1) and q. Thus for any path P ∈ Pk(λ), wt(P ) = (q −1) P q P for some aP , bP ∈ Z≥0. It follows that

X X aP bP aQ bQ dim Zk,n = (q − 1) q (q − 1) q

(λ,k)∈Λ(n) P,Q∈Pk(λ) is a polynomial in q. 54

Some dimensions of centralizer subalgebras Zk,n for q = 2 are shown below.

1 1 k = 1 k = 1 2 k = 2 k = 2 2 k = 3 n = 2 2 4 8 16 32

n = 3 3 10 36 136 528

n = 4 4 19 105 676 4600

n = 5 5 31 235 2257 24125

⊗k When the centralizer subalgebra Zk,n is the full centralizer algebra EndUn (V ), we have nice formulas for its dimension.

Proposition 5.3.4. For q = 2, dim End (ResUn (V )) = 3n(n − 1)/2 + 1, the nth centered Un−1 Un−1 triangular number (sequence A005448 in OEIS).

Proof. Let q = 2. By the branching rules 3.2.5, the restriction ResUn (V ) is Un−1

X ResUn (V ) = n1 + χi_k Un−1 1≤i

Un 2 X 3n(n − 1) dim EndU (Res (V )) = n + 1 = + 1. n−1 Un−1 2 1≤i

Proposition 5.3.5. For q = 2, dim End (V ⊗k) = s(2k − 1) and dim End (ResU3 (V ⊗k)) = s(2k) U3 U2 U2 where s(n) = 2n−1(1 + 2n) (sequence A007582 in OEIS).

Proof. Let q = 2 and n = 3. By the branching rules 3.1.10, V ⊗k is given by

V ⊗k =∼ 2k−2(2k−1 + 1)1 + 2k−2(2k−1 + 1)χ2_3 + 22k−2χ1_3

+ 2k−2(2k−1 − 1)χ1_2 + 2k−2(2k−1 − 1)χ1_2_3, and so the dimension of the centralizer algebra is

⊗k k−2 k−1 2 2k−2 2 k−2 k−1 2 2k−2 2k−1 dim EndU3 (V ) = 2(2 (2 + 1)) + (2 ) + 2(2 (2 − 1)) = 2 (1 + 2 ). 55

Similarly, the restriction of V ⊗k is

ResU3 (V ⊗k) =∼ 2k−1(2k + 1)1 + 2k−1(2k − 1)χ1_2, U2 which implies

dim End (ResU3 V ⊗k) = (2k−1(2k + 1))2 + (2k−1(2k − 1))2 = 22k−1(1 + 22k). U2 U2

5.3.2 The path basis

Since summing the product of the weights of pairs of paths in the Bratteli diagram gives the dimension of the centralizer subalgebra, these paths also help index a basis. In this section we will explicitly construct part of this basis of the centralizer subalgebra.

Recall, from the proof of the supermodule version of the Double Centralizer Theorem 5.3.2 that if mλ ⊗k ∼ M M λ λ V = Un ⊗ i (λ,k)∈Λ(n) i=1 a basis for Zk,n is given by the maps

λ λ λ λ λ πij : Un ⊗ j ! Un ⊗ i .

λ ⊗k We seek to determine each copy of Un ⊗i inside V in order to provide explicit formulas for these maps.

More precisely, since

⊗k 1 ⊗k 1 ⊗k ∼ n−1 ⊗k V = (CUn ⊗CUn−1 ) = (CUn/CUn−1 ⊗CUn−1 ) = (CFq ) , a standard basis for V ⊗k is

n−1 {v1 ⊗ v2 ⊗ · · · ⊗ vk | v1, v2, . . . , vk ∈ Fq }.

λ We work towards expressing Un ⊗ i as

λ ∼ λ,i λ,i λ,i Un ⊗ i = -span{u , u , . . . , u λ } C 1 2 dim Un 56 where X uλ,i = ds v ⊗ v ⊗ · · · ⊗ v , and ds ∈ . s (v1,...,vk) 1 2 k (v1,...,vk) C n−1 k (v1,...,vk)∈(Fq )

λ,i −1 By writing each basis element us in terms of the standard basis we obtain the maps σ and σ as shown below.

V ⊗k V ⊗k σ−1 X X λ,j λ,j cs us vj ⊗ · · · ⊗ vj 1 k (λ,k)∈Λ(n) 1≤j≤mλ λ 1≤s≤dim Un

λ πij

V ⊗k V ⊗k dim U λ λ n dim Un X X λ,j s σ X c d vi ⊗ · · · ⊗ vi λ,j λ,i s (i1,...,ik) 1 k cs us s=1 n−1 k (v1,...,vk)∈(Fq ) s=1

This induces projection maps of V ⊗k that form a basis for the centralizer subalgebra and act on the standard basis of V ⊗k. While determining these maps appears to be generally unattainable, such as in the classical or partition algebra versions of Schur–Weyl duality, it seems within reach in this case.

