TWO PROBLEMS IN REPRESENTATION THEORY: AFFINE LIE ALGEBRAS AND ALGEBRAIC COMBINATORICS
By
ALEJANDRO GINORY
A dissertation submitted to the School of Graduate Studies Rutgers, The State University of New Jersey In partial fulfillment of the requirements For the degree of Doctor of Philosophy Graduate Program in Mathematics
Written under the direction of Siddhartha Sahi And approved by
New Brunswick, New Jersey May, 2019 ABSTRACT OF THE DISSERTATION
Two Problems in Representation Theory: Affine Lie Algebras and Algebraic Combinatorics
By ALEJANDRO GINORY Dissertation Director: Siddhartha Sahi
In this dissertation, we investigate two topics with roots in representation theory. The first topic is about twisted affine Kac-Moody algebras and vector spaces spanned by their characters. Specifically, the space spanned by the characters of twisted affine Lie algebras admit the action of certain congruence subgroups of SL(2, Z). By embedding the characters in the space spanned by theta functions, we study an SL(2, Z)-closure of the space of characters. Analogous to the untwisted affine Lie algebra case, we construct a commutative associative algebra (fusion algebra) structure on this space through the use of the Verlinde formula and study important quotients. Unlike the untwisted cases, some of these algebras and their quotients, which relate to the trace of diagram automorphisms on conformal blocks, have negative structure constants with respect to the (usual) basis indexed by the dominant integral weights of the Lie algebra. We give positivity conjectures for the new structure constants and prove them in some illuminating cases. We then compute formulas for the action of congruence subgroups on these character spaces and give explicit descriptions of the quotients using the affine Weyl group.
The second topic concerns algebraic combinatorics and symmetric functions. In statistics, zonal polynomials and Schur functions appear when taking integrals over
ii certain compact Lie groups with respect to their associated Haar measures. Recently, a conjecture, related to certain integrals of statistical interest, was proposed by D. Richards and S. Sahi. This conjecture asserts that certain linear combinations of Jack polynomials, a one-parameter family of symmetric polynomials that generalizes the zonal and Schur polynomials, are non-negative when evaluated over a certain cone. In the second part of this dissertation, we investigate these conjectures for Schur polyno- mials and give a refined version of the conjecture. In addition, we prove some cases and arrive at certain seemingly new combinatorial results. In an important instance, we give an analogous result for Jack polynomials.
iii Acknowledgements
First, I would like to express my deepest gratitude to my advisor, Siddhartha Sahi, for introducing me to and guiding me through so many areas in mathematics. I am especially grateful for his intellectual generosity, abundant patience, and razor sharp wit. It has been a great honor to work with him and witness his tireless curiosity up close.
I would like to thank the dissertation committee members Jim Lepowsky, Yi-Zhi Huang, and Jiuzu Hong for their help and comments on my thesis. I would further like to thank Jim for the many hours of discussion of many interesting problems and his guidance in my graduate studies in general. Thank you to Yi-Zhi for many useful conversations that opened up new avenues of inquiry. Thank you to Jiuzu for great discussions and hosting me during my visit to the University of North Carolina at Chapel Hill.
I would also like to thank all of those professors who took time to discuss mathe- matics with me. Let me mention a few. Thanks to Chris Woodward, who was generous enough to start working with me as a first year graduate student. Thanks to Vladimir Retakh, who was willing to listen to any idea I would come up with throughout the years. Thanks to Lisa Carbone, for also serving on my oral qualifying examination committee and discussing potential projects we might work on. Thanks to Doron Zeil- berger, whose opinions gave me a different perspective on many topics and who helped me appreciate the relevance of experimental mathematics. Thanks to Roger Nussbaum who honored me by taking interest in a certain discovery that I came across and was patient enough to guide me through the wonderful, yet alien, world of non-linear anal- ysis. Thanks to Angela Gibney, who mentored me, challenged me, and worked with me intensely in the short time we have known each other. Let me also thank some of the
iv important professors who helped me during my undergraduate years: Tedi Draghici, Laura De Carli, Miroslav Yotov, Gueo Grantcharov, and Thomas Leness.
A special thanks goes out to all of my peers and colleagues for helping me grow while staying sane. In particular, thanks to Ed Karasiewicz, Zhuohui Zhang, Sjuvon Chung, Pat Devlin, Fei Qi, Richard Voepel, Rebecca Coulson, Jorge Cantillo, Mingjia Yang, Matthew Russell, Doug Schultz, Semeon Artamonov, Johannes Flake, Robert Laugwitz, Jason Saied, Sven Moeller, Jongwon Kim, Henry Zorrilla, and Eric Wawerczyk. I could go on, but I won’t.
I would like to thank Kafung Mok for her endless support and patience during these years. I could not do anything without her. She was and continues to be the light of my life. I would also like to thank my family for their support and encouragement throughout my journey. Finally, I would like to thank Alejandro Ortiz-Burgos and his wife Dagmar, kind strangers who helped a lowly waiter discover mathematics.
As for non-human ‘actors’, I would like to thank the warmth of my hometown Miami for recharging me every time I visited, the summer season in New Jersey for being the only reprieve I got from the dreariness of the rest of the year, and the Middlesex Greenway, a former railway track converted into a beautifully green trail, where I would attempt to remain healthy.
v Table of Contents
Abstract ...... ii
Acknowledgements ...... iv
1. Introduction ...... 1
2. Fusion Algebras, the Verlinde Formula, and Twisted Affine Kac-Moody Algebras ...... 4
2.1. Introduction ...... 4
2.1.1. Structure of Chapter ...... 10
2.2. Preliminaries ...... 11
2.2.1. Affine Kac-Moody Lie Algebras ...... 11
2.2.2. Integrable Highest Weight Modules ...... 14
2.2.3. Theta Functions, Modular Invariance, and Characters ...... 15
2.3. Verlinde Algebras of Untwisted Affine Algebras ...... 19
2.4. Verlinde Algebras of Twisted Affine Algebras ...... 24
(2) 2.5. Verlinde Algebras for A2` ...... 29 2.6. Applications ...... 33
2.6.1. Quotients of Vk(g)...... 33 2.6.1.1. More Negative Structure Constants ...... 37
(2) 2.6.2. The A2` Quotient Structure ...... 38 2.6.3. Congruence Group Action ...... 42
2.7. Some Data ...... 44
3. Richards-Sahi Conjectures for Schur Polynomials ...... 52
3.1. Introduction ...... 52
vi 3.2. Preliminaries ...... 54
3.2.1. Partitions ...... 54
3.2.2. Symmetric Polynomials ...... 56
r 3.3. αµ Conjecture ...... 58 3.4. Special Cases of Richards-Sahi Conjecture ...... 59
3.4.1. Case µ = ni ...... 59
3.4.2. Case µ = m11n ...... 61
3.5. Another Partial Order on Partitions ...... 65
3.6. Certain Chains of Partitions ...... 70
3.7. Refinement of Conjecture ...... 73
3.7.1. Positive Functions γµλ ...... 74 3.8. Kostka-Content Identities ...... 77
3.8.1. An Inductive Argument ...... 78
3.8.2. Case λ = µ ...... 80
3.8.3. Jack Hook-Length Recursion Formula ...... 81
References ...... 88
vii 1
Chapter 1
Introduction
Representation theory is a broad field with many beautiful branches and applications. In this dissertation, we will explore two topics in representation theory that at first glance seem very far apart, but at their core are concerned with representations of Lie algebras. Both address conjectures involving mysterious identities, giving concrete answers in certain important families. Both give rise to even deeper questions about their connections to other fields. Finally, both involve computational and experimental techniques, though ultimately the results are purely conceptual.
This dissertation has two main stand-alone parts, Chapters 2 and 3, each an essay in its own right. In first part, we try to determine what sort of hidden algebraic structures exist among certain classes of representations of twisted affine Kac-Moody Lie algebras. More precisely, we try to find an analogue to the Verlinde fusion algebras, which exist for untwisted affine Kac-Moody algebras, for the twisted case. In the second part, we investigate curious conjectures about positivity arising from statistics, but with representation theoretic connections, give partial results confirming them in many cases, and provide a further refinement of the conjectures. The identities are formulated in the generality of Jack polynomials, and, in this work, we focus on the special case of Schur polynomials, where the conjectures are no easier to prove. Many of the results for Schur polynomials generalize easily and directly to Jack polynomials, and the Jack polynomial case will be the object of future study.
Each part has an introduction which will motivate and describe the results in that chapter, but let us a give a broad outline of what is contained there. In the first part, we give a background on Verlinde fusion algebras for untwisted affine Kac-Moody algebras and then discuss what we do for the twisted Kac-Moody algebras. Specifically, 2 we study an embedding of the space spanned by the characters of twisted Kac-Moody algebras into a SL2(Z)-module and then immediately show that, in most cases, this space is isomorphic to a Verlinde fusion algebra. One case stands out among the rest, (2) that is the case of Kac-Moody algebras of type A2` , where we show that this gives rise to a new fusion algebra. This fusion algebra has special properties that differ from the usual Verlinde fusion algebras. Just like the classical untwisted case, the structure (2) constants of the fusion algebra associated to A2` , with respect to a natural basis, are integers. Unlike the classical untwisted case, these integers are sometimes negative. We proved exactly when they are negative for certain Kac-Moody algebras and give a precise conjecture as to when negative structure constants appear for the remaining ones. This answers questions brought up in [Ho2] and gives evidence that these fusion algebras might not directly compute the dimensions of modules. Further results are given for related fusion algebras defined in [Ho2] and certain identities with respect to important generators of congruence subgroups are proved.
In the second part, we investigate the Richards-Sahi conjectures for Schur polynomi- als [RS]. These conjectures assert that certain linear combinations of Schur polynomials in d variables x1, . . . , xd evaluate to non-negative real numbers whenever 1 ≥ x1 ≥ x2 ≥
· · · ≥ xd ≥ 0. As mentioned earlier, a stronger conjecture can be formulated for Jack polynomials with an arbitrary parameter α, and we plan to address Jack case in future work. To give an idea of the types of linear combinations we refer to (see section 3.2 for precise definitions), given a partition µ and and certain integers r−, r, related to µ,
r X αµ(x) = Rµ(ν, r)sν(x) ≥ 0 ν∈H(µ)
where H(µ) is the set of partitions ν such that ν ⊆ µ and µ \ ν is a horizontal strip, sν
is the Schur polynomial corresponding to the partition ν, Rµ(ν, r) is a certain rational
function in r depending on µ and ν, and 1 ≥ x1 ≥ x2 ≥ · · · ≥ xd ≥ 0. (The coefficients
Rµ(ν, r) are often negative.) We verify these conjectures for several important families, indexed by partitions, and give combinatorial proofs. Later, we give a stronger, refined conjecture that exploits the combinatorics of partitions. More precisely, we show that
≤n in all of the known cases, given n ∈ N, the set P of partitions of integers no greater 3 than n (partially) ordered by inclusion can be partitioned into certain chains λ0 ⊂ λ1 ⊂ · · · ⊂ λk in such a way so that
k X i Rµ(λ , r)sλi (x) ≥ 0. i=0 In many cases, proving the positivity of these expressions is easier and uses combina- torial arguments. Surprisingly, in proving special cases of these refined conjectures, we arrive at interesting recursions for important values known as Kostka coefficients. In particular, they give a new recursion for the dimensions of irreducible representations of symmetric groups.
During the course of this research, we wrote software in order to compute many examples and give us insight into these mysteries. These programs, written in Maple, are too bulky to include in this text but can be downloaded from the link below. http://math.rutgers.edu/~ag930 4
Chapter 2
Fusion Algebras, the Verlinde Formula, and Twisted Affine Kac-Moody Algebras
2.1 Introduction
Fusion algebras associated to affine Lie algebras can be traced back to the works of physicists working in conformal field theory in the early 1980’s (see for example [BPZ], [KZ], [W]). In particular, they investigated the so-called Wess-Zumino-Witten (WZW) model (sometimes called the WZNW model where N stands for Novikov) which is directly related to the representation theory of affine Kac-Moody Lie algebras. Specifi- cally, they worked with untwisted affine Lie algebras, which are universal central exten-
−1 sions of the the loop algebra Lg = g⊗C[t , t] where g is a finite dimensional simple Lie algebra. Of critical importance in the theory are the so-called fusion rules which give a special commutative and associative product structure on the Grothendieck group of a certain category of integrable highest weight representations of an untwisted affine Lie algebra. The resulting ring is called the fusion ring or fusion algebra. One moti- vation for such a product structure arises from the fact that the usual tensor product of vector spaces does not preserve the action of a distinguished element in the affine Lie algebra: the canonical central element (which acts as a scalar, called the level, on these representations). This led to great interest in realizing such a product struc- ture in representation-theoretic terms. Indeed, for untwisted affine Lie algebras, the representation-theoretic product structure was finally constructed using vertex opera- tor algebra theory by Y.-Z. Huang and J. Lepowsky in 1997 (see [HL4]). In this work, they constructed a vertex tensor category structure and, in particular, a braided tensor
category structure, that gives rise to the fusion algebra. (For future reference, let Ok(g) denote the category of level k integrable highest weight representations of an affine Lie 5 algebra g.)
Rather early on, the group SL(2, Z) was shown to act on the complex vector space spanned by the characters of certain modules for affine Lie algebras (for example, [FS]). In 1987-8, the endeavor to understand fusion algebras gained further significance after Erik Verlinde conjectured that the modular group transformation S, corresponding to the M¨obiustransformation τ 7→ −1/τ on H, diagonalizes the fusion rules [V]. (Since the left multiplication, with respect to the fusion product, can be represented by commuting symmetric matrices, these matrices can be simultaneously diagonalized.) He presented the Verlinde formula which describes how the matrix corresponding to the S action can be used to compute the fusion rules. Since then, many people have presented proofs of the Verlinde formula, most notably G. Faltings [Fal] in 1994 for some affine Lie algebras, and C. Teleman [Te] for affine Lie algebras in general around 1994, with the latest proof, in much greater generality than affine Lie algebras, given by Huang [Hu3] in 2004 based on vertex tensor category theory (referred to above). In fact, the last proof showed that a suitable tensor product structure on Ok(g), where k ∈ Z>0 and g is an untwisted affine Lie algebra, (see [HL1], [HL2], [HL3], [Hu1]) gives rise to the same fusion algebra predicted by the Verlinde formula.
The entire story cannot be directly applied to twisted affine Lie algebras. Twisted affine Lie algebras can be defined in two equivalent ways: the first via their generalized Cartan matrices and the Chevalley-Serre relations for Kac-Moody algebras, and the second as fixed-point subalgebras of untwisted affine Lie algebras (with respect to so- called diagram automorphisms). We will use the former definition exclusively, but acknowledge the importance of understanding the connection between our results here and diagram automorphisms.