5.3.3 Decomposing V ⊗k

⊗k i_n We start decomposing V by considering the modules Un inside of V . From the con-

i_n struction of the supercharacters of Un in Section 2.3.2, for 1 ≤ i ≤ n, the supermodule Un is

i_n ∼ ∼ n X ×o Un = C-span aen,i | a ∈ Bn = C-span ajej,i | ai+1, . . . , an−1 ∈ Fq, an ∈ Fq . i

n_n ∅ 1_3 Note that Un corresponds to the trivial module Un . For example, the module U3 is given by 57

  

 0 0 0        1_3 ∼   × U3 = C-span a 0 0 a ∈ Fq, b ∈ Fq .          b 0 0 

We embed Un−1 into Un by

ε : Un−1 ,−! Un     1 ∗ · · · ∗ 0   1 ∗ · · · ∗      . .  . 0 1 . .  .   0 1 . . .    7! . .. ∗ 0 . .    . .. ∗         0 ··· 0 1 0   0 ··· 0 1   0 ··· 0 0 1 so that

Un−1 = {u ∈ Un | (u − 1)ij 6= 0 implies i < j < n}.

n_n When restricting to Un−1, the trivial supermodule Un is itself and we denote the restriction as

ResUn (U n_n) = U n_n. For i < n, the supermodule U i_n decomposes as Un−1 n n−1 n

M ResUn (U i_n) = (q − 1) U i_m Un−1 n n−1 i≤m

i_i by Corollary 3.1.12, where Un−1 corresponds to the trivial supermodule. The following proposition explicitly determines these supermodules inside the restricted supermodule.

× Proposition 5.3.6. Fix an ∈ Fq . The map

R : U i_m , ResUn U i_n n−1 −! Un−1 n P P i

× Proof. Fix an ∈ Fq . For g ∈ Un−1, we have

X X X R g ajej,i = R ϑ( ajgi,j) akgj,kej,i i

X X X R g ajej,i = ϑ( ajgi,j) anen,i + akgj,kej,i i

+ × where ε is the embedding of Un−1 into Un and ϑ : Fq ! C is a nontrivial homomorphism.

It follows that each a ∈ × speciﬁes a copy of the supermodule U i_m ⊆ ResUn U i_n given by n Fq n−1 Un−1 n

n X ×o C-span anen,i + ajej,i | ai+1, . . . , am−1 ∈ Fq, am, an ∈ Fq . i

1_3 1_3 1_3 For instance, the copy of U3 ⊗ 1 indexed by the dummy variable 1 corresponding to the

× element 1 ∈ Fq is      0 0 0 0          a 0 0 0  1_3   × U4 1_4 R(U ) = -span a ∈ q, b ∈ ⊆ Res (U ). 3 C   F Fq U3 4    b 0 0 0         1 0 0 0    By the branching rules 3.2.5, V is given by

M V = IndUn ResUn (1) =∼ U i_n. Un−1 Un−1 n i≤n

i_n We construct this isomorphism in order to realize the supermodules Un inside V . For vectors 59

s t s t u = (u1, . . . , us) ∈ Fq and v = (v1, . . . , vt) ∈ Fq deﬁne the stack function stk : Fq × Fq ! Us+t+1 as   1 0 0 0 0 ··· u1    . . . .   ......   . .       1 0 0 ··· u   s   stk(u, v) =   .  1 0 ··· v1    . . .   .. .. .   .       1 v   t    1

As an example, for q = 5 stacking the vectors (2, 3) and (4) yields   1 0 0 2       0 1 0 3 stk((2, 3), (4)) =   .   0 0 1 4     0 0 0 1

+ × Proposition 5.3.7. Let ϑ : Fq ! C be a nontrivial homomorphism. The map

L i_n 1 I : i≤n Un −! CUn ⊗CUn−1 X 0 7! stk(u, ()) ⊗ 1 n−1 u∈Fq X X ajej,i 7! ϑ(anui) stk(u, (ai+1, . . . , an−1)) ⊗ 1 i i

+ × Proof. Fix a nontrivial homomorphism ϑ : Fq ! C . Let g ∈ Un. Then the image of g0 is

X X I(g0) = I(0) = stk(u, ()) ⊗ 1 = g stk(u, ()) ⊗ 1 = gI(0), n−1 n−1 u∈Fq u∈Fq and for i < n, we have

X X X I g ajej,i = I ϑ( ajgi,j) akgj,kej,i i

X X ϑ( ajgi,j)I akgj,kej,i i

1_3 Example 5.3.8. When q = 2, U3 is given by      0 0 0 0 0 0          1_3 ∼     U3 = C-span 0 0 0, 1 0 0          1 0 0 1 0 0  and           1 0 0 1 0 1 1 0 0 1 0 1                    1_3 ∼   1   1   1   1 I(U3 ) = C-span 0 1 0 ⊗ − 0 1 0 ⊗ , 0 1 1 ⊗ − 0 1 1 ⊗ .                    0 0 1 0 0 1 0 0 1 0 0 1 

Now that we have decomposed V as a direct sum of Un-supermodules, we work towards

⊗k decomposing V as a direct sum of Un-supermodules. The branching rules specify the coeﬃcients mλ for which M V ⊗k =∼ (IndUn ResUn )k(1) =∼ m U λ Un−1 Un−1 λ n λ`[n]

λ ⊗k so we aim to determine each copy of Un inside V .

Using Lemma 5.3.1 we construct a map from V ⊗ InfUn ResUn (V ⊗k−1) to V ⊗k. Let Un−1 Un−1

U[i]×n = {u ∈ Un | (u − 1)j,k 6= 0 implies 1 ≤ j ≤ i, k = n}. 61

∼ The group U[n]×n is a normal subgroup of Un with Un/U[n]×n = Un−1 so that

Un λ λ 1 X Inf (Un−1) = CzU ⊗Un−1 Un−1, where zU = u. Un−1 [n]×n [n]×n qn−1 u∈U[n]×n

1 ⊗k−1 Proposition 5.3.9. For a Un−1-module M ⊆ (CUn ⊗CUn−1 ) , the map

Ψ:( U ⊗ 1) ⊗ InfUn (M) , ( U ⊗ 1)⊗k C n CUn−1 Un−1 −! C n CUn−1 1 1 (g ⊗ ) ⊗ (zU[n]×n ⊗ m) 7! (g ⊗ ) ⊗ gm is an injective homomorphism.

Proof. Applying Lemma 5.3.1, gives

1 τ −1 (CUn ⊗CUn−1 ) ⊗ (CzU[n]×n ⊗CUn−1 M) −! CUn ⊗CUn−1 CzU[n]×n ⊗CUn−1 M

1 −1 (g ⊗ ) ⊗ (zU[n]×n ⊗ m) 7! g ⊗ g zU[n]×n ⊗ m.

1 ∼ 1 Since CUn ⊗CUn−1 = CUn/CUn−1 ⊗CUn−1 , we can assume g ∈ U[n]×n to obtain

−1 g ⊗ g zU[n]×n ⊗ m = g ⊗ zU[n]×n ⊗ m.

In the tensor product CUn ⊗CUn−1 CzU[n]×n ⊗CUn−1 M, any h ∈ Un−1 passes through CzU[n]×n as

h ⊗ zU[n]×n ⊗ m = 1 ⊗ h zU[n]×n ⊗ m = 1 ⊗ zU[n]×n h ⊗ 1 = 1 ⊗ zU[n]×n ⊗ hm.

This implies there is an isomorphism φ : CUn ⊗CUn−1 CzU[n]×n ⊗CUn−1 M ! CUn ⊗CUn−1 M given by removing zU[n]×n . Therefore, we have

φ CUn ⊗CUn−1 CzU[n]×n ⊗CUn−1 M −! CUn ⊗CUn−1 M

g ⊗ zU[n]×n ⊗ m 7! g ⊗ m

ι U ⊗ ResUn (( U ⊗ 1)⊗k−1) −! C n CUn−1 Un−1 C n CUn−1

7! g ⊗ m

τ 1 1 ⊗k−1 −! (CUn ⊗CUn−1 ) ⊗ (CUn ⊗CUn−1 )

7! (g ⊗ 1) ⊗ gm, 62 where ι is the inclusion and τ is the map in Lemma 5.3.1. Hence Ψ = τ ◦ ι ◦ φ ◦ τ −1 is an injective homomorphism.

Using Proposition 5.3.9 we can start to recursively build up the decomposition of V ⊗k as a

Un-supermodule in some cases. The following theorem gives a copy of certain Un−1 submodules

M ⊆ ResUn−1 (V ⊗k−1). As a corollary we obtain a copy of U i_n ⊗ InfUn−1 (M) in V ⊗k. Un n Un

Theorem 5.3.10. Let (i1 _ l1, i2, _ l2, . . . ik _ lk) be a sequence such that 1 ≤ ir ≤ lr ≤ n, and

k lr = n if and only if ir = n. Let (b1, b2, . . . , bk) ∈ Fq where br = 0 if and only if ir = n. The supermodule k O U ir_lr ⊗ ir_lr ⊆ ResUn ( U ⊗ 1)⊗k n−1 br Un−1 C n CUn−1 r=1 indexed by the dummy variables i1_l1 , . . . , ik_lk is isomorphic to b1 bk

 0   n−ir−1   ar ∈ Fq with   k r !   X O Y  -span ϑ(b (u ) 0 ) stk(u , a ) ⊗ 1 C r r ir s s (ar)j 6= 0 if j = lr − ir  0 i0 r=1 s=1   i1 k  (u1,...,uk)∈Fq ×···×Fq   (ar)j = 0 if lr − ir < j < n 

0 where (ar)j denotes the jth entry of the rth vector and ir = min{ir, n − 1}.

Proof. We induct on k. If k = 1, consider the copy of U i1_l1 ⊆ ResUn (U i1_n) indexed by the n−1 Un−1 n dummy variable i1_l1 for some b ∈ . If i = l = n and b = 0, we have b1 1 Fq 1 1 1 ( ) X U i1_l1 ⊗ i1_l1 =∼ -span ϑ(b (u ) ) stk(u , ()) ⊗ 1 n−1 b1 C 1 1 i1 1 n−1 u1∈Fq

× by Proposition 5.3.7. For 1 ≤ i1 ≤ l1 < n and b1 ∈ Fq , we have  

 a ∈ n−i1−1 with   1 Fq    i1_l1 i1_l1 ∼ X U ⊗ = -span ϑ(b1(u1)i ) stk(u1, a1) ⊗ 1 n−1 b1 C 1 (a1)j 6= 0 if j = l1 − i1  i1  u1∈Fq     (a1)j = 0 if l1 − i1 < j < n  by Propositions 5.3.6 and 5.3.7.