In this paper, we study analogous structures for twisted affine Lie algebras and investigate their properties. Let us briefly summarize the contents. To begin with, we use the definition of twisted affine Lie algebras using generalized Cartan matrices, i.e., in terms of generators and relations (the generalized Chevalley-Serre relations). We
then proceed to find the most suitable SL(2, Z)-module in which to embed the space of characters of twisted affine algebras. This space is then endowed with the structure of 6 a commutative associative algebra, which we refer to as a Verlinde algebra, using the action of S ∈ SL(2, Z) in a fashion analogous to the construction of fusion algebras for (2) untwisted affine algebras. For affine Lie algebras of type A2` , the space of characters admits the action of SL(2, Z) so we define an algebra structure on it directly. In the remaining cases of twisted affine Lie algebras we consider a strictly larger space on which SL(2, Z) acts. Using the algebra structure and the affine Weyl group, we construct a natural family of quotients, in which, quite surprisingly, negative structure constants appear. These quotients are isomorphic to the fusion algebras constructed in [Ho2] which encode the trace of diagram automorphisms on conformal blocks. In [Ho2], the author discovers that the structure constants are non-negative in certain families and asks how generally this property holds. Our results show that the non-negativity of the structure con- stants holds in about ‘half’ the cases (depending on level), while in the other ‘half’ we determine when negative structure constants appear. Ultimately, we hope that these Verlinde algebras lead to a greater understanding of the structure of certain categories of modules for twisted affine Lie algebras.
Now we will describe the contents of this paper in more technical detail. The space (r) of characters of an affine Lie algebra of type XN admits an action of the congruence subgroup Γ1(r) ⊆ SL(2, Z). Untwisted affine algebras correspond to the case when r = 1. Therefore, Γ1(1) = SL(2, Z) acts on the space of characters of untwisted 0 −1 affine algebras and, using the action of the element S = , one can define a 1 0 commutative associative algebra structure on this space explicitly using the so-called Verlinde formula, which we will define shortly.
The construction of the algebras associated to twisted affine algebras is as follows.
First, suppose that V is a finite dimensional SL(2, Z)-module with a fixed basis {eλ}λ∈I , where 0 ∈ I is a distinguished index. If the 0th column of the action of S ∈ SL(2, Z), with respect to this basis, has no zero entries, then one can put a commutative asso-
ciative algebra structure on V, with e0 as the identity element, by defining its structure 7 constants as −1 ν X SφλSφµ(S )νφ Nλµ = (2.1) Sφ0 φ∈I (see [Ka, ex. 13.34]). In this paper, the above formula will be referred to as the Verlinde formula. Notice that any nonzero scalar multiple of S will yield the same
ν structure constants Nλµ. We will now construct such a space associated to the character space of twisted affine algebras. (See section 2.2.1 for definitions.) Consider the space of characters
Chk(g) = Chk of integrable highest weight modules with fixed positive integer level k of an affine Lie algebra g. This complex vector space is spanned by χλ = Aλ+ρ/Aρ, where Aµ are alternating sums of theta functions (which we refer to as alternants). The − space spanned by Aλ+ρ, for λ ranging over dominant level k weights, is denoted Chk ∨ ∨ and is a subspace of T hk+h∨ , the space of theta functions of level k+h , where h is the − − dual Coxeter number of g. In fact, Chk is a subspace of T hk+h∨ ⊂ T hk+h∨ , the space of alternating theta functions (alternating with respect to the action of the associated
− finite Weyl group of g). The space T hk+h∨ admits an action of SL(2, Z) different from the usual Mp(2, Z) action (see section 2.2.3 for the definition), and the injection above intertwines the action of SL(2, Z) up to scalar. In the untwisted cases and in the case (2) that g is of type A2` , the map χλ 7→ Aλ+ρ = χλAρ is an injection. For the remaining − twisted cases, Chk does not admit an action of SL(2, Z), but its image under S is − naturally contained in a certain subspace V ⊆ T hk+h∨ . This subspace V is the smallest − − subspace of T hk+h∨ that contains Chk , is closed under S, and has a basis {Aµ}µ∈X , where X is a subset of the set of regular dominant weights (not necessarily integral of
− course). The space V is called the SL(2, Z)-closure of Chk . (The condition that V be spanned by Aµ reflects the hope that this space will correspond to integrable highest weight modules for some affine Lie algebra and thus can be interpreted representation- theoretically.) More precisely, we have the following description of V .
(2) Theorem 2.1. If g is a twisted affine Lie algebra, but not of type A2` , then the − − t t SL(2, Z)-closure of Chk (g) is equal to Chk (g ), where g is the transpose affine Lie (2) − algebra. In the case where g is of type A2` , Chk (g) is an SL(2, Z)-module. 8
If g is untwisted, then the map χλ 7→ Aλ+ρ is an algebra isomorphism (since the (2) Verlinde formula only depends on the action of S up to scalar). If g is of type A2` , we get a completely new algebra defined on the space of characters. For the remaining twisted cases, we get an algebraic structure on a strictly larger space (the precise increase in size will be discussed in detail in section 2.4). In both cases, the identity element is chosen to be the alternant corresponding to the ‘smallest’ weight. Our discussion motivates the following unified definition.
Definition 2.2. Let g be an affine Kac-Moody Lie algebra and k be a positive integer.
The Verlinde algebra Vk(g) is the algebra whose underlying vector space is the SL(2, Z)- closure V with basis {Aλ}λ∈X , where X is a subset of regular dominant weights, (see the discussion above Theorem 2.1) and whose structure constants, with respect to the given basis, are given by the Verlinde formula (eq. 2.1).
This definition has several advantageous features. The first is that the action of
Γ1(r) on the space of characters can be exhibited directly by the SL(2, Z) action on the untwisted Verlinde algebra. Indeed, explicit formulas for the action of congruence subgroups on the linear space of characters of twisted affine algebras are given in terms of evaluations of characters of irreducible finite dimensional modules for finite dimensional simple Lie algebras similar to the untwisted case. The second is that the algebra has structure constants that can be computed via the Verlinde formula. Another, as we shall later see, is that the Verlinde algebras can be defined as quotients of the representation ◦ ring R(gt) of the transpose of the underlying simple Lie algebra (the transpose Lie
algebra of Bn is Cn, Cn is Bn, and the remaining are self-transpose). (2) In all cases except for A2` , these Verlinde algebras turn out to be isomorphic to the Verlinde algebras of untwisted affine algebras at different levels. By construction, this
isomorphism is compatible with the SL(2, Z) action. More specifically, the following (2) theorem is proved. (Note that in the case of A2` this is a tautology since these algebras are isomorphic to their transposes.)
(r) Theorem 2.3. Let g be a twisted affine Lie algebra of type XN , k a positive integer, 9 gt its transpose algebra. Then there is an algebra isomorphism
t ϕk : Vk(g) → Vk+h∨−h(g ) that intertwines the action of SL(2, Z) up to scalar.
Verlinde algebras have striking features, including the following algebra grading structure whose proof relies on the Kac-Walton algorithm (see section 2.3).
(1) Theorem 2.4. Let g be an affine Lie algebra of type XN ,V be its Verlinde algebra at ◦ ◦ level k, and set A = P/Q. The algebra V has an A-grading
M V = Va a∈A + where Va = span{χλ : λ ∈ Pk and λ ∈ a}.
In certain cases, the grading structure highlights the subspace of characters of twisted affine algebras, but not in general. The grading is not immediately evident from studying the S-matrix but can be observed after computing the structure con- stants. The grading seems to play a significant role in the Verlinde algebras of type (2) A2` and in certain quotients of the other types (with multiple root lengths). (2) In all but the case when the affine Lie algebra is of type A2` with even level, the structure constants in these Verlinde algebras are known to be non-negative integers. (2) In the A2` even level case, all known examples have some negative integral structure (2) constants. We prove that in the special case A2 , at even levels, the Verlinde algebra always has some negative structure constants and the grading helps us determine the signs of all structure constants. The author has not found this phenomenon cited in any of the literature. Based on a large amount of evidence, we conjecture (see 2.5) ◦ ◦ that ‘twisting’ the basis by a character on P/Q gives nonnegative integral structure (2) constants in type A2 .
Remark. There is some difficulty in working with this definition of the algebra since computing the structure constants is quite tedious if done ‘by hand’, involving large sums of roots of unity. We overcame this problem by writing and using the Maple pack- age Affine Lie Algebras, which can efficiently compute these quantities and helped 10 reveal much of the phenomena studied here. This program helped immensely in testing our conjectures for many families of affine Lie algebras.
The same algebraic structure can be defined in terms of a certain action of the affine Weyl group on the representation ring of an associated finite dimensional simple Lie algebra. While this sidesteps the problem of the aforementioned computations and is more conceptual, working with the algebraic structure in this setting involves finding specific representatives in the infinite orbits of the affine Weyl group, again a difficult task. Nonetheless, in some new cases arising from our methods, we present a recursive method of computing these representatives by finding a natural family of affine Weyl group elements that move the vectors they act on closer to a certain fundamental domain of the action of the group (see section 2.6.2).
2.1.1 Structure of Chapter
In section 2.2, we go over preliminaries, notation, and conventions. Of particular impor- tance is the subsection on theta functions, modular invariance, and characters (section
2.2.3) where we define and discuss the actions of SL(2, Z) that feature throughout this work. In section 2.3, the well-known Verlinde algebras for untwisted affine Lie algebras are reviewed and their structure is discussed. In section 2.4, the definition of Verlinde algebras is investigated for all twisted affine Lie algebras. The grading structure and linear subspace spanned by the twisted affine algebra characters are discussed. In sec- (2) tion 2.5, the Verlinde algebras of type A2` are more extensively studied. When the level is odd, these algebras are well understood and are isomorphic to an untwisted Verlinde algebra. When the level is even, as mentioned earlier, the structure constants are sometimes negative. Finally, in section 2.6, we show that J. Hong’s fusion algebras for affine Kac-Moody Lie algebras are isomorphic to natural quotients of the Verlinde algebras presented here and thereby answer some open questions about non-negative structure constants posed in [Ho2]. The quotient structure of the Verlinde algebra of (2) type A2` is then explicitly computed in terms of the affine Weyl group. In the process, we produce an algorithm for bringing weights that lie in 2kA1 into kA1, where A1 is the fundamental alcove of the affine Weyl group of type C and k is a positive integer. The 11 algorithm is a technical tool can be used to study the negative structure constants but is of interest in its own right. In addition, explicit formulas for the action of congruence subgroups on the space of characters are given, proving certain identities such as the following one (which is very similar to the classical S-matrix action).
◦ − 2πi(ν+ρ) ◦ 2πi(ν+ρ) ◦ − 2πi(µ+ρ) X −2πimν 2 k+h∨ k+h∨ 2πi(mλ+mµ) k+h∨ e (SkΛ0,ν) χλ e χµ e = e SkΛ0,µχλ e ◦ ν∈P
r The action of the element u21 of Γ1(r) is computed and shown to be expressible as multiples of evaluations of summands of the characters of finite dimensional simple Lie algebras. In the 2.2, we introduce notation and briefly go over the basic representation theory of affine Kac-Moody Lie algebras.
2.2 Preliminaries
2.2.1 Affine Kac-Moody Lie Algebras
This section very briefly reminds the reader about the theory of affine Kac-Moody Lie algebras. For greater detail, the reader is encouraged to consult [Ka]. For the sake of simplicity, affine Kac-Moody Lie algebras will often be referred to as affine Lie algebras or affine algebras.
∨ Let A = (aij)0≤i,j≤` be an irreducible affine (generalized) Cartan matrix and (h, Π, Π ) be a realization of A, with
∗ ∨ ∨ ∨ ∨ Π = {α0, α1, . . . , α`} ⊂ h and Π = {α0 , α1 , . . . , α` } ⊂ h.
◦ ◦ ◦ Let (h, Π, Π∨) be a realization of the associated finite root system whose Cartan matrix ◦ is A = (aij)1≤i,j≤`. In particular,
◦ ◦ ◦ ◦ ∗ ∨ ∨ ∨ Π = {α1, . . . , α`} ⊂ h and Π = {α1 , . . . , α` } ⊂ h.
∗ ∨ ∨ The vector spaces h and h have bases Π ∪{d} and Π∪{Λ0}, resp., where hΛ0, αi i = δ0i
and hΛ0, di = 0. The affine Kac-Moody Lie algebra g = g(A) is generated by h and ◦ ei, fi, 0 ≤ i ≤ `, subject to the Chevalley-Serre relations. The Lie algebra g is the Lie 12
◦ subalgebra of g generated by h and the generators ei, fi, 1 ≤ i ≤ `, along with their ◦ corresponding Chevalley-Serre relations. The algebra g is a finite dimensional simple Lie algebra and is referred to as the underlying simple Lie algebra of g. The transpose At has the realization (h, Π∨, Π) and has associated affine Kac-Moody algebra gt := g(At) called the transpose algebra of g(A).
The irreducible affine Kac-Moody algebras are classified (up to isomorphism) by the affine Dynkin diagrams given in [Ka, Ch. 4]. More precisely, there are three tables of affine algebras: Aff 1, Aff 2, and Aff 3.
(2) WARNING 2.2.1. Unless stated otherwise, the labeled Dynkin diagram of type A2` will be assumed to be the transpose of the one appearing in the table Aff 2. This is consistent with the construction of twisted affine algebras via non-trivial diagram automorphisms of simply laced, untwisted affine Lie algebras. Moreover, it allows us to uniformly give a definition of Verlinde algebras for twisted affine algebras. Note that in [Ka, pg. 132] there is a related warning.
If an affine algebra is isomorphic to one from table Aff r, it is said to be of type (r) XN for X ∈ {A, B, C, D, E, F, G} and N ∈ Z>0. The affine algebras in Aff 1 are called untwisted affine Lie algebras while those in Aff 2 and Aff 3 are called twisted affine Lie algebras. For any affine Lie algebra g, the Dynkin diagram of its underlying simple Lie ◦ algebra g can be obtained from the affine Dynkin diagram of g by removing the 0 node (and all edges attached to the 0 node).
For an affine irreducible Cartan matrix A, let D(A) denote its Dynkin diagram.
The numerical labels of D(A) are a0, a1, . . . , a` and can be found in the aforementioned
∨ ∨ ∨ t tables. The dual labels a0 , a1 , . . . , a` of D(A) are the labels of D(A ). The Coxeter P` ∨ P` ∨ number h = i=0 ai and dual Coxeter number h = i=0 ai play very important roles in the theory.