Let (i1 _ l1, i2, _ l2, . . . ik _ lk) be a sequence such that 1 ≤ ir ≤ lr ≤ n, and lr = n if

k and only if ir = n. Let (b1, b2, . . . , bk) ∈ Fq where br = 0 if and only if ir = n. By the inductive 63 hypothesis the tensor product

k O U ir_lr ⊗ ir_lr ⊆ ResUn ( U ⊗ 1)⊗k−1 n−1 br Un−1 C n CUn−1 r=2 indexed by the dummy variables i2_l2 , . . . , ik_lk is isomorphic to b2 bk

 0   n−ir−1   (ar)j ∈ Fq with   k r !   X O Y  -span ϑ(b (u ) 0 ) stk(u , a ) ⊗ 1 C r r ir s s (ar)j 6= 0 if j = lr − ir  0 i0 r=2 s=2   i2 k  u∈Fq ×···×Fq   (ar)j = 0 if lr − ir < j < n 

0 where u = (u2, . . . , uk), and ir = min{ir, n − 1}.

i1_n 1 If Un ⊆ CUn ⊗CUn−1 , then for i < n we obtain the isomorphism ( ) X U i1_n ∼ -span ϑ(b (u ) ) stk(u , a ) ⊗ 1 a ∈ n−i1−1, b ∈ × n = C 1 1 i1 1 1 1 Fq 1 Fq i1 u1∈Fq and for b1 = 0, ( ) n_n ∼ X 1 Un = C-span ϑ(b1(u1)i1 ) stk(u1, ()) ⊗ n−1 u1∈Fq by Proposition 5.3.7 . Thus, a basis element of

k O U i1_n ⊗ InfUn U ir_lr ⊗ ir_lr n Un−1 n−1 br r=2 is !! k r X 1 X O Y 1 ϑ(b1(u1)i1 ) stk(u1, a1) ⊗ ⊗ zU[n]×n ⊗ ϑ(br(ur)ir ) stk(us, as) ⊗ i0 i i r=2 s=1 1 u∈ 2 ×···× k u1∈Fq Fq Fq

0 0 n−i1−1 where b1 ∈ Fq with b1 = 0 if and only if i1 = n, i1 = min{i1, n − 1}, and a1 ∈ Fq . Applying 1 ⊗k the map Ψ in Proposition 5.3.9 to send this element to (CUn ⊗CUn−1 ) , we obtain

k r ! X X O Y ϑ(b (u ) 0 ) stk(u , a )⊗1 ⊗ ϑ(b (u ) 0 ) stk(u , a )ϑ(b (u ) 0 ) stk(u , a )⊗1 1 1 i1 1 1 1 1 i1 1 1 r r ir s s i0 0 i0 r=2 s=2 1 i2 k u1∈Fq u∈Fq ×···×Fq

k r ! X O Y 0 1 = ϑ(br(ur)ir ) stk(us, as) ⊗ . 0 i0 r=1 s=1 i1 k (u1,...,uk)∈Fq ×···×Fq 64

Therefore, k O U i1_n ⊗ InfUn U ir_lr ⊗ ir_lr ⊆ ( U ⊗ 1)⊗k n Un−1 n−1 br C n CUn−1 r=2 is isomorphic to      (ar)j ∈ Fq with   k r !   X O Y  -span ϑ(b (u ) 0 ) stk(u , a ) ⊗ 1 C r r ir s s (ar)j 6= 0 if j = lr − ir  0 i0 r=1 s=1   i1 k  (u1,...,uk)∈Fq ×···×Fq   (ar)j = 0 if lr − ir < j < n  such that l1 = n and b1 ∈ Fq with b1 = 0 if and only if i1 = n.

Restricting to Un−1, gives

k O ResUn U i1_n ⊗ InfUn U ir_lr ⊗ i1_l1 Un−1 n Un−1 n−1 b1 r=2 k O =∼ ResUn (U i1_n) ⊗ ResUn InfUn U ir_lr ⊗ ir_lr Un−1 n Un−1 Un−1 n−1 br r=2 k O =∼ ResUn (U i1_n) ⊗ U ir_lr ⊗ ir_lr . Un−1 n n−1 br r=2

Recall by Propositions 5.3.6 and 5.3.7, inside the restricted module ResUn (U i1_n) we have Un−1 n ( ) X U i1_l1 ⊗ i1_l1 =∼ -span ϑ(b (u ) ) stk(u , ()) ⊗ 1 n−1 b1 C 1 1 i1 1 n−1 u1∈Fq

× for i1 = l1 = n and b1 = 0, and for i1 < n and some ﬁxed b1 ∈ Fq the tensor product is  

 a ∈ n−i1−1 with   1 Fq    i1_l1 i1_l1 ∼ X U ⊗ = -span ϑ(b1(u1)i ) stk(u1, a1) ⊗ 1 . n−1 b1 C 1 (a1)j 6= 0 if j = l1 − i1  i1  u1∈Fq     (a1)j = 0 if l1 − i1 < j < n  This gives the inclusion

k k O O ι :(U i1_l1 ⊗ i1_l1 ) ⊗ U ir_lr ⊗ ir_lr , ResUn U i1_n ⊗ InfUn U ir_lr ⊗ ir_lr n−1 b1 n−1 br −! Un−1 n Un−1 n−1 br r=2 r=2 X X ϑ(b (u ) 0 ) stk(u , a ) ⊗ 1 ⊗ v 7! ϑ(b (u ) 0 ) stk(u , a ) ⊗ 1 ⊗ z ⊗ v . 1 1 i1 1 1 1 1 i1 1 1 U[n]×n 0 0 i1 i1 u1∈Fq u1∈Fq 65

Composing the maps ι and Ψ, we obtain

Ψ ResUn U i1_n ⊗ InfUn Nk U ir_lr ⊗ ir_lr ResUn ( U ⊗ 1)⊗k Un−1 n Un−1 r=2 n−1 br Un−1 C n CUn−1

ι Ψ◦ι (U i1_l1 ⊗ i1_l1 ) ⊗ Nk U ir_lr ⊗ ir_lr n−1 b1 r=2 n−1 br shows that the module

k O U ir_lr ⊗ ir_lr ⊆ ResUn ( U ⊗ 1)⊗k n−1 br Un−1 C n CUn−1 r=1 indexed by the dummy variables i1_l1 , . . . , ik_lk is isomorphic to b1 bk

 0   n−ir−1   (ar)j ∈ Fq with   k r !   X O Y  -span ϑ(b (u ) 0 ) stk(u , a ) ⊗ 1 . C r r ir s s (ar)j 6= 0 if j = lr − ir  0 i0 r=1 s=1   i1 k  (u1,...,uk)∈Fq ×···×Fq   (ar)j = 0 if lr − ir < j < n 

In proof of Proposition 5.3.10 we showed the following corollary.

Corollary 5.3.11. Let 1 ≤ i1 ≤ n. Let (i2, _ l2, . . . ik _ lk) be a sequence such that 1 ≤ ir ≤ lr ≤

k n, and lr = n if and only if ir = n. Let (b2, . . . , bk) ∈ Fq where br = 0 if and only if ir = n. The supermodule k O U i1_n ⊗ InfUn U ir_lr ⊗ ir_lr ⊆ ( U ⊗ 1)⊗k n Un−1 n−1 br C n CUn−1 r=2 indexed by the dummy variables i2_l2 , . . . , ir_lr is isomorphic to b2 bk

 0   n−ir−1   ar ∈ Fq with   k r !   X O Y  -span ϑ(b (u ) 0 ) stk(u , a ) ⊗ 1 C r r ir s s (ar)j 6= 0 if j = lr − ir  0 i0 r=1 s=1   i1 k  (u1,...,uk)∈Fq ×···×Fq   (ar)j = 0 if lr − ir < j < n 

0 such that b1 ∈ Fq with b1 = 0 if and only if i1 = n, and ir = min{ir, n − 1}. 66

Example 5.3.12. When q = 2, U 1_2 ⊗ 1_2 is the only copy of U 1_2 ⊆ ResU3 (U 1_3) so we 2 1 2 U2 3 1_2 supress the dummy variables and simply write U2 . Then

U 1_3 ⊗ InfU3 (U 1_2) =∼ -span{v , v } 3 U2 2 C 1 2 where              1 0 0 1 0 0 1 0 0 1 0 1                             1   1   1    1 v1 = 0 1 0 ⊗  ⊗ 0 1 1 ⊗  + 0 1 0 ⊗  ⊗ − 0 1 1 ⊗                            0 0 1 0 0 1 0 0 1 0 0 1                1 0 1 1 0 1 1 0 1 1 0 0                                  1   1    1    1 + − 0 1 0 ⊗  ⊗ 0 1 1 ⊗  + − 0 1 0 ⊗  ⊗ − 0 1 1 ⊗                                0 0 1 0 0 1 0 0 1 0 0 1 and              1 0 0 1 0 0 1 0 0 1 0 1                             1   1   1    1 v2 = 0 1 1 ⊗  ⊗ 0 1 0 ⊗  + 0 1 1 ⊗  ⊗ − 0 1 0 ⊗                            0 0 1 0 0 1 0 0 1 0 0 1                1 0 1 1 0 1 1 0 1 1 0 0                                  1   1    1    1 + − 0 1 1 ⊗  ⊗ 0 1 0 ⊗  + − 0 1 1 ⊗  ⊗ − 0 1 0 ⊗  .                               0 0 1 0 0 1 0 0 1 0 0 1

Restricting to U2, gives

1_1 1_2 ∼ 1_2 1_2 ∼ U2 ⊗ U2 = C-span{v1} and U2 ⊗ U2 = C-span{v2}.

Corollary 5.3.11 gives a copy of tensor products of supermodules inside V ⊗k, which leads to a decomposition of V ⊗k. Recall that V = IndUn ResUn (1) =∼ L U i_n. Thus Un−1 Un−1 i≤n n

M V ⊗2 =∼ (IndUn ResUn )2(1) =∼ IndUn ResUn U i_n Un−1 Un−1 Un−1 Un−1 n i≤n M =∼ IndUn m U i_l Un−1 i_l n−1 i≤l≤n M M =∼ U j_n ⊗ InfUn m U i_l , n Un−1 i_l n−1 j≤n i≤l≤n 67 where   1 if i = n, mi_l =  q − 1 if i < n. More generally, we have

k M M O V ⊗k =∼ (IndUn ResUn )k(1) =∼ U i1_n ⊗ InfUn m U ir_lr . Un−1 Un−1 n Un−1 ir_lr n−1 i1≤n 1

If m = q − 1, we index the m copies of U ir_lr by dummy variables ir_lr for ir._lr ir_lr n br

1 ≤ br ≤ q − 1 otherwise let br = 0. When each

k O U i1_n ⊗ InfUn U ir_lr ⊗ ir_lr n Un−1 n−1 br r=2

λ ⊗k is isomorphic to Un for some λ ` [n], then we have a decomposition of V as a direct sum of

Un-supermodules. This occurs when the supercharacters coincide with irreducible characters.