Using these labels and the Cartan matrix A = (aij)0≤i,j≤`, define the following symmetric nondegenerate bilinear forms on h and h∗,
∨ ∨ ∨−1 ∨ (αi , αj ) = ajaj aij, (αk , d) = a0δ0k, and (d, d) = 0 (i, j, k = 0, . . . , `)
∨ −1 −1 (αi, αj) = ai ai aij, (αk, Λ0) = a0 δ0k, and (Λ0, Λ0) = 0 (i, j, k = 0, . . . , `), 13 respectively. The bilinear form on h induces an isomorphism
ν : h → h∗
∨ ∨−1 ν(αi ) = aiai αi, ν(d) = a0Λ0 which will be used to identify both vector spaces (with little to no warning). We will also
∨ ∗ ∨ 2 −1 be using the map α 7→ α where α is a nonisotropic element of g and α = (α,α) ν (α). Define the important isotropic elements
` ` X ∗ X ∨ ∨ δ = aiαi ∈ h and K = ai αi ∈ h. i=0 i=0 The element K is called the canonical central element of g. Also define the element ◦ θ = δ − a0α0, which is a dominant root of g. With these new elements define the new bases
∗ {α1, . . . , α`, Λ0, δ} ⊂ h
∨ ∨ {α1 , . . . , α` , d, K} ⊂ h
and the projection map ◦ π : h∗ → h∗
α 7→ α with respect to the new basis.
The Weyl group W of g is the group of orthogonal transformations on h∗ generated by the reflections
∨ si(λ) = λ − hλ, αi iαi for i = 0, . . . , `. Using the isomorphism ν, W also acts by orthogonal transformations ◦ ◦ on h. Let W be the subgroup generated by si, 1 ≤ i ≤ `, then W is the Weyl group of ◦ the underlying simple Lie algebra g. Moreover, the Weyl group W can be decomposed as the semidirect product ◦ ◦ ∼ W = W n T = W n M ◦ −1 where M is the lattice spanned by the set W (a0 θ), T is the abelian group of trans- ◦ lations by elements of M (acting on h∗), and W acts on M in the natural way. The 14
(r) elements of T are written as tα, α ∈ M. For g of type XN , in the case r = a0 the ◦ ◦ ∨ lattice M = ν(Q ) and in the case r > a0, M = Q (see the next subsection for the definitions of relevant lattices).
2.2.2 Integrable Highest Weight Modules
Throughout this work, we will be dealing with the category of integrable highest weight modules of level k, where k is a positive integer. This category is endowed with the usual direct sum structure. Quite significantly, the usual tensor product does not preserve the level. In order to describe these modules, let us very rapidly recall the various lattices and sets that feature in the representation theory.
Let g be an affine Kac-Moody algebra. Recall that the root lattice Q is the lattice
∗ ∨ in h generated by the simple roots α0, . . . , α`. The coroot lattice Q is the lattice in h
∨ ∨ ∗ generated by α0 , . . . , α` . Define the fundamental weights to be elements Λi ∈ h such ∨ ∗ that Λi(αj ) = δi,j, for i, j = 0, . . . , `. The weight lattice P is the lattice in h generated + by the fundamental weights Λ0,..., Λ`. The dominant integral weights P is the set of
∨ ++ λ ∈ P such that hλ, αi i ≥ 0 for i = 0, . . . , `. The regular dominant integral weights P ∨ are the dominant integral weights such that the above inequality is strict for all αi . When decorated above by the symbol ◦, these lattices are the corresponding lattices ◦ for g.
A very important element of P ++ is
ρ := Λ0 + Λ1 + ··· + Λ`.
The level k of an element λ ∈ h∗ is the quantity hλ, Ki. Note that the level of ρ is h∨. Define the sets
Pk = {λ ∈ P : hλ, Ki = k and (λ, α) ∈ Z for all α ∈ M},
◦ k ∨ ∨ ∨ P = {λ ∈ P : hλ, Ki = k and hλ, α i ∈ Z for all α ∈ Q },
+ + ++ ++ k+ k + k++ k ++ Pk = Pk ∩ P ,Pk = Pk ∩ P ,P = P ∩ P ,P = P ∩ P . 15
For an affine algebra g, an integrable highest weight module of highest weight λ is a non-trivial g-module V with decomposition
M V = Vµ, µ∈h∗
Vµ = {v ∈ V : h · v = µ(h)v for all h ∈ h},
+ such that λ ∈ P , there is an element v0 ∈ Vλ (unique up to scaling) that generates
V , and eiv0 = 0 for all i = 0, . . . , `. The irreducible integrable highest weight modules
+ of highest weight λ ∈ P are denoted Lλ. The central element K acts by a scalar k, called the level, on Lλ. The terminology is consistent since the level of Lλ is the level of λ as an element of h∗.
2.2.3 Theta Functions, Modular Invariance, and Characters
This subsection discusses characters and theta functions associated with affine Lie al- gebras. For a full account of the now classical theory, see [KP] or [Ka]. Significantly,
− at the end of this section, we define an action of SL(2, Z) on the space T hk+h∨ that features prominently in the definition of the Verlinde algebras.
Let g be an affine Lie algebra, k ∈ Z>0, and λ ∈ Pk, then the classical theta function of level k with characteristic λ is defined
(λ,λ) − δ X tα(λ) Θλ = e 2k e , α∈M
(recall the notation tα given at the end of 2.2.1). It can be shown that λ ≡ µ
(mod kM + Cδ) implies Θλ = Θµ. These functions are defined on the half-space
Y = {λ ∈ h| Re(λ, δ) > 0}. The space of theta functions T hk is a subspace of the vec- tor space of functions on Y certain invariance properties with respect to a Heisenberg group. Due to the following result, which can be found in [Ka,§13], we will not need the formal definition.
Proposition 2.5. Let k > 0, then the set of functions {Θλ|λ ∈ Pk (mod kM + Cδ)} is a C-basis of T hk. 16
The space T hk admits a right action of the metaplectic group Mp(2, Z), which is a double cover of a b SL(2, Z) = : a, b, c, d ∈ Z and ad − bc = 1 . c d
The group SL(2, Z) has the generators 1 1 0 −1 T = and S = 0 1 1 0
which play a prominent role. The action of X ∈ Mp(2, Z) on Θλ is denoted Θλ|X . The same notation will be used when referring to the action of SL(2, Z) when applicable.
For later use, we record the action certain elements of Mp(2, Z) on T hk,
πi|λ|2/h Θλ|(T,j1) = γ(T, j1, `)e Θλ
`/2 ∗ −1/2 X −2πi(w(λ),µ)/k Θλ|(S,j2) = γ(S, j2, `)(−i) |M /kM| (w)e Θµ ◦ µ∈Pk,w∈W where ` is the rank of M, λ ∈ Pk, j1, j2 are holomorphic functions whose squares are 1, τ, resp., and the function γ takes values in {1, −1}.
For r = 1, 2, 3, the congruence subgroups a b a b 1 ∗ Γ1(r) = : a, b, c, d ∈ Z and ≡ (mod r) , c d c d 0 1
are also important. The group Γ1(r) is generated by the matrices 1 −1 1 0 −1 r u12 = T = and u21 = 0 1 r 1
3 r 2r which satisfy the relations (u12u21) = −1, (u12u21) = 1 for r ∈ {2, 3}, and S =
u12u21u12 [IS].
If V is a g-module with weight space decomposition
M V = Vµ µ∈h∗ 17
such that dim Vµ < ∞, the formal character of V is the formal sum
X µ ch(V ) = (dim Vµ)e . µ∈h∗ For any λ ∈ P k+, define the normalized character or simply character
−mλδ χλ = e ch(Lλ)
where mλ is the so-called modular anomaly of λ |λ + ρ|2 |ρ|2 m = − . λ 2(k + h∨) 2h∨
The complex vector space spanned by the characters χλ of g of level k will be denoted
Chk := Chk(g). The characters can be expressed in terms of classical theta functions. To explain this, first define the alternants
X Aλ = (w)Θw(λ) ◦ w∈W − + and let T hk be the space spanned by the Aλ for λ ∈ Pk . The following result can also be found in [Ka,§12].
Proposition 2.6. If k > 0 and λ ∈ P k+, then
χλ = Aλ+ρ/Aρ.
In the case when r ≤ a0, the space of characters is invariant under the action of
SL(2, Z), i.e., the action of Mp(2, Z) factors through SL(2, Z). In the case r > a0, Chk
is invariant under Γ1(r). The explicit actions of these subgroups will be discussed in the upcoming sections.
− In the case that r ≤ a0, the spaces Chk and T hk+h∨ are linearly isomorphic. By − transport of structure, this defines an action of SL(2, Z) on T hk+h∨ that is different
from the (restriction of the) Mp(2, Z) action defined on T hk+h∨ . In the case that r > a0, − there is an embedding Chk ,→ T hk+h∨ defined by χλ 7→ Aλ+ρ and, again by transport
of structure, Γ1(r) acts on the image of this embedding. In fact, we can define an − SL(2, Z) action on T hk+h∨ , for k ≥ 0, that extends the action of Γ1(r) up to scalar in the following way. 18
− Proposition 2.7. Let g be an affine Kac-Moody algebra, k ∈ N, and T hk+h∨ be as − above. Let U ⊆ T hk+h∨ be the image of Chk under the embedding χλ 7→ χλAρ, equipped
with the linear action αr of Γ1(r) induced by this embedding. Then then there is a linear − action α1 of SL(2, Z) on T hk+h∨ such that for any F ∈ U,
α1(u12) · F = c12αr(u12) · F
r r α1(u21) · F = c21αr(u12) · F
where c12 and c21 are nonzero scalars.
Proof. If r ≤ a0, then, by the above discussion, we are done. Suppose that r > a0, ◦ ∨ + then M = Q. Let ` be the rank of g (or, equivalently, M) and let ρ ∈ Ph be defined ∨ − (modulo Cδ) by (ρ , αj) = 1, for j = 1, . . . , `. Note that Aρ∨ 6= 0 and T hh = CAρ∨ . − The action of Mp(2, Z) on T hh is given by
πi|ρ∨|2/h ∨ ∨ Aρ |(T,j1) = γ(T, j1, `)e Aρ
`/2 ∗ −1/2 X −2πi(w(ρ∨),ρ∨)/h ∨ ∨ Aρ |(S,j2) = γ(S, j2, `)(−i) |M /hM| (w)e Aρ ◦ w∈W where j1, j2, γ are defined earlier in this section. Notice that only γ depends on j1, j2, and `.
Now, let M Wk+h∨ = CAλ+ρ/Aρ∨ , + λ∈Pk − ∼ then T hk+h∨ = Wk+h∨ . The natural action of Mp(2, Z) is easily seen (due to the cancellation of the γ terms) to descend to an action of SL(2, Z) on Wk+h∨ . Using the − isomorphism, we equip T hk+h∨ with that action of SL(2, Z). Denote this action by α1.
To show that α1 and αr differ by scalars, first note that αr(T ) acts diagonally with
k++ respect to the basis Aλ, λ ∈ P . Second, note that u21 = TST. The Mp(2, Z) action r 2 of (u21, j), for a holomorphic j such that j = rτ +1, on the numerator of χλ = Aλ+ρ/Aρ r is a scalar multiple of the αr action of u21 on Aλ+ρ. Similarly, also by definition, the r Mp(2, Z) action of (u21, j), as above, on the numerator of Aλ+ρ/Aρ∨ is a scalar multiple r of the α1 action of u21 on Aλ+ρ. The result follows. 19
2.3 Verlinde Algebras of Untwisted Affine Algebras
This section introduces the well-known Verlinde algebras of untwisted affine Lie al- (1) gebras, i.e., affine Kac-Moody Lie algebras of type XN , and discuss their grading structure, including the proof of theorem 2.4.
Let g be an untwisted affine Lie algebra and k a positive integer that will serve
as the level. Recall that in this case Chk is invariant under the action of SL(2, Z). More importantly, the action of the element S ∈ SL(2, Z) defined earlier 2.2.3 encodes
information about a commutative associative algebra structure on Chk. The S-matrix, which gives the action of S with respect to the basis of characters, encodes the structure constants of this algebra via the Verlinde formula, which we restate for this particular
P ν case, χλχµ = + N χν where ν∈Pk mod Cδ λµ
−1 X SλφSµφ(S )νφ N ν = . λµ S + kΛ0,φ φ∈Pk mod Cδ
ν The structure constants Nλµ are known as the fusion rules, are well-defined, and corre- spond to a tensor product structure on an appropriate category of modules for g (which will not be discussed here).
First, let us give an explicit description of the SL(2, Z) action on Chk, which can be found in [Ka]. Afterwards, it will be apparent that the Verlinde algebras are well- defined. Surprisingly, the S-matrix can be expressed in terms of evaluations of char- ◦ ◦ acters of irreducible highest weight g-modules. In what follows, the function χλ is the ◦ character of the irreducible highest weight g-module with highest weight λ. By [KP],
Chk is invariant under the action of SL(2, Z). The action, denoted A : χλ 7→ χλ|A where A ∈ SL(2, Z), is described explicitly in terms of the generators T and S.
(1) (2) Theorem 2.8 (Kac-Peterson). Let g be an affine Lie algebra of type XN or A2` and + k be a positive integer. For any λ ∈ Pk ,
2πim χλ|T = e λ χλ X χλ|S = Sλµχµ + µ∈Pk mod Cδ 20 where
◦ 2πi(λ+ρ,w(µ+ρ)) |∆+| ∗ ∨ −1/2 X − ∨ Sλµ = i |M /(k + h )M| (w)e k+h ◦ w∈W ◦ − 2πi(λ+ρ) k+h∨ = cλχµ e
∗ ∨ −1/2 Q π(µ+ρ,α) and cλ = SkΛ0,λ = |M /(k + h )M| ◦ 2 sin k+h∨ . α∈∆+
Note that the constant cλ = SkΛ0,λ in the above theorem is a positive real number. Furthermore, the theorem shows that the S-matrix is symmetric, therefore the product
in Vk is commutative. The associativity follows from the fact that χkΛ0 is decreed to be the identity element and the columns of the S-matrix are assumed to be, up to
nonzero scalars, a system of orthogonal idempotents of Vk and, under this assumption, the Verlinde formula simply computes the structure constants of the resulting algebra. t It is worth noting that S−1 = S = S. We now proceed to prove theorem 2.4, i.e., that ◦ ◦ these Verlinde algebras are graded by P/Q.
Proof of 2.4. Let us denote by P (µ) the set of weights (regarded as a multiset) of the ◦ irreducible finite dimensional g-module with highest weight µ. By the Kac-Walton
+ algorithm [W], the fusion rules for λ, µ, ν ∈ Pk are of the form X N ν = (w )N φ λµ φ λµ φ ◦ where φ ranges over a finite multiset of integral weights of g and wφ are certain elements in W . Therefore, N ν = 0 implies that N ν = 0. Using the Racah-Speiser algorithm λµ λµ [FH,§25.3], the finite fusion rules have the form
X N ν = (w ) λµ φ φ where the sum ranges over all φ ∈ P (µ), such that wφ(λ + φ + ρ) − ρ = ν for some ◦ wφ ∈ W . Note that such wφ are unique (when they exist).