5.3.4 Projections for pairs of paths

In the case when there are no conﬂicting arcs, we have the isomorphism

k O U i1_n ⊗ InfUn U ir_lr ⊗ ir_lr =∼ U λ n Un−1 n−1 br n r=2 and we can construct basis elements of the centralizer subalgebra Zk,n. For simplicity, we establish this part of the basis when q = 2.

Consider the set STgk of paths in the Bratteli diagram corresponding to shell tableaux T =

(ς1, ς2, . . . , ςk) of length k such that for 1 ≤ r < k, ςr = {(n _ n; a)} or |ςr| = 2. Then each strictly labeled shell has no inner whorls, so by conditions 4 and 5, there are no conﬂicts with the outer whorls of each shell. For example, suppose that

1 3 5 7 ! T = , , , 6 2 4 . 68

1 2 k If T = (ς , ς , . . . , ς ) ∈ STgk is given by   {(ir _ n; ar)} ∪ {(ir ^ lr; ar + 1)} for lr 6= n, ςr =  {(ir _ lr; ar)} for lr = n, let br = 1 if ir < n and br = 0 if ir = n for r > 1. Then the module corresponding to T is

k O U sh(T ) =∼ U i1_n ⊗ InfUn U ir_lr ⊗ ir_lr n n Un−1 n−1 br r=2 for dummy variables i2_l2 , . . . , ik_lk . b2 bk While the dummy variables are superﬂuous when q = 2, we can generalize to arbitrary q by

r coloring the maximally labeled arcs in each strictly labeled shell ς by br ∈ Fq such that br = 0 if and only if ςr = {(n _ n; a)}. Then the module corresponding to the colored tableau T is

k O U sh(T ) =∼ U i1_n ⊗ InfUn U ir_lr ⊗ ir_lr n n Un−1 n−1 br r=2 for dummy variables i2_l2 , . . . , ik_lk . b2 bk sh(T ) Moreover, a basis for Un is given by Corollary 5.3.11. Thus for each pair of shell tableau

0 0 (T,T ) ∈ STgk × STgk with sh(T ) = sh(T ), we can obtain explicit formulas for the projection map

sh(T 0) sh(T ) πT,T 0 from Un to Un . In the next section we illustrate this when q = 2 and n = 2 to produce a basis for the full centralizer algebra.

⊗k 5.3.5 The centralizer algebra EndU2 (V )

Let n = 2 and q = 2. Then     

 a1 ak  ⊗k ∼     ∼ k V = C-span   ⊗ · · · ⊗   a1, . . . , ak ∈ F2 = CF2.    1 1 

k We may identify this basis F2 with the k-dimensional hypercube by taking vertices labeled by

k (a1, . . . , ak) ∈ F2 and joining two vertices (a1, . . . , ak) and (b1, . . . , bk) by an edge if and only if   1 1   k |{i | ai 6= bi}| = 1. The generator g =   of U2 acts on F2 by 0 1

g · (a1, . . . , ak) = (a1 + 1, . . . , ak + 1) 69 to send each vertex to its antipode.

⊗k Decomposing V as a U2-module, the ﬁrst few rows of the Bratteli diagram are given by

and

V ⊗k =∼ 2k−11 + 2k−1χ1_2.

⊗k Since the centralizer algebra EndU2 (V ) is the subalgebra Zk,2, the dimension of the centralizer algebra is

(2k−1)2 + (2k−1)2 = 22k−1.

Suppose a path in the Bratteli diagram is given by the shell tableau T = (ς1, ς2, . . . , ςk) ∈

STgk = ST k where   {(ir _ 2; ar)} ∪ {(ir ^ lr; ar + 1)} for lr 6= 2, ςr =  {(ir _ lr; ar)} for lr = 2.

Note that we have   if l1 = 1, sh(T ) =  if l1 = 2.

Let b = 1 if i = 1, and b = 0 if i = 2. Then we may index U i1_2 by the dummy variable i1_2 1 r 1 r 2 b1 so that k sh(T ) O U =∼ (U i1_2 ⊗ i1_2) ⊗ InfU2 U ir_lr ⊗ ir_lr . 2 2 b1 U1 1 br r=2 k When n = 2, we have br = 1 if and only if ir = 1. Thus each tuple (b1, . . . , bk) ∈ F2 uniquely

k determines a shell tableau T ∈ ST k. Identify T with (b1, . . . , bk) ∈ F2. 70

k For b = (b1, . . . , bk), a = (a1, . . . , ak) ∈ F2, deﬁne the “twisted” dot product as

a ~ b = (a1 + a2, a2 + a3, . . . , ak−1 + ak, ak) · (b1, b2, . . . , bk).

sh(T ) This dot product counts the number of ai = bi and ai+1 = bi. Then the basis vector uT of U2 is written in the standard basis with coeﬃcients given by the twisted dot product.