◦ ◦ ◦ ◦ ◦ In these cases, M = Q∨ ⊆ Q ⊆ M ∗ = P so let A = M ∗/Q. Notice that W acts trivially on A. Define the subspaces
◦ Va = span{χλ|λ ≡ a (mod Q)} 21 for a ∈ A. Recall that whenever φ ∈ P (µ), µ − φ is a sum of positive roots. Therefore, from the above discussion we know that
◦ ν = wφ(λ + φ + ρ) − ρ ≡ λ + µ (mod Q),
ν L proving that Nλµ = 0 whenever λ + µ 6= ν in A. This proves that V = a∈A Va is a grading.
Remark 1. The theorem above indicates a relationship between the center of the asso- ciated simply connected compact Lie group G and the grading structure. There may be an interpretation of this mystery in terms of an action of Z(G) on the Verlinde algebra or on conformal blocks.
(1) Example 2.9. Let us observe the grading structure in the case when g is of type A2
with level k = 2. Consider the ordered basis χλ1 , χλ2 , . . . , χλ6 where the λi are
0, Λ1 + Λ2, Λ1, 2Λ2, 2Λ1, Λ2, respectively. The weights have been paired off so that they lie in the classes 0, Λ1, 2Λ1 ∈ ◦ ◦
P/Q respectively. The matrices for Li, the operator of left multiplication by χλi , are 1 0 0 1 0 1 1 1 1 0 1 1 L1 = L2 = 0 1 1 0 1 0 0 1 0 1 1 1
0 1 1 0 1 1 0 1 1 1 0 1 L3 = L4 = 0 1 1 0 1 0 0 1 1 1 1 0 22
0 1 1 0 1 0 1 1 0 1 1 1 L5 = L6 = . 1 0 0 1 1 0 0 1 0 1 1 1
(1) Example 2.10. Let us observe the grading structure in the case when g is of type B3 with level k = 2. Consider the ordered basis χλ1 , χλ2 , . . . , χλ7 where the λi are
0, Λ1, 2Λ1, Λ2, 2Λ3,
Λ3, Λ1 + Λ3
◦ ◦ respectively. There are only two classes {0, Λ3} = P/Q and so Vk has a Z2-grading. (1) (Indeed, this is always the case for type B` , with the system of representatives {0, Λ`}.)
The first row lies in the class of 0 and the second row lies in the class of Λ3. The matrices
for the Li, the operator of left multiplication by χλi , are L1 = id 0 1 0 0 0 0 0 1 0 0 1 0 1 1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 L2 = 0 1 0 0 1 L3 = 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 1 1 0 1 1 1 1 0
0 0 0 1 0 0 0 0 0 1 0 1 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 L4 = 1 0 1 0 1 ,L5 = 0 1 0 1 0 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 1 1 1 23
1 0 0 1 1 1 1 1 0 1 1 0 L6 = 1 1 L7 = 1 1 . 1 1 1 1 1 1 0 1 1 0 1 1 1 1 0 1 1 1 1 1 1 0 1 1
This grading structure also applies to the case where k = ∞, i.e., when the Verlinde ◦ algebra is equal to the representation algebra of g, since it depends only on the Racah- Speiser algorithm for computing the tensor product decomposition of finite dimensional ◦ simple g-modules.
A method of describing this algebra more conceptually is via the representation ring ◦ ◦ R(g) of the finite dimensional simple Lie algebra g. (This interpretation of the Verlinde algebra will be used later on.) There is a map
◦ Fk : R(g) → Chk(g)
+ ◦ (w)χw(λ+ρ)−ρ if there exists w ∈ W, w(λ + ρ) − ρ ∈ Pk χλ 7→ 0 otherwise
+ where λ ∈ Pk (note that the w referenced in the definition of Fk is unique). There is also a canonical section ◦ Sk : Chk(g) → R(g)
◦ χλ 7→ χλ.
The following proposition was proved in [W].
Theorem 2.11 (Walton). The Verlinde algebra Vk has the product structure
χλ · χµ = Fk(Sk(χλ) ⊗ Sk(χµ))
+ for λ, µ ∈ Pk . 24
k+h∨ Instead of using W in the definition of Fk, we can use the affine Weyl group Waff = ◦ ◦ ◦ ∨ ∨ ∗ aff + W n (k + h )Q , which acts on h . The fundamental alcove Pk+h∨ is defined to be ◦ the set of λ ∈ h∗ such that
0 ≤ (λ, αi) for i = 1, . . . , `
(λ, θ) ≤ k + h∨,
k+h∨ aff ++ this is a fundamental domain for the action of Waff . The set Pk+h∨ consists of aff + ∗ all λ ∈ Pk+h∨ such that the above inequalities are all strict. For any λ ∈ h of level aff + aff ++ k, λ + kΛ0 ≡ λ (mod Cδ). Using λ 7→ kΛ0 + λ, the sets Pk and Pk can be + ++ identified with Pk and Pk , respectively. Therefore, we can also define the map Fk by the rule
k+h∨ aff ++ (w)χ if there exists w ∈ W , w(λ + ρ) ∈ P ∨ ◦ kΛ0+w(λ+ρ)−ρ aff k+h χλ 7→ . 0 otherwise
Later on in 2.6, this definition will prove very useful.
Remark 2. While the representation ring description of these Verlinde algebras show that the structure constants are integers, these algebras have an interpretation as the Grothendieck ring of a tensor category of modules for untwisted affine Lie algebras and, therefore, the structure constants are all non-negative integers [Hu]. For the twisted cases, the Verlinde algebras will be described as a quotient of the representation ring of a finite dimensional simple Lie algebra and non-negativity of structure constants will (2) not hold in exactly the case A2` when the level is even.
2.4 Verlinde Algebras of Twisted Affine Algebras
(r) The affine Kac-Moody Lie algebras of type XN for r > 1 are twisted affine Lie algebras and their spaces of characters behave very differently from those of untwisted affine Lie ◦ k algebras. In these cases, M = Q and P ( Pk, therefore we cannot use the same (1) techniques used for the algebras of type XN . In particular, there is no action of the S- matrix. Nonetheless, to each twisted affine Lie algebra g of this type, we will associate 25
a Verlinde algebra Vk(g) that contains space of characters of g at level k and admits the action of the modular group. In particular, we prove theorems 2.1 and 2.3 in this section by comparing a twisted affine algebra g to its transpose gt. Since the Verlinde (2) algebras for type A2` are fundamentally different from the others, we will more deeply explore them in the next section.
The lattice P k is the dual of the root lattice, therefore it has the basis
a1 a2 a` ∨ Λ1, ∨ Λ2,..., ∨ Λ`. a1 a2 a`
+ ∨ + If λ ∈ Pk then (λ, θ) = hλ, θ i ≤ k. Conversely, any λ ∈ P satisfying the above + inequality has unique preimage λ ∈ Pk modulo Cδ.
In [Ka], Kac relates each g of the type considered here to another affine algebra
0 denoted g (this will be discussed later). For the purpose of investigating Chk, it will (1) be helpful to instead relate g to an affine Lie algebra of type XN . To each affine Lie t t algebra with r > a0, associate the transpose algebra g = g(A ), where A is the Cartan matrix of g. The following table gives the explicit correspondences.
g gt g0
(2) (1) (2) A2`−1 B` D`+1 (2) (1) (2) D`+1 C` A2`−1 (2) (1) (2) E6 F6 E6 (3) (1) (3) D2`−1 G` D2`−1
The superscript t will be used when dealing with data associated with gt, e.g., the
t t t labels of g are denoted ai, its Cartan subalgebra is denoted h , the invariant bilinear form is (·, ·)t, etc. In order to compare the two algebras more readily, define the isometry
T : h∗ → ht
∨t αi 7→ αi
t Λ0 7→ d 26 which in turn induces an isometry h∗ → (ht)∗, also denoted T , given by
∨ ai t t Λi 7→ Λi and δ 7→ δ . ai
This map ‘transposes’ the roots and coroots and is crucial in the construction of the
Verlinde algebra for twisted affine algebras. The restriction map T |M is an isomorphism of lattices M =∼ M t and T (θ) = θt,T (θ∨) = θ∨t.
k k t t ∼ The latter fact shows that T P ,→ (P ) = (Pk) = Pk. Recall that {Θλ|λ ∈ Pk
(mod kM + Cδ)} is a basis of T hk for k > 0. Therefore, T induces a canonical map between the theta functions of g and gt.
∼ t Proposition 2.12. The isometry T induces an W -invariant isomorphism T hk = T hk.
◦ ◦ t t t ∼ Proof. A direct check shows that W = W n M = W n M and
t e(λ,µ) = e(T (λ),T (µ)) .
t ∗ t ∗ Therefore, the map Θλ 7→ ΘT (λ) is an inclusion. Furthermore, since T (M ) = (M ) , t proposition 2.5 implies that T (T hk) = T hk.
Since the finite Weyl groups of g and gt are isomorphic and commute with isometries
− between their Cartan subalgebras, the space T hk , which is spanned by the alternants X Aλ = (w)Θw(λ) ◦ w∈W
t − for λ ∈ Pk, can be identified with the space T hk .
At this point, one should notice that T (ρ) 6= ρt and so, instead of using the map T between theta functions given above, consider the map
• t T : Pk → Pk+h∨−h
λ 7→ T (λ + ρ) − ρt and the map it induces on the spaces of characters, denoted by the same symbol,
• t T : Chk → Chk+h∨−h 27
t χλ 7→ χ • . T (λ) • ∨ + Note that this map is a well-defined injection, i.e., h − h ≥ 0 and T (λ) ∈ Pk+h∨−h, • • and that T (χλ) = AT (λ+ρ)/Aρt . Furthermore, T gives a tight relationship between the S-matrices of the two Lie algebras. Before describing it, recall that for any g of the type considered in this section, the adjacent algebra g0 is given by the above table. The S-matrix of g, as defined in [Ka], does not act on the space of characters but, rather, is a map between different character spaces. Nonetheless, its image can be described in terms of the characters of the adjacent algebra g0. For more details, please refer to [Ka,§13.9]. In that section, Kac discusses a map α 7→ α0 between the Cartan subalgebras of g and g0 that is very similar to the map T in this paper.
Proposition 2.13. Let S(g) be the S-matrix acting on the space of theta functions at level k and S(gt) be the analogous S-matrix for gt and level k + h∨ − h. Then
◦ 1/2 ∨ t S(g) = Q /M S(g ) • λ,µ ∨ t T (λ),T (µ)+(h −h)Λ0
k λ ∈ P , µ ∈ Pk.
Proof. The result follows directly from the dictionary between the map given in [Ka] • and the construction of the two maps T and T in this paper. Since ρt = T (ρ0) and ◦ M 0 = Q∨, comparing the two quantities below
◦ 2πi(w(λ+ρ),µ+ρ0) |∆+| ∗ ∨ −1/2 0 1/2 X − ∨ S(g)λ,µ = i |M /(k + h )M| |M /M| (w)e k+h ◦ w∈W
◦ t t 2πi(w(T (λ+ρ)),T (µ)+ρ ) t |∆ +| t∗ ∨ t −1/2 X − ∨ S(g ) • = i | M /(k + h )M | (w)e k+h T (λ),T (µ)+(h∨−h)Λt 0 ◦ w∈W gives the result.
t− t ∼ Theorem 2.1 asserts that the space V is equal to T hk+h∨ . Notice that Chk+h∨−h = t− t t T hk+h∨ via the map χλ 7→ χλAρt . Using this isomorphism, we can study the action of t− S on T hk+h∨ and prove the theorems 2.1 and 2.3. 28
t− t− Proof of 2.1. The space V is a subspace of T hk+h∨ . The action of S on T hk+h∨ differs t from its action on Chk+h∨−h by a nonzero scalar, see Theorem 2.8. The S-matrix of t t+ Chk+h∨−h has SkΛ0,,λ = Sλ,kΛ0 6= 0 for all λ ∈ Pk (see Theorem 2.8) and so the coefficient of Aρt in the expansion of Aλ+ρ|S (with resect to the alternant basis) is − t+ nonzero. It follows that Aρt ∈ V . Similarly, for any µ ∈ Pk the coefficient of Aµ+ρt t− in the expansion of Aρt |S is nonzero. It follows that T hk+h∨ ⊆ V . t Note that ρe = ρ and that the S matrix has nonzero entries in the column corre- sponding to A . Therefore, we can use the Verlinde formula to give V the structure of ρe a commutative associative algebra as discussed in section 2.1.
Theorem 2.3 is a direct consequence of above proof. Using the Verlinde algebra
t t ∨ Vk+h∨−h of g at level k + h − h, one can study the action of the modular group on
Chk(g) directly.
k Theorem 2.14. Let g be a twisted affine Lie algebra, λ ∈ P , and X ∈ Γ1(r) then
t T (Aλ+ρ|X ) = AT (λ+ρ)|X .
Furthermore, there is a nonzero scalar v(k, X) such that
• • T (χλ|X ) = v(k, X)T (χλ)|X .
Proof. T is an isometry on the underlying Cartan subalgebras and induces an isomor-
t phism between the spaces of theta functions T hk+h∨ and T hk+h∨ . Therefore, the mod- ular group action (actually, Mp(2, Z) action) commutes with T . The second identity follows from the fact that Aρ|X ∈ CAρ \{0} and Aρt |X ∈ CAρt \{0}.
Let us investigate the structure of Vk relative Chk. The grading structure (intro- (2) duced in 2.4) on Vk(g), where g is a twisted affine Lie algebras different from type A2` , is easy to describe
(2) (2) Vk(A2`−1) = V0 ⊕ V1,Vk(D`+1) = V0 ⊕ V1
(2) (3) Vk(E6 ) = V0,Vk(D4 ) = V0. 29
• Using the grading and comparing it to the map T , it can be shown that the image of (2) (2) Chk(g) lies in VT (ρ)−ρt for g of type A2`−1. The image of Chk for type D`+1 is generally not concentrated in a homogeneous component.
(2) • Proposition 2.15. If g is of type A2`−1, then Vk = V0 ⊕ V1 and T (Chk) = V1.
Proof. The first part is clear. For the second part, note that
◦ ◦ t (2) (1) t t t t t Q (A2`−1) = Q(B` ) = hΛ1,..., Λ`−1, 2Λ`i and T (ρ) − ρ = Λ`, ◦ t (2) where h, i means ‘spanned by’. Therefore, VT (ρ)−ρt = V1. The lattice Q (A2`−1) is self-dual which implies the equality
• t T (Chk) = VT (ρ)−ρt = V1.
(2) 2.5 Verlinde Algebras for A2`
(2) The case when g is of type A2` has fascinating features that differ substantially from the other cases. On one hand, at odd integer level k the space of characters has the (1) structure of a fusion algebra that is isomorphic to the Verlinde algebra for C` at k−1 the level 2 [Ho2]. (A proof is included here.) On the other hand, more interesting features reveal themselves when the level is an even integer. Among them is the fact that the structure constants are no longer non-negative integers.