Theorem 5.3.13. The map σ : V ⊗k ! V ⊗k from the basis of V ⊗k indexed by shell tableaux to the standard basis is deﬁned for the shell tableau T = (b1, . . . , bk) ∈ F2 by

T X (b1,...,bk)~(v1,...,vk) σ(u ) = (−1) v1 ⊗ · · · ⊗ vk. k (v1,...,vk)∈F2 Proof. Corollary 5.3.11 gives

sh(T ) ∼ T U2 = C-span{u }, where k r T X O Y u = ϑ(brvr) stk(vs, ()) ⊗ 1 k r=1 s=1 (v1,...,vk)∈F2 + × for ϑ : F2 ! C given by ϑ(0) = 1 and ϑ(1) = −1. If vs, vt ∈ F2, we have       1 vs 1 vt 1 vs + vt       stk(vs, ()) stk(vt, ()) =     =   = stk(vs + vt, ()). 0 1 0 1 0 1

Therefore, uT is given in the standard basis as

k r T X O X u = ϑ(brvr) stk vs, () ⊗ 1 . k r=1 s=1 (v1,...,vk)∈F2

k Since the sum runs over all (v1, . . . , vk) ∈ F2, we can reindex to obtain

k T X O u = ϑ(brvr + br+1vr) stk(vr, ()) ⊗ 1 k r=1 (v1,...,vk)∈F2 k k X Y O = ϑ(brvr + br+1vr) stk(vr, ()) ⊗ 1 k r=1 r=1 (v1,...,vk)∈F2 k k X Y O = ϑ(brvr)ϑ(br+1vr) stk(vr, ()) ⊗ 1 , k r=1 r=1 (v1,...,vk)∈F2 71

brvr where bk+1 = 0. Because ϑ(brvr) = (−1) , it follows that k T X (b1+b2,...,bk+bk+1)·(v1,...,vk) O u = (−1) stk(vr, ()) ⊗ 1 . k r=1 (v1,...,vk)∈F2

Identifying stk(vr, ()) ⊗ 1 with vr, we obtain

T X (b1,...,bk)~(v1,...,vk) u = (−1) v1 ⊗ · · · ⊗ vk. k (v1,...,vk)∈F2

−1 ⊗k ⊗k Inverting this maps yields σ : V ! V given for v1, . . . vk ∈ F2 by

1 X σ−1(v ⊗ · · · ⊗ v ) = (−1)(b1,...,bk)~(v1,...,vk)uT . 1 k 2k k (b1,...,bk)∈F2 −1 ⊗k The maps σ and σ allow us to construct a basis for EndU2 (V ) that acts on the standard basis of V ⊗k.

Corollary 5.3.14. Let σ be the transition map between the basis of V ⊗k indexed by shell tableaux

⊗k and the standard basis. A basis for the centralizer algebra EndU2 (V ) is

−1 0 0 {(σ ◦ πT,T 0 ◦ σ ) | (T,T ) ∈ ST k × ST k with sh(T ) = sh(T )}

0 k k where for T = (b1, . . . , bk),T = (a1, . . . , ak) ∈ F2, and v = (v1, . . . , vk) ∈ F2, the action on the standard basis is given by

−1 1 X (a ,...,a ) v+(b ,...,b ) (u ,...u ) (σ ◦ π 0 ◦ σ )(v ⊗ · · · ⊗ v ) = (−1) 1 k ~ 1 k ~ 1 k u ⊗ · · · ⊗ u . T,T 1 k 2k 1 k k (u1,...,uk)∈F2 k −1 Proof. Suppose S ∈ ST k corresponds to (u1, . . . , uk) ∈ F2. Composing the maps σ ◦ πT,T 0 ◦ σ yields

−1 1 X (u ,...,u ) v S (σ ◦ π 0 ◦ σ )(v ⊗ · · · ⊗ v ) = (σ ◦ π 0 ) (−1) 1 k ~ u T,T 1 k T,T 2k k (u1,...,uk)∈F2 1 = σ · (−1)(a1,...,ak)~vuT 2k 1 X = · (−1)(a1,...,ak)~v (−1)(b1,...,bk)~(u1,...,uk)u ⊗ · · · ⊗ u 2k 1 k k (u1,...,uk)∈F2 1 X = (−1)(a1,...,ak)~v+(b1,...,bk)~(u1,...uk)u ⊗ · · · ⊗ u 2k 1 k k (u1,...,uk)∈F2 72 as desired. Since

⊗k ∼ M sh(T ) V = U2 , T ∈ST k the supermodule analogue of the Double Centralizer Theorem 5.3.2 says a basis of the centralizer subalgebra Zk,2 is

0 0 {πT,T 0 | (T,T ) ∈ ST k × ST k with sh(T ) = sh(T )}.

⊗k 0 In this case Zk,2 = EndU2 (V ) so the πT,T form a basis for the centralizer algebra. Conjugating

⊗k this basis by σ produces the basis of EndU2 (V )

−1 0 0 {(σ ◦ πT,T 0 ◦ σ ) | (T,T ) ∈ ST k × ST k with sh(T ) = sh(T )} that acts on the standard basis of V ⊗k.