It can be verified, see 2.6, that the fusion rules in this case are well-defined and de- fine a commutative associative unital algebra with integral structure constants. Similar to the untwisted Verlinde algebras, one can define an algebraic structure on the repre- ◦ sentation ring of g via a map from the representation ring of g. Unlike the untwisted case, the fusion rules can only be shown to be non-negative at odd level (see prop 2.16 below).
(2) Proposition 2.16. Let g be of type A2` and fix the level 2k + 1, where k ∈ N, then the Verlinde algebra of g is isomorphic to the Verlinde algebra of the untwisted affine Lie (1) algebra of type C` at level k. 30
Proof. It can be checked directly that the S-matrices of both affine Lie algebras are
0 (1) equal, up to conjugation by a permutation matrix. More explicitly, if g is of type C` with Cartan subalgebra h0,
∨ ∨ ∨0 ∨0 0 2(αi , αj ) = (αi , αj )
1 0 0 (Λ , Λ ) = (Λ , Λ )0 2 i j i j for 1 ≤ i, j ≤ `. Therefore, we can define isometries
◦ ◦ ϕ : h0 → h
√ α 7→ 2α and the dual map ◦ 0 ◦ ϕ∗ : h ∗ → h∗ 1 α 7→ √ α 2 √ such that hϕ(α), ϕ∗(λ)i = hα, λi. Note that 2M = ϕ(M 0) and √1 M ∗ = ϕ∗(M 0∗), 2 therefore |M 0∗/(k + `)M 0| = |M ∗/(2k + 2`)M|.
The S-matrix of g0 at level k is defined by
◦ 2πi(w(λ+ρ),µ+ρ)0 |∆+| 0∗ 0 −1/2 X − Sλµ = i |M /(k + `)M | (w)e k+` ◦ w∈W while the S-matrix of g at level 2k + 1 is defined by
◦ 2πi(w(λ+ρ),µ+ρ) |∆+| ∗ −1/2 X − Sλµ = i |M /(2k + 2`)M| (w)e 2k+2` . ◦ w∈W Since the structure constants of the Verlinde algebras are computed by the coefficients of the corresponding S-matrices, the algebras are isomorphic.
In the cases where the level is even, one can observe that the fusion rules are often negative. In fact, it is true in every case we could compute. 31
(2) Conjecture 2.17. Let g be an affine Lie algebra of type A2` and fix the level 2n for n ∈ Z>0. For any λ, µ ∈ P2n,
X [λ]+[µ]+[ν] ν χλχµ = (−1) |Nλµ|χν + ν∈P2n mod Cδ ∼ in V2n(g), where [φ] is the class of φ in A = {0, 1}. In particular, the structure con-
[λ] stants of V2n(g), with respect to the basis {χλ} + , where χλ = (−1) χλ, are e λ∈P2n mod Cδ e nonnegative.
We can show exactly when negative structure constants appear in the following case. (2) Let g be of type A2 and fix a level 2n, n > 0. The underlying simple Lie algebra is ∨ ∨ isomorphic to sl2. Set α = α1, α = α1 , Λ1 = Λ then
θ = α, θ∨ = α∨, and ρ = α∨.
Note also that h∨ = 3. The of dominant integral weights of level 2n project to
2n n P + = {mΛ}m=0.
The set of dominant integral weights P+ of sl2 is indexed by N and the map between character spaces is
F2n : {χm}m∈P → {χλ} 2n + λ∈P+ 0 0 ∨ 0 χm Λ if 0 ≤ m ≤ n and m ≡ m (mod 2n + h ) ∨ χm 7→ 0 if m ≡ −1 (mod 2n + h ) . 0 0 ∨ −χm0Λ if 0 ≤ m ≤ n and m + m + 2 ≡ 0 (mod 2n + h ) The proof of the above map follows from the fact that the affine Weyl group of type
(2) 0 ∨ A2` is S2 n Z and so, for any m ∈ N, there is a unique m modulo 2n + h satisfying the conditions in the piecewise definition of F2n. More importantly, there is a canonical section
S2n : {χλ} 2n → {χm}m∈P λ∈P+ +
χmΛ 7→ χm
c and the fusion rules Nab are given by the coefficients of the irreducible character χc in
F2n(S2n(χaΛ) ⊗ S2n(χbΛ)). 32
Th product S2n(χaΛ) ⊗ S2n(χbΛ) is the direct sum of χcΛ where |a − b| ≤ c ≤ a + b and a + b + c ≡ 0 (mod 2). The tuples (a, b, c) with the above property are known as admissible triples. More explicitly, the tensor product of sl2-modules is given by‘
χaΛ ⊗ χbΛ = χ|a−b|Λ ⊕ χ(|a−b|+2)Λ ⊕ · · · ⊕ χ(a+b)Λ.
Assume that a ≥ b. Since a + b ≤ 2n we have two cases: a + b ≤ n or n < a + b ≤ 2n. The above discussion proves the following.
(2) Proposition 2.18. For g of type A2 at level 2n, the product structure of V2n(g) is explicitly given by
X X χaΛ ⊗ χbΛ = χ(a−b+2i)Λ − χ(2n+1+b−2i−a)Λ. n+b−a n+b−a 0≤i≤ 2 2 c It follows that Nab < 0 whenever (a, b, c) is admissible, a + b > n, and c > n. In 1 2 particular, Nn,n = −1 when n is even and Nn,n = −1 when n is odd. Later on, in section (2) 2.6.2, we will give a more explicit description of the product structure of V2n(A2` ) for any rank `. The following curious fact shows that when the level is 2n and 2n + 1, the underly- ing character spaces are equal while the Verlinde algebra structures are quite different. ν Moreover, I observed that, at even level, if the structure constants Nλµ are replaced by ν their absolute values |Nλµ|, then the resulting algebra is also associative and commu- tative. 2n 2n+1 Lemma 2.19. Let k ≥ 1, then P + = P + . 2n 2n+1 2n+1 Proof. It is a straightforward check to show that P + ⊆ P + . Suppose that λ ∈ P+ P` and λ = i=0 ciΛi, then ci ∈ N for 0 ≤ i ≤ ` and * ` + ∨ X ∨ hλ, Ki = λ, α0 + 2 αi = 2n + 1. i=1 0 + 0 This equation implies that c0 > 0 and so we can define λ ∈ P2n by λ = λ − Λ0. 2n 2n+1 Therefore, the earlier inclusion and P + ⊇ P + implies the lemma. 33 Another different feature in this case is that the grading structure (as an algebra) (1) only exists when the level is odd (since it exists in Vk(C` )). Nevertheless, when ◦ ◦ ∼ the level is even we can still define the same quotient A = P/Q = Z2 and get the L decomposition Vk(g) = V0 V1 as a vector space. Empirically, we have observed the following phenomenon and conjecture that Vk(g) can be realized as a commutative associative algebra with nonnegative integral structure constants. 2.6 Applications 2.6.1 Quotients of Vk(g) The definition of Verlinde algebras for the various types of affine Lie algebras gives an explicit way of computing the structure constants of these algebras. In a recent paper by J. Hong [Ho2], the author attaches a fusion ring to all affine Lie algebras and he expects that the corresponding structure constants are all non-negative. Here we show that Hong’s fusion algebras can be realized as quotients of the Verlinde algebras presented here. With this realization, we can answer some questions posed in [Ho2]. (2) In the case of A2` , the Verlinde algebras and Hong’s fusion algebras are isomorphic, proving that the structure constants are sometimes negative (specifically, when the level is even they are negative in all known cases). In the remaining cases, computations show that the structure constants are also sometimes negative (depending on the level and type). First, let us describe the quotients of the Verlinde algebras to be considered. Let k be a positive integer throughout this section. Recall the map ◦ Fk : R(g) → Chk(g) given in section 2.3 where g is an untwisted affine Lie algebra of rank `. This map depends on the affine Weyl group ◦ ◦ ∨ W n Q ◦ (where, as usual, Q∨ is identified with a lattice in h∗ using the invariant bilinear form.) ◦ ◦ 0 0 Consider, instead, the group W := W nQ. When g is simply laced, W = W and so we 34 (1) (1) (1) (1) 0 may assume that g is of type B ,C ,F , or G . The group W is generated by s 0 ` ` 4 2 αi 0 0 ◦ where αi = αi for i = 1. . . . , ` and α0 = δ − θs, where θs is the highest short root of g. 0 t0 The αi give a root system (that is equivalent to that of g ) and there are corresponding 0 0 0 P` 0 notions for fundamental weights Λ0, Λ1,..., the element ρ = i=0 Λi, the weight 0 L` 0 0+ lattice P = i=0 ZΛi, the level k dominant weights Pk , the space of ‘primed’ level k 0 L characters Ch (g) = 0+ CAλ0+ρ0 /Aρ0 , etc. The corresponding dual Coxeter k λ∈Pk mod Cδ number h∨0 coincides with the Coxeter number h. Note that ρ0 = ρ. In a little while, we will see why the ‘prime’ notation is suitable. ◦ ◦ ∨ The map Fk+h−h∨ uses the action of W n (k + h)Q in its definition. Using the ◦ ◦ 0k+h ‘primed’ version of the affine Weyl group Waff = W n (k + h)Q, define the map 0 ◦ 0 Fk : R(g) → Chk(g) 0k+h 0aff ++ ◦ (w)χ 0 if there exists w ∈ W , w(λ + ρ) ∈ P kΛ0+w(λ+ρ)−ρ aff k+h χλ 7→ . 0 otherwise ◦ ◦ ∨ k+h 0k+h 0 Since Q ⊆ Q, Waff ⊆ Waff and Fk factors through Chk+h−h∨ (g), i.e., there is a map 0 Gk : Chk+h−h∨ (g) → Chk(g), 0 ∼ such that Fk = Gk ◦ Fk+h−h∨ . Recall that we may identify Chk+h−h∨ (g) = Vk+h−h∨ (g) ∼ t and also, by 2.3, we identify Vk+h−h∨ (g) = Vk(g ). We use the same notation for the corresponding map t 0 Gk : Vk(g ) → Chk(g). Let us now give briefly describe Hong’s fusion algebras for twisted affine Lie algebras. (2) Let g be a twisted affine Kac-Moody Lie algebra that is not of type A2` and let k a positive integer, then Hong’s fusion algebra Rk(g) is defined to be the vector space M ◦ Rk(g) = Cχλ λ∈P k mod Cδ with structure constants X cν = (w)N ν λµ λ,µ † w∈Wk 35 ◦ † k+h∨ where Wk is the set of minimal representations of the left cosets of W in Waff and ◦ ◦ ◦ N ν is the coefficient of χ in χ χ . When g is of type A(2), we use the same definition λ,µ ν λ µ 2` ◦ ◦ ∨ ∨ of Rk(g) except that we use the affine Weyl group W n (k + h )Q . See [Ho2] for more details. The vector spaces Chk(g) and Rk(g) can be naturally identified using the map ◦ k+ χλ 7→ χλ, λ ∈ P . Let us now compare our Verlinde algebras with Rk(g). Let g be a twisted affine Kac-Moody algebra. Recall the correspondence between adjacent algebras g ↔ g0 given in the table in section 2.4. Note that the underlying simple Lie algebra of g0 is isomorphic to the underlying simple Lie algebra of gt. Identify their Cartan subalgebras ◦ ◦ 0 t 0 0∗ h = h and notice that since the θ element of g is the highest short root, α0 ∈ h t 0 t∗ and (α0) ∈ h can also be identified. This identification induces a natural bijection 0 ∼ t 0 0∨ ∨t 0 t Pk = (Pk) . Finally, note that h = (h ) = h . All of this shows that there is a 0 ∼ 0 t natural linear isomorphism Chk(g ) = Chk(g ) given by χλ 7→ χλ. (2) t Now, let g be a twisted affine algebra, not of type A2` , then g is untwisted. Also, ∼ t ∨ t ∨t Vk(g) = Vk+h∨−h(g ) and h = h , h = h . The maps ◦ t t Fk+h∨−h : R(g ) → Chk+h∨−h(g ) ◦ 0 t 0 t 0 Fk : R(g ) → Chk(g ) = Chk(g ) t 0 t Gk : Chk+h∨−h(g ) → Chk(g ), give rise to maps ◦ t Fk+h∨−h : R(g ) → Vk(g) ◦ 0 t 0 Fk : R(g ) → Rk(g ) 0 Gk : Vk(g) → Rk(g ), by abuse of notation, using the identifications discussed earlier. The following theorem 0 appears in [Ho2] and will be used to show that Rk(g ) is a quotient of Vk(g). This will ν ν give a method of computing the structure constants cλµ from the fusion rules Nλµ. 36 Theorem 2.20. (Hong) Let g be a twisted affine Kac-Moody algebra, then, using the 0 above conventions, Fk is a surjective homomorphism. Theorem 2.21. Let g be a twisted affine Kac-Moody algebra and k a positive integer, 0 then there is a surjective homomorphism Vk(g) → Rk(g ). In particular, the fusion 0 algebra Rk(g ) is isomorphic to a quotient of Vk(g). (2) 0 Proof. If g is of type A2` , then g = g and the vector spaces Vk(g) and Rk(g) can be ◦ k+ identified via Aλ 7→ χλ, λ ∈ P . In [Ho2], the structure constants may be computed by the same Verlinde formula as the one found here and so the algebra structures coincide. (2) t Now, let g be a twisted affine algebra not of type A2` , then g is untwisted. Since the map ◦ 0 t 0 Fk : R(g ) → Rk(g ) is surjective, so is 0 Gk : Vk(g) → Rk(g ). ◦ ◦ For χλ, χµ ∈ Vk(g), recall that χλχµ = Fk+h∨−h(χλχµ). Therefore, by 2.20, ◦ ◦ 0 ◦ 0 ◦ Gk(χλχµ) = Gk(Fk+h−h∨ (χλχµ)) = Fk(χλ)Fk(χµ) = Gk(χλ)Gk(χµ). ν Corollary 2.22. Let g be a twisted affine Lie algebra, then the structure constants cλµ 0 of Rk(g ) are given by −1 ν X X SλφSµφ(S )νφ cλµ = , SkΛ0,φ w∈Xν φ∈P 0k+ mod Cδ † t ∨ where Xν = {w ∈ Wk :(w(ν + ρ) − ρ, θ ) ≤ k + h − h}. Example 2.23. The left multiplication matrices of Gk(Vk(g)) are given by ‘truncating and folding’ the left multiplication matrices of Vk(g). More precisely, one removes all columns corresponding to weights λ such that hλ+ρ, θ∨i ≥ k+h∨ (here θ∨ is the highest 37 ◦ coroot of g), removes all rows corresponding to λ such that hλ + ρ, θ∨i is an integral multiple of k + h∨, and then adds all rows that are in the same orbit under the dot action w · λ = w(λ + ρ) − ρ with appropriate sign (w). In particular, consider the data from example 2.10 where the weights are 0, Λ1, 2Λ1, Λ2, 2Λ3, Λ3, Λ1 + Λ3. (2) In this case, g is of type A5 and k = 1. The only λ + ρ, for λ from the list above, ∨ ∨ satisfying hλ + ρ, θ i < k + h = 1 + 6 are 0 and Λ3. The weights Λ1 + ρ, Λ2 + ρ, and ∨ 2Λ3 satisfy hλ + ρ, θ i = 7 and so are thrown out. Finally, adding the remaining rows (all zeros) we get 1 0 0 1 0 0 L1 = and L2 = . 