Since we have found formulas for the projections of V ⊗k when q = 2 and n = 2, it seems tractable to be able to determine these formulas in general. Bibliography

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Sage Code

sage: R = QQ[’q’].fraction_field() sage: q = R.gen() sage: NCSym = SymmetricFunctionsNonCommutingVariables(R) sage: chi = NCSym.chi(q) sage: def crs(x, y):

crs = 0

for i in range(len(x)):

for j in range(len(y)):

if x[i][0] < y[j][0] < x[i][1] < y[j][1]:

crs = crs+1

return crs sage: def nesting(x, y):

if y[0] < x[0] < x[1] < y[1]:

return True sage: def res_coeff(x, y):

return (q-1)^(Set(x).difference(Set(y)).cardinality())*q^(crs(Set(x).

intersection(Set(y)),Set(x).difference(Set(y))))/q^(crs(Set(y).

intersection(Set(x)),Set(y).difference(Set(x)))) sage: def ind_coeff(x, y): 76

return (q-1)^(Set(x).difference(Set(y)).cardinality())*q^(crs(Set(x)

.difference(Set(y)),

Set(x).intersection(Set(y))))/q^(crs(Set(y).difference(Set(x)),Set(y)

.intersection(Set(x)))) sage: def getKey(item):

return item[1] sage: def Res(n,self): sage: # self = supercharacter given by [[partition], coeff]

arcsort = sorted(self[0], key=getKey)

if arcsort == []:

return [self]

if arcsort[-1][1] != n:

return [self]

elif arcsort[-1][1] == n:

spiral = arcsort.pop(-1)

l = []

l.append(spiral)

for _ in self[0]:

if _[0] < spiral[0]:

l.append(_)

P = Poset((self[0], nesting), cover_relations=False)

C = list(P.chains(exclude=[_ for _ in l]))

S = [_+[spiral] for _ in C]

res = []

re = [max(_) for _ in self[0]]

for i in range(len(S)):

arcs = self[0][:] 77

for j in range(len(S[i])):

arcs.remove(S[i][j])

for j in range(len(S[i])-1):

arcs.append((S[i][j+1][0], S[i][j][1]))

res.append(arcs)

for k in range(S[i][0][1]-S[i][0][0]-1):

if S[i][0][1]-k-1 not in re:

res.append(arcs+[(S[i][0][0],S[i][0][1]-k-1)])

return [[res[i], res_coeff(self[0], res[i])*self[1]] for i in

range(len(res))] sage: def Ind(n, self): sage: # self = supercharacter given by [[partition], coeff]

P = Poset((self[0], nesting), cover_relations=False)

C = list(P.chains())

ind = [self[0]]

le = [min(_) for _ in self[0]]

for j in range(1,n):

if j not in le:

ind.append(self[0]+[(j,n)])

for j in range(1,len(C)):

arcs = self[0][:]

for k in range(len(C[j])):

arcs.remove(C[j][k])

for k in range(len(C[j])-1):

arcs.append((C[j][k][0], C[j][k+1][1]))

arcs.append((C[j][len(C[j])-1][0],n))

ind.append(arcs) 78

for k in range(C[j][0][1]-C[j][0][0]-1):

if C[j][0][0]+k+1 not in le:

ind.append(arcs+[(C[j][0][0]+k+1,C[j][0][1])])

return [[ind[i],ind_coeff(self[0], ind[i])*self[1]] for i in

range(len(ind))] sage: def ResSum(n, self):

l = map(Res, [n]*len(self), self)

d = [[tuple(sorted(item[0], key=getKey)), item[1]] for sublist in l

for item in sublist]

simplify = {}

for k, v in d:

simplify[k] = simplify.get(k, 0) + v

return [[list(item[0]), item[1]] for item in list(simplify.items())] sage: def IndSum(n, self):

l = map(Ind, [n]*len(self), self)

d = [[tuple(sorted(item[0], key=getKey)), item[1]] for sublist in l

for item in sublist]

simplify = {}

for k, v in d:

simplify[k] = simplify.get(k, 0) + v

return [[list(item[0]), item[1]] for item in list(simplify.items())] sage: def IndRes(n,k):

def funk(y):

return reduce(lambda x, _: IndSum(n,ResSum(n,x)), xrange(k), y)

return funk sage: IndRes(4,3)([[[],1]]) 79

[[[(2, 3), (3, 4)], 2*q^3 - q^2 - 2*q + 1],

[[(1, 2)], 2*q^3 - q^2 - 2*q + 1],

[[(2, 3)], 2*q^3 - q^2 - 2*q + 1],

[[(1, 2), (2, 3)], q^3 - q^2 - q + 1],

[[(2, 4)], 2*q^4 - q^2],

[[(3, 4)], 3*q^3 - 3*q + 1],

[[(1, 3)], q^4 - q^2],

[[(1, 4)], q^5 + q^4 - q^3],

[[(1, 2), (2, 3), (3, 4)], q^3 - q^2 - q + 1],

[[(1, 3), (3, 4)], q^4 - q^2],

[[(1, 3), (2, 4)], q^4 - q^2],

[[], 3*q^3 - 3*q + 1],

[[(1, 2), (2, 4)], q^4 - q^2],

[[(2, 3), (1, 4)], q^5 - q^3],

[[(1, 2), (3, 4)], 2*q^3 - q^2 - 2*q + 1]]