0 1 1 0 2.6.1.1 More Negative Structure Constants In the remaining twisted cases, negative structure constants also appear in a pattern very similar to the appearance of negative structure constants for the Verlinde algebra (2) (2) of A2` . Unlike the conjectured state of affairs for A2` , experimental data shows that it is not possible to ‘twist’ the basis by a character in order to make the structure constants non-negative. Nonetheless, a grading structure plays a role in determining when the structure constants are non-negative. (2) Let g be a twisted affine Lie algebra that is not of type A2` . The algebra R(g) 0 defined in the previous section does not inherit the algebra grading of Vk(g ). Instead, ◦ ◦ there is a vector space grading indexed by P/Q analogous to the untwisted case, i.e., M R(g) = R(g)a a∈A + (2) where R(g)a = span{χλ : λ ∈ Pk and λ ∈ a}. Similar to the A2` case, this vector space grading seems to play a role (albeit different) in determining the non-negativity of the ν structure constants of R(g). The structure constants Nλµ of R(g) can be observed to behave according to the following 2/3’s rule. 38 Conjecture 2.24. Let k be an even positive integer and g be a twisted affine Lie ◦ (2) + algebra that is not of type A2` . If λ, µ, ν ∈ Pk and at least 2 out of λ, µ, ν lie in Q, ν ν then Nλµ ≥ 0, where Nλµ are the structure constants of R(g) with respect to the basis {χλ} + . λ∈Pk (2) Example 2.25. Let us consider the case when g is of type A5 with level k = 2. In this case there are negative structure constants. Consider the ordered basis χλ0 , χλ1 , χλ2 , . . . , χλ4 where the λi are 0, Λ1, 2Λ1, Λ2, Λ3, ◦ ◦ respectively. There are only two classes {0, Λ3} = P/Q. The matrices for the Li, the operator of left multiplication by χλi , are L0 = id 1 1 1 1 1 1 L1 = 1 L2 = 1 1 1 1 1 1 1 1 1 1 1 L3 = 1 L4 = 1 1 1 1 −1 1 −1 1 1 −1 (2) 2.6.2 The A2` Quotient Structure (2) In the case where g is of type A2` , the Verlinde algebra Vk(g) can be realized as a (1) quotient of a Verlinde algebra Vk0 (C` ). In this section, we use the affine Weyl groups in question to explicitly describe the quotient Vk(g). We then explain the relationship between the grading structures of these vector spaces. 39 (2) Lemma 2.26. Let g be of type A2` and fix a positive integer k. Let eg be an affine (1) algebra of type C` , then there exist a surjective homomorphism F : Vk+`(eg) → Vk(g) ◦ given by F (χλ) = Fkχλ, where Fk is the homomorphism defined at the end of section 2.3. Proof. The underlying simple Lie algebras of g and eg coincide and so we will identify their Cartan subalgebras and their duals. The map F is clearly well-defined as a linear map. To show that it is a homomorphism, it is enough to show that the homomorphism ◦ Fk factors through Vk+`(eg). Indeed, Vk+`(eg) is the quotient of the character ring R(g) ◦ ◦ ∨ ∗ by the action of W n (k + 2` + 1)νe(Q ) where νe is the map h → h induced by the normalized bilinear form of eg. If ν is the normalized bilinear form of g, then note that ◦ ◦ ◦ ◦ ∨ ∨ ∨ νe(Q ) = 2ν(Q ) ⊂ Q ⊂ ν(Q ). ◦ ◦ ◦ ∨ Since Vk(g) is the quotient of R(g) by the action of W n (k + 2` + 1)ν(Q ), it follows that Fk factors through Vk+`(eg). Remark 3. Let A1 be the fundamental alcove of the affine Weyl group. Note that the ◦ ◦ ∨ fundamental domain of W n (k + 2` + 1)ν(Q ) is (k + 2` + 1)A1 and the fundamental ◦ ◦ ∨ domain of W n (k + 2` + 1)νe(Q ) is 2(k + 2` + 1)A1. We will soon present an algorithm mapping elements from 2(k + 2` + 1)A1 into (k + 2` + 1)A1. (2) Let g be of type A2` as before. For the next result, let us use the notation w0 := id and wi := si−1si−2 ··· s0 for 0 < i ≤ ` where si ∈ W are the Coxeter generators of the Weyl group W of g. For any list of integers I = (i1, . . . , im), also define wI = wim wim−1 ··· wi1 . In the proof of this next theorem, we define an explicit recursive algorithm defining the quotient indicated by the previous lemma. 40 (2) Theorem 2.27. Let g be of type A2` , fix a positive integer k, and let eg be an affine (1) algebra of type C` . If λ is a dominant weight of eg of level k + ` such that λ + ρ ◦ ◦ ∨ has trivial stabilizer with respect to the affine Weyl group W n ν(Q ), then there is a weight µ of g of level k and a sequence of nonnegative integers I = (i1, . . . , im), where m satisfies 1 ≤ m ≤ k + 2` + 1, such that µ + ρ = wI (λ + ρ). In particular, for the homomorphism given in the previous lemma and λ as above, ◦ F (χ ) = (−1)i1+···+im F (χ ). λ k wI ·λ Proof. Let k be as in the statement and recall that h∨ = 2` + 1. Also recall that θ = ◦ ◦ ∨ ∨ ∨ ∼ ∨ ∨ 2α1 +···+2α`−1 +α` and θ = α1 +···+α` . We identify W = W n(k+h )ν(Q ). The ◦ ◦ ∨ ∨ Weyl group W of g contains the subgroup W n2(k+h )ν(Q ) and maps every dominant ◦ + ∨ 1 ∨ weight λ ∈ P to a unique dominant weight µ such that hµ, θ i = 2 (µ, θ) ≤ k + h . Therefore, for the purpose of describing F , we only need to consider dominant weights µ such that (µ + ρ, θ) < 2(k + h∨). + Let λ ∈ Pk+`(eg) be a regular dominant weight, i.e., a weight such that λ + ρ has ◦ ◦ ∨ ∨ trivial stabilizer with respect to W n (k + h )ν(Q ), then there is a unique element ◦ ◦ ◦ ∨ ∨ + ∨ wλ ∈ W n (k + h )ν(Q ) such that wλ(λ + ρ) ∈ P , and (wλ(λ + ρ), θ) < k + h . If ∨ (λ + ρ, θ) < k + h , then we can take wλ = w0 = id. Suppose that k + h∨ < (λ + ρ, θ) < 2(k + h∨). Write ` ` X ∨ X λ + ρ = biαi = ciΛi i=1 i=1 ◦ ∨ ∗ (where αi is mapped into h via ν) such that bi > 0, ci > 0 for i = 1, . . . , `, then ∨ ∨ k + h < 2b1 < 2(k + h ) since θ = 2Λ1. It is straightforward to check that b1 = 41 c1 + ··· + c`. Applying the affine generator s0, we get ∨ ∨ s0(λ + ρ) = sθ(λ + ρ) + (k + h )θ = λ + ρ − hλ + ρ, θ∨iθ + (k + h∨)θ∨ ∨ ∨ = λ + ρ + (k + h − 2b1)θ ` ∨ X = (k + h − 2b1 + c1)Λ1 + ciΛi. i=2 ∨ ∨ ∨ It follows that (s0(λ+ρ), θ) = 2(k+h −b1) < k+h and, therefore, if k+h −2b1 +c1 > 0, then wλ = w1 = s0. ∨ Suppose that k + h − 2b1 + c1 < 0. Compute the following weights ∨ s1s0(λ + ρ) = − (k + h − 2b1 + c1)Λ1 ∨ + (k + h − 2b1 + c1 + c2)Λ2 + ··· + c`Λ` ∨ s2s1s0(λ + ρ) =c2Λ1 − (k + h − 2b1 + c1 + c2)Λ2 ∨ + (k + h − 2b1 + c1 + c2 + c3)Λ3 + ··· ∨ s3s2s1s0(λ + ρ) =c2Λ1 + c3Λ2 − (k + h − 2b1 + c1 + c2 + c3)Λ3 + ··· s`−1 ··· s1s0(λ + ρ) =c2Λ1 + c3Λ2 + ··· + c`−1Λ`−2 ∨ ∨ − (k + h − 2b1 + c1 + ··· + c`−1)Λ`−1 + (k + h − b1)Λ`. ◦ + If we set λi = si−1si−2 ··· s0(λ + ρ) = wi(λ + ρ), then it must be the case that λi1 ∈ P for some i1 such that 0 ≤ i1 ≤ `. In fact, λj is a regular dominant weight as soon as Pj ∨ P` ∨ 2b1 − i=1 ci ≤ k + h and, by assumption, b1 = 2b1 − i=1 ci < k + h . Moreover, for ∨ i > 1, (λi, θ) = 2(b1−c1) < 2b1. If the element wi1 (λ+ρ) satisfies (wi1 (λ+ρ), θ) < k+h then take wλ = wi1 and we are done. Otherwise, we may continue in this fashion, producing a sequence (λ + ρ, θ) > (wi1 (λ + ρ), θ) > (wi2 wi1 (λ + ρ), θ) > ··· ∨ ∨ which must eventually lead to (wI (λ + ρ), θ) < k + h in at most k + h steps for some finite sequence I = (i1, . . . , im). 42 Corollary 2.28. Let g, k, eg, λ, I, m etc. be as in the previous theorem. Recall that ◦ ◦ ∼ V = Vk(g) has a Z2 = P/Q grading as a vector space. If k is even, then F (χλ) ∈ Vm+[λ] mod 2 ◦ ◦ if all ij > 0 for 1 ≤ j ≤ m, where [λ] ∈ Z2 is the class of λ in P/Q. ◦ ◦ ◦ ∨ 1 ∨ Proof. The lattice ν(Q ) is spanned by α1, . . . , α`−1, 2 α`, 2ν(Q ) ⊆ Q, and ◦ 1 ◦ 1 ◦ W α + Q = α + Q. 2 ` 2 ` We can express ∨ wi = si−1 ··· s0 = tsi−1···s1θ si−1 ··· s1sθ = t ∨ ∨ si−1 ··· s αi +···+α` θ for any i > 0, and so wiλ is in class [λ] + 1. The result follows immediately. 2.6.3 Congruence Group Action (r) As a another application, let g be a twisted affine Lie algebra of type XN and k a positive integer, then Chk(g) is a module for the congruence subgroup Γ1(r) (see 2.2.3) [Ka]. Proposition 2.29. Let g be as above and λ ∈ P k, then 2πi(µ+ρ) 2πi(µ+ρ) X −2rπim t 2 ◦ t − ◦ t χ | r = v e µ (S ) χ e k+h∨ χ e k+h∨ χ λ u21 k 0µ λ ν ν k µ∈Pk mod Cδ,ν∈P mod Cδ where vk is a nonzero scalar depending only on k and g. In particular, when r = 1 X ◦ − 2πi(µ+ρ) 2πi(mλ+mµ) k+h∨ χλ|u21 = e SkΛ0,,µχλ e χµ. µ∈Pk mod Cδ r r −1 Proof. These follow directly from 2.14 and the identities u21 = Su12S and u21 = −1 −1 u12 Su12 , respectively. 43 ◦ ◦ Corollary 2.30. Let g be a finite dimensional simple Lie algebra with weight lattice P ◦ and χλ be the character of the irreducible finite dimensional module of highest weight ◦ ◦ λ ∈ P +, then, for any λ, µ ∈ P +, ◦ − 2πi(ν+ρ) ◦ 2πi(ν+ρ) ◦ − 2πi(µ+ρ) X −2πimν 2 k+h∨ k+h∨ 2πi(mλ+mµ) k+h∨ e (SkΛ0,ν) χλ e χµ e = e SkΛ0,µχλ e , ◦ ν∈P where mφ is defined the same way as the modular anomaly is for affine algebras and ∗ ∨ −1/2 Q π(φ+ρ,α) SkΛ0,φ = |M /(k + h )M| ◦ 2 sin k+h∨ . α∈∆+ ◦ The evaluations the characters of gt in 2.29 can be replaced by multiples of the ◦ characters of g. It is worth that noting that the action of S on untwisted Verlinde algebras gives the strikingly similar identity ◦ − 2πi(ν+ρ) ◦ 2πi(ν+ρ) X 2 k+h∨ k+h∨ (SkΛ0,ν) χλ e χµ e = δλµ. ◦ ν∈P Indeed, the above transformation property is used to define a nondegenerate Hermitian form on the fusion algebras in other works such as [B], [Ho2]. Pursuing this idea further, one can say more about the above identity in 2.29. Let us adopt the notation D ◦ ◦ E ◦ − 2πi(ν+ρ) ◦ 2πi(ν+ρ) X −2rπimν t 2 t k+h∨ t k+h∨ χλ, χµ = e (S0ν) χλ e χµ e . r + ν∈Pk mod Cδ (r) k+ Theorem 2.31. Let g be of type XN , k be a positive integer, and λ, µ ∈ P , then ◦ there exists a β = β(g, k) ∈ h∗ and a nonzero scalar v = v(g, k) such that 2 πi|β| πi 2 − ∨ X ∨ |µ+ρ−w(λ+ρ)| χ | r = ve r(k+h ) ε(w)e r(k+h ) χ λ u21 µ (µ,w)∈P (g,k,λ) mod Cδ where the sum is over ◦ P (g, k, λ) = {(µ, w) ∈ P k+ ×W | µ+ρ−β = w(λ+α)(mod (k+h∨)M) where α ∈ rM ∗}. In particular, if µ + ρ − β ∈ P k++, D ◦ ◦ E πi (|µ+ρ|2+|λ+ρ|2−|β|2) − 2πi (µ+ρ,w(λ+ρ)) r(k+h∨) X r(k+h∨) χλ, χµ = ve ε(w)e r w ◦ ∗ where the sum is over w ∈ W such that µ + ρ − w(λ + ρ) − β ∈ rM = rP k, and D ◦ ◦ E χλ, χµ = 0 otherwise. r 44 Proof. By the transformation rules given in [KP,§3], level k theta functions satisfy X πik−1(α,rα+2β) Θ | r = v e Θ λ u21 λ+rα+β α∈M ∗, rα (mod kM) ◦ ∨ ∗ ∨ −1/2 ∗ for some scalar v ∈ C such that |v| = |((k + h )M + rM )/(k + h )M| and β ∈ h such that 2 kr|α| ≡ 2(α, β) (mod 2Z) 1 1 ∗ P for all α ∈ M ∩ M . Applying this to Aλ+ρ = ◦ (w)Θw(λ+ρ) and finding the r k w∈W k+ coefficient of Θ in the expansion of A | r for µ ∈ P gives the first result. Note µ+ρ λ+ρ u21 that for (µ, w) ∈ P (g, k, λ), w(α) = µ + ρ − β − w(λ + ρ) ∈ rM ∗ and so (w(α), w(α) + 2β) = |µ + ρ − w(λ + ρ)|2 − |β|2. The second identity in the theorem statement follows immediately from the first and 2.29. D ◦ ◦ E ◦ Notice that when r = 1, one has that β = 0 and χλ, χµ is a scalar multiple of χλ, 1 D ◦ ◦ E matching proposition 2.30. Indeed, χλ, χµ is in general a multiple of a summand of r ◦ 2πi(µ+ρ) the character χλ evaluated at r(k+h∨) . Specifically, the summand is X (w)ew(λ+ρ). ◦ w∈W : µ+ρ−w(λ+ρ)−β∈rM ∗ 2.7 Some Data Below we give the multiplication matrices of certain twisted affine Lie algebras of type (2) A2` with the indicated level. The vectors v1, v2,... in the row labeled by ”Weights” give the ordered basis Aλ1 ,Aλ2 ,... of the corresponding Verlinde algebra such that X λj = vjiΛi i≥1 where vj = (vj1, vj2,...). The matrices L1,L2,... are defined to be λk Li = Nλ λ , i j i,j,k 45 ν where Nλµ are the fusion rules. (2) Type: A4 Level: 2 Weights: (0, 0), (0, 1), (1, 0) 1 0 0 0 1 0 0 0 1 0 1 0 , 1 0 −1 , 0 −1 1 0 0 1 0 −1 1 1 1 −1 (2) Type: A4 Level: 3 Weights: (0, 0), (0, 1), (1, 0) 1 0 0 0 1 0 0 0 1 0 1 0 , 1 0 0 , 0 0 1 0 0 1 0 0 1 1 1 0 (2) Type: A4 Level: 4 Weights: (0, 0), (2, 0), (0, 1), (0, 2), (1, 0), (1, 1) 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1 1 1 −1 −1 0 0 1 0 0 0 0 1 1 0 0 −1 , , 0 0 0 1 0 0 0 1 0 1 0 −1 0 0 0 0 1 0 0 −1 0 0 1 1 0 0 0 0 0 1 0 −1 −1 −1 1 2 46 0 0 1 0 0 0 0 0 0 1 0 0 0 1 1 0 0 −1 0 1 0 1 0 −1 1 1 0 1 0 0 0 0 1 0 0 −1 , , 0 0 1 0 0 −1 1 1 0 0 −1 0 0 0 0 0 1 1 0 0 0 −1 0 1 0 −1 0 −1 1 1 0 −1 −1 0 1 1 0 0 0 0 1 0 0 0 0 0 0 1 0 −1 0 0 1 1 0 −1 −1 −1 1 2 0 0 0 0 1 1 0 −1 0 −1 1 1 , 0 0 0 −1 0 1 0 −1 −1 0 1 1 1 1 1 0 0 0 0 1 1 1 0 −1 0 1 1 1 0 −1 1 2 1 1 −1 −2 (2) Type: A4 Level: 5 Weights: (0, 0), (2, 0), (0, 1), (0, 2), (1, 0), (1, 1) 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 1 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 1 1 0 1 0 0 , , , 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 , , 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 1 0 0 0 47 (2) Type: A6 Level: 2 Weights: (0, 0, 0), (0, 1, 0), (1, 0, 0), (0, 0, 1) 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 1 1 −1 −1 0 −1 1 1 0 −1 1 0 , , , 0 0 1 0 0 −1 1 1 1 1 −1 0 0 1 0 −1 0 0 0 1 0 −1 1 0 0 1 0 −1 1 0 −1 0 (2) Type: A6 Level: 3 Weights: (0, 0, 0), (0, 1, 0), (1, 0, 0), (0, 0, 1) 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 1 1 0 0 0 0 1 1 0 0 1 0 , , , 0 0 1 0 0 0 1 1 1 1 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 (2) Type: A6 Level: 4 Weights: (0, 0, 0), (2, 0, 0), (0, 1, 0), (0, 2, 0), (1, 0, 1), (0, 0, 2), (1, 0, 0), (1, 1, 0), (0, 0, 1), (0, 1, 1) 48 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 , 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 −1 −1 0 0 0 1 1 0 1 0 0 −1 0 0 0 1 0 1 1 1 0 −1 0 −1 0 0 1 1 2 0 0 −1 −1 −1 , 0 0 0 1 0 1 0 0 0 −1 0 −1 0 0 0 0 1 1 0 0 0 −1 −1 −1 −1 0 1 2 1 1 0 0 0 0 −1 0 0 1 1 0 0 0 0 −1 −1 −1 0 1 0 2 0 0 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 −1 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 −1 0 −1 0 1 1 1 1 1 0 −1 0 −1 , 0 0 0 0 1 0 0 0 0 −1 0 0 0 0 0 0 1 1 1 0 0 −1 0 −1 −1 0 1 2 1 1 0 0 0 0 0 0 1 1 0 1 0 0 0 −1 −1 −1 0 1 1 1 49 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 1 0 −1 0 −1 0 0 1 1 1 0 0 −1 0 −1 1 1 1 1 1 1 −1 −1 −1 −1 0 1 1 1 2 0 0 −2 −1 −1 , 0 1 0 1 0 0 0 −1 0 0 0 0 0 −1 0 0 0 1 0 1 0 −1 −1 −1 −2 −1 1 2 1 2 0 0 0 −1 −1 0 0 1 1 0 0 −1 −1 −1 −1 0 1 2 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 1 2 0 0 −1 −1 −1 0 1 1 1 1 1 0 −1 0 −1 0 1 1 1 2 0 0 −2 −1 −1 1 2 1 2 1 1 −1 −2 0 −2 , 0 0 1 0 1 0 0 −1 −1 0 0 0 0 0 −1 0 0 1 1 1 0 −1 −1 −2 −2 −1 1 3 1 2 0 −1 0 −1 0 −1 1 1 0 1 0 −1 −1 −1 −2 0 1 2 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 −1 0 0 0 0 1 0 0 0 0 −1 0 1 0 1 0 0 0 −1 0 0 0 0 1 0 1 0 0 −1 −1 0 , 1 1 0 0 0 0 −1 0 0 0 0 0 0 0 0 −1 0 0 0 1 0 0 0 −1 −1 0 0 1 0 1 0 0 0 0 −1 0 0 0 1 0 0 −1 −1 0 0 0 1 1 0 0 50 0 0 0 0 0 0 1 0 0 0 0 −1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 −1 0 0 0 1 0 1 0 0 0 0 −1 0 0 1 1 1 , 0 0 0 0 0 −1 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 −1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 −1 0 0 0 0 0 0 0 1 0 0 0 −1 −1 −1 −1 0 1 2 1 1 0 −1 0 −1 −1 0 1 2 1 1 0 −1 −1 −1 −2 −1 1 2 1 2 0 −1 −1 −2 −2 −1 1 3 1 2 , 0 0 0 −1 −1 0 0 1 0 1 0 1 1 1 1 0 0 −1 0 0 1 2 2 2 3 1 −1 −3 −1 −2 0 1 1 1 1 0 0 −1 0 −1 0 1 1 2 2 1 0 −2 −1 −2 0 0 0 0 0 0 0 0 1 0 0 0 0 0 −1 0 0 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 −1 −1 0 0 1 1 0 0 −1 0 −1 0 −1 1 1 0 1 , 0 0 0 0 −1 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 1 1 1 1 0 0 −1 0 −1 1 1 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 −1 0 −1 51 0 0 0 0 0 0 0 0 0 1 0 0 0 −1 −1 −1 0 1 0 2 0 0 0 −1 −1 −1 0 1 1 1 0 −1 −1 −1 −1 0 1 2 0 1 0 −1 −1 −1 −2 0 1 2 1 1 0 −1 −1 0 0 0 1 1 0 0 0 0 0 1 1 1 0 0 0 −1 0 1 1 2 2 1 0 −2 −1 −2 0 0 1 0 1 0 0 −1 0 −1 1 2 1 1 1 0 −1 −2 −1 0 52 Chapter 3 Richards-Sahi Conjectures for Schur Polynomials 3.1 Introduction Jack polynomials are a family of symmetric polynomials, depending on a parameter usually denoted by α, that generalize Schur and zonal polynomials. They are of great importance in representation theory, combinatorics, and statistics, to name a few areas. For example, they appear in computing integrals of polynomials over the orthogonal and unitary groups. Jack polynomials are further generalized by the important two- parameter family of symmetric polynomials called Macdonald polynomials. Recently, D. Richards and S. Sahi introduced a conjecture asserting the positivity of certain expressions involving Jack polynomials [RS]. The version of the conjecture, r which we call the αµ conjecture, that we investigate in this dissertation can be found in section 3.3. There has been little progress in resolving this conjecture, even in the case of Schur polynomials, i.e., the case when α = 1. In this chapter, we develop a stronger version of the conjecture for Schur polynomials and prove it for several important cases. In the process we discover new identities involving Kostka coefficients. One of these identities is a seemingly new recursion for the dimensions of irreducible representations of the symmetric group. As a bonus, we give a proof of the analogue of this identity for Jack polynomials. The Richards-Sahi conjecture has its roots in statistics. More precisely, in the study of parameter matrices. Nonetheless, the problem can be expressed in purely algebraic and combinatorial terms (the reader is not expected to be familiar with statistics). Let us briefly describe the background. Statisticians often try to determine how “good” certain estimators of a statistic are with respect to some criteria. Suppose that we have an n × n positive definite random matrix X whose probability distribution is given by 53 an n × n (symmetric) positive definite parameter matrix Σ (e.g. a covariance matrix). Suppose further that the distribution is invariant with respect to the conjugation action by the orthogonal group On(R). Many significant multivariate distributions have this property, such as the Wishart distribution and the multivariate F-distribution. To estimate Σ, it is natural to use estimators Σb that are also invariant under the same action of the orthogonal group. Estimators of this type can be shown to be given by Σ(b X) = GΨ(L)GT T where X = GLG , G ∈ On(R), L = diag(l1, . . . , ln) such that l1 > ··· > ln > 0 a.e. and Ψ(L) = diag(ψ1(L), . . . , ψn(L)). Such an estimator is called order-preserving if ψ1(L) > ··· > ψn(L) > 0 a.e. In a 2005 paper (see [Sh]), Yo Sheena conjectured that any non-order-preserving estimator Σb can be modified (for example, by simply 0 0 rearranging the ψi in decreasing order) into an new estimator Σb so that Σb is a better estimator (with respect to a certain entropy loss function called Stein’s loss function). To be more precise, in statistics there is a notion of admissibility of estimators. Given some criteria that allows one to compare estimators, an estimator Σ is inadmissible if one can find another estimator Σ0 that is at least as good as Σ is all instances and strictly better in at least one instance. Sheena’s conjecture states that non-order-preserving estimators are inadmissible. In the same paper, Sheena shows that this conjecture can be framed in terms of the non-negativity of certain integrals of linear combinations of zonal polynomials. Zonal polynomials are special cases of Jack polynomials where the Jack parameter α = 2. Allowing the parameter α to take the values 1 and 1/2, respectively, we may restate the same (integral form of the) conjecture for the unitary and symplectic group. Over the three compact groups, integration of polynomials (with respect to their corresponding Haar measures) can be accomplished via combinatorial methods involving Jack poly- nomials (see [CS]). In this work, we consider the case when α = 1, i.e., the Schur polynomial case, which corresponds to Hermitian matrices (as opposed to the symmetric matrices considered by Sheena). Many of the results and methods generalize directly to the case α = 2, i.e., 54 the zonal polynomial case. 3.2 Preliminaries 3.2.1 Partitions We will first do a quick review of basic combinatorial concepts that we will be using. A partition λ = (λ1, λ2, ..., λl) of n ∈ N, is a tuple of non-negative integers such that P i λi = n and λ1 ≥ λ2 ≥ · · · ≥ λl. The entries of a partition λ are referred to as parts. For a partition λ = (λ1, ..., λl) of n, the length of λ, which we denote by `(λ), is the total number of strictly positive entries in λ. The weight of λ is the sum of its entries and is denoted |λ|. We write λ ` n if |λ| = n. Given a partition λ, we define the conjugate λ0 to be the partition whose ith row equals the number of boxes in the ith column of the Young diagram of λ. Sometimes we use the so-called frequency notation for partitions. Let λ be the partition λ = (α, . . . , α, β, . . . , β,...) | {z } | {z } m1 many m2 many then in frequency notation we write λ as λ = (αm1 , βm2 ,...). In this work, we seldom use frequency notation, but when it is used we will state it explicitly. Let P denote the set of all partitions, where were identify λ and µ if `(λ) = `(µ) and λj = µj for all j = 1, . . . , `(λ). For any positive integer d, we denote by Pd the subset of P of all partitions of length at most d. Therefore, we naturally identify Pd d with the corresponding subset of N . There are two important partial orders defined on the set of partitions that we will be discussing. The first is the inclusion order, defined by: λ ⊆ µ if and only if λi ≤ µi 55 for all i. We write λ ⊂: µ when λ ⊆ µ and |λ| + 1 = |µ|. The second is the dominance order, defined by: λ ≤ µ if and only if |λ| = |µ| and m m X X λj ≤ µj j=1 j=1 for all m ≥ 1, where we assume that λj = 0, (resp. µj = 0) when j > `(λ) (resp. j > `(µ)). A Young diagram of a partition λ is a diagram: a left aligned collection of boxes whose ith row has exactly λi many boxes. For example, for λ = (4, 3, 1) has the Young diagram In this way, we think of a partition λ also as a set of boxes λ := {(i, j) : 1 ≤ i ≤ `(λ), 1 ≤ j ≤ λi}. For example, the above figure is the set of boxes {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1)}. This notion is especially convenient when dealing with the inclusion order. In particular, for any λ, µ ∈ P, λ ⊆ µ is the set of boxes of λ is a subset of the set of boxes of µ. It is also convenient when dealing with the dominance order. In particular, λ <: µ if we can raise a single box in λ to a higher row and produce µ. Often we use language that alludes to the Young diagram when referring to par- titions. For example, ‘removing a part of size 2’ from (4, 2, 2, 1) yields the partition (4, 2, 1, 0). We will need the following two notions. Definition 3.1. Let λ, µ ∈ P such that λ ⊆ µ, then the skew partition µ \ λ is set of boxes that lie in µ but do not lie in λ, i.e., µ \ λ where µ and λ are treated as sets of boxes as above. 56 Definition 3.2. Let λ, µ ∈ P such that λ ⊆ µ. A skew partition µ \ λ is called a horizontal strip if for every pair of distinct boxes (a, b), (c, d) ∈ µ \ λ, b 6= d. The Young diagram of λ0 is simply the transpose of the Young diagram of λ. For example, for λ = (4, 3, 1) as above, the conjugate is λ0 = (3, 2, 2, 1) and its Young diagram is Let λ be a partition, then a standard Young tableaux T of shape λ is a filling of the boxes in the Young diagram of λ with the integers {1,..., |λ|} so that the integers are strictly increasing along rows (read from left to right) and also strictly increasing along columns (read from top to bottom). Let λ and µ be partitions, then a (column-strict) semi-standard Young tableaux T of shape λ and weight µ, is a filling of the boxes in the Young diagram of λ with µ1 many 1’s, µ2 many 2’s, and so on, so that the integers are non-decreasing along rows (read from left to right) and also strictly increasing along columns (read from top to bottom). 3.2.2 Symmetric Polynomials Let d be a positive integer. Let ΛZ,d := Z[x1, . . . , xd] be the ring of polynomials in d variables with coefficients in Z. For any commutative ring R containing Z, i.e., Z ,→ R, define ΛR,d := R ⊗Z ΛZ,d. When the ring is understood from context or specified beforehand, we will write Λd for simplicity. The symmetric group Sd acts on Λd by permuting the variables, i.e., for any σ ∈ Sd, k ∈ N, and i1, . . . , ik ∈ {1, . . . , n}, σ · xi1 ··· xik = xσ(i1) ··· xσ(ik). Let us adopt the notation α α1 α2 αd x = x1 x2 ··· xd (3.1) 57 d where α ∈ Z . Note that Sd acts on Zd by σ · (α1, . . . , αd) = (ασ−1(1), . . . , ασ−1(d)), so that σxα = xσ·α. Assume that we are working over Q, then Λd has several important bases indexed by partitions of length at most d. We will quickly introduce just a few. The most natural is the monomial basis X µ mλ = x µ∈Sd·λ where λ ∈ Pd and Sd · λ is the orbit of λ. A second important basis is the power sum basis, whose elements are pλ = pλ1 pλ2 ··· pλd where p0 = 1 and, for r > 0, d X r pr := xi . i=1 A third, and very important, basis is comprised of the Schur polynomials, defined by λj +n−j det(xi )1≤i,j≤n sλ = n−j . det(xi )1≤i,j≤n Note that the denominator is the Vandermonde determinant and that sλ is a polynomial over Z. In fact, both the monomial basis and the Schur basis are bases of ΛZ,d (but not the power sum basis). There is a purely combinatorial formula for the Schur polynomials that proves that the entries are non-negative linear combinations in the monomial basis. It is well-known that X T sλ = x T where the sum is over all semi-standard Young tableaux (see previous subsection) of T t1 t2 td shape λ and x = x1 x2 ··· xd , such that T has tj many j’s. 58 r 3.3 αµ Conjecture In this section, we will present the Richards-Sahi conjectures and set notation for the remainder of this work. The conjectures are about the positivity of certain linear combinations of Schur polynomials with polynomial coefficients over a prescribed region d of R . Let d be a positive integer and Pd be the set of all partitions of length less than d. For any µ ∈ Pd, define G (µ) = {1 ≤ i ≤ d | µi > µi+1} H (µ) = {ν ∈ Pd | µ \ ν is a horizontal strip} . For ν ∈ H (µ) and x a variable. we define the polynomial Y gµ (ν, x) = (x + νi − i) . i∈G(µ) Next define P (µ) = {(i, j) ∈/ µ | µ ∪ {(i, j)} is a partition} R (µ) = {i − j :(i, j) ∈ P (µ)} . We will refer to elements of these sets as Pieri boxes and Pieri roots of µ, respectively. For r ∈ R (µ) and ν ∈ H (µ) we define r gµ(ν, r) kµ (ν) = Q . s∈R(µ)\{r} (r − s) d For a ∈ R define |µ|−|ν| s˜ν (a) = a1 sν(a2, . . . , ad), where sν is the Schur function in d − 1 variables. For r ∈ R (µ) we define r X r ψ (a) = k (ν)s ˜ν (a) µ ν∈H(µ) µ r 1 r ϕµ(a) = r ψµ(a) (d − r)kµ (µ) 59 − The most negative Pieri root is r = −µ1 corresponding to the Pieri box (1, µ1 + 1) and we define R− (µ) = R (µ) \ r− . Finally, define r r− r αµ(a) = ϕµ (a) − ϕµ(a), The Richards-Sahi conjecture for Schur polynomials is the following. d Conjecture 3.3. If µ is a partition and a ∈ R satisfies a1 ≥ · · · ≥ ad ≥ 0 then for all r ∈ R− (µ) we have r r αµ(a) ≥ 0 and βµ(a) ≥ 0. If a1 = ... = ad = 0, then the conjecture is true. Assume that ad > 0. Note that |µ| by setting yi = ai+1, for i = 0, . . . , d − 1, then dividing by y0 and defining xi = yi/y0, we may replace the polynomialss ˜ν(a) by the Schur polynomials sν(x1, . . . , xd−1). The r r conjecture then asserts the non-negativity for the corresponding αµ, βµ on the domain x1 ≥ · · · ≥ xd−1 ≥ 0. 3.4 Special Cases of Richards-Sahi Conjecture In this section, we prove the conjecture for some important families of partitions. The techniques here do not seem to be directly adaptable for general partitions. These are basically ‘brute-force’ arguments that take advantage of ad hoc manipulations that are possible in these special cases. Afterward, we give a refined version of the conjecture. 3.4.1 Case µ = ni r In order to show positivity for αµ we need only show that the i X r aµλj mλj (1) ≤ 0 j=0 60 for λ ≤ µ, i = 0, . . . , k, k = k(λ), C(λ) = {λ0, . . . , λk(λ)}, with equality when i = k. r r Indeed, it suffices to show that aµλj ≥ 0 implies that aµλj+1 ≥ 0 and k X r aµλj mλj (1) = 0. (3.2) j=0 i In the special case µ = n , we have that Gµ = {i},Rµ = {−n, i}, and every ν ⊆ µ such that µ \ ν is a horizontal strip is of the form ν = ni−1, n − j for j = 0, . . . , n. The coefficients of the relevant Schur polynomials are g(ν, r−) g(ν, i) kr = − µν (d + n)g(µ, r−) (d − i)g(µ, i) −n + n − i − j i + n − i − j = − (d + n)(−n + n − i) (d − i)(i + n − i) (dj − in)(n + i) = . in(d − i)(d + n) Let µj = ni−1, n − j. Since r r aµν = Kµ|µ|−|ν|,νkµ,µ|µ|−|ν| , r r r for all ν appearing in the monomial expansion of αµ and kµµj ≥ 0 =⇒ kµµj+1 ≥ 0, we need only to show equation (2). Note that if a particular monomial mν appears with nonzero coefficient, then d > `(ν). in particular, for λ ≤ µ b1(dj1 − in) + b2(dj2 − in) + ··· + bk(djk − in) = din − `in > 0 m where jm = |λ| − |λ |, bm is the associated multiplicity in λ of the removed row, and ` is the length of λ. After clearing denominators, we get that for any λ ≤ µ we must show that k X j mλj (1)Kµ|µ|−|λj |,λj d(|λ| − |λ |) − in = 0. j=0 In this case, Kµ|µ|−|λj |,λj = Kµ,λ for all j = 0, . . . , k and so this reduces to proving k X j mλj (1) d(|λ| − |λ |) − in = 0 j=0 61 or the more combinatorially friendly equation k k X j X d(|λ| − |λ |)mλj (1) = in mλj (1). (3.3) j=0 j=0 The mλj (1) are multinomials and a direct check shows that equation 3.3 is true. Indeed, k d − 1 k d − 1 X j X d(|λ| − |λ |) = in (3.4) j=0 bk, . . . , bj − 1, . . . b0 j=0 bk, . . . , bj − 1, . . . b0 where b0, . . . , bk − 1 are the multiplicities of the parts of λ and b0 + ··· + bk = d. By factoring and clearing denominators we get k X j d(|λ| − |λ |)bj = ind (3.5) j=0 which is true since, by definition, λ = (|λ| − |λk|)bk ,..., (|λ| − |λ1|)b1 , (|λ| − |λ0|)b0 i Pk j and in = |n | = j=0(|λ| − |λ |)bj. 3.4.2 Case µ = m11n We now prove the conjecture for some hooks of the form µ = m11n. Let’s first study these quantities a little more closely. µ = m11n−10d−n. 1 n−2 m 1 n−1 m We have ν ∈ H(µ) = {k 1 }k=1 ∪ {k 1 }k=1 for n > 1, and gµ(ν, −m) = (ν1 − 1 − m)(νn − n − m) gµ(ν, 0) = (ν1 − 1)(νn − n) gµ(ν, n) = (ν1 − 1 + n)νn. More precisely, 1 n−2 gµ(k 1 , −2) = (k − m − 1)(−n − m) = (m − k + 1)(n + m) 1 n−1 gµ(k 1 , −2) = (k − m − 1)(1 − n − m) = (m − k + 1)(n + m − 1) 62 1 n−2 gµ(k 1 , 0) = (ν1 − 1)(νn − n) = −(k − 1)n 1 n−1 gµ(k 1 , 0) = (ν1 − 1)(νn − n) = −(k − 1)(n − 1) 1 n−2 gµ(k 1 , n) = (ν1 − 1 + n)νn = 0 1 n−1 gµ(k 1 , n) = (ν1 − 1 + n)νn = n + k − 1 r Now, we compute kµ(ν): (m − k + 1)(n + m) m − k + 1 k−2(k11n−2) = = µ −m(−m − n) m (m − k + 1)(n + m − 1) (m − k + 1)(n + m − 1) k−2(k11n−1) = = µ −m(−m − n) m(m + n) −(k − 1)n k − 1 k0(k11n−2) = = µ −mn m −(k − 1)(n − 1) (k − 1)(n − 1) k0(k11n−1) = = µ −mn mn n 1 n−2 kµ(k 1 ) = 0 n + k − 1 kn(k11n−1) = µ (n + m)n Putting these expressions together we arrive at the following. m 0 m(m + n) X (m − k + 1)(n + m − 1) α = s˜ 1 n−1 µ (d + m)(n + m − 1) m(m + n) k 1 k=1 m ! X m − k + 1 + s˜ 1 n−2 m k 1 k=1 m m ! mn X (k − 1)(n − 1) X k − 1 − s˜ 1 n−1 + s˜ 1 n−2 d(m − 1)(n − 1) mn k 1 m k 1 k=1 k=1 m n m(m + n) X (m − k + 1)(n + m − 1) α = s˜ 1 n−1 µ (d + m)(n + m − 1) m(m + n) k 1 k=1 m ! X m − k + 1 + s˜ 1 n−2 m k 1 k=1 m (n + m)n X n + k − 1 − s˜ 1 n−1 (d − n)(m + n − 1) (n + m)n k 1 k=1 63 r We rescale αm111 as follows: m 0 X (m − k + 1)(n + m − 1) αf = m(m + n)d(m − 1)(n − 1) s˜ 1 n−1 µ m(m + n) k 1 k=1 m ! X m − k + 1 + s˜ 1 n−2 m k 1 k=1 m m ! X (k − 1)(n − 1) X k − 1 −mn(d + m)(n + m − 1) s˜ 1 n−1 + s˜ 1 n−2 mn k 1 m k 1 k=1 k=1 m X = d(m − 1)(n − 1) (m − k + 1)(n + m − 1)˜sk11n−1 k=1 m X +(m + n)d(m − 1)(n − 1) (m − k + 1)˜sk11n−2 k=1 m X −(d + m)(n + m − 1) (k − 1)(n − 1)˜sk11n−1 k=1 m X −n(d + m)(n + m − 1) (k − 1)˜sk11n−2 k=1 m n X (m − k + 1)(n + m − 1) α = m(m + n)(d − n) s˜ 1 n−1 fµ m(m + n) k 1 k=1 m ! X m − k + 1 + s˜ 1 n−2 m k 1 k=1 m X n + k − 1 −(n + m)n(d + m) s˜ 1 n−1 (n + m)n k 1 k=1 m X = (d − n) (m − k + 1)(n + m − 1)˜sk11n−1 k=1 m X +(m + n)(d − n) (m − k + 1)˜sk11n−2 k=1 m X −(d + m) (n + k − 1)˜sk11n−1 k=1 Simplifying, we get m 0 X αfµ = m(n − 1)(m + n − 1) (−dk + dm − k + 1)˜sk11n−1 k=1 m X +(m + n)d(m − 1)(n − 1) (m − k + 1)˜sk11n−2 k=1 64 m X −n(d + m)(n + m − 1) (k − 1)˜sk11n−2 k=1 m n X αfµ = (m + n) ((d − n)(m − k) − n − k + 1)˜sk11n−1 k=1 m X +(m + n)(d − n) (m − k + 1)˜sk11n−2 . k=1 Recall that X sλ = Kλνmν, νλ so we can set a1 = 1 and rewrite the above quantities in terms of monomials m 0 X X αfµ = m(n − 1)(m + n − 1) (−dk + dm − k + 1) Kk11n−1νmν k=1 νk11n−1 m X X +(m + n)d(m − 1)(n − 1) (m − k + 1) Kk11n−2νmν k=1 νk11n−2 m X X −n(d + m)(n + m − 1) (k − 1) Kk11n−2νmν k=1 νk11n−2 n m αfµ X X = ((d − n)(m − k) − n − k + 1) K 1 n−1 m m + n k 1 ν ν k=1 νk11n−1 m X X +(d − n) (m − k + 1) Kk11n−2νmν. k=1 νk11n−2 Let’s revisit the m = 2 case. We have n 2 αfµ X X = ((d − n)(2 − k) − n − k + 1) K 1 n−1 m 2 + n k 1 ν ν k=1 νk11n−1 2 X X +(d − n) (2 − k + 1) Kk11n−2νmν k=1 νk11n−2 X = (d − 2n)m1n − (n + 1) K211n−1νmν ν211n−1 X +2(d − n)m1n−1 + (d − n) K211n−2νmν ν211n−2 = (d − 2n)m1n − (n + 1)m211n−1 − n(n + 1)m1n+1 +2(d − n)m1n−1 + (d − n)m211n−2 + (n − 1)(d − n)m1n 65 = n(d − n − 1)m1n − (n + 1)m211n−1 − n(n + 1)m1n+1 +2(d − n)m1n−1 + (d − n)m211n−2 Replacing the largest negative monomials by the corresponding inequalities that ‘make sense’, i.e., that are in the up-set of the existing smaller partitions, we get n αfµ ≥ n(d − n − 1)m n − t (n + 1)m 1 n−2 − t n(n + 1)m n 2 + n 1 1 2 1 2 1 d − n − 1 −n(n + 1) m n + 2(d − n)m n−1 + (d − n)m 1 n−2 n + 1 1 1 2 1 (n + 1)(d − n) = −t m 1 n−2 − t (n + 1)(d − n)m n−1 1 n − 1 2 1 2 1 +2(d − n)m1n−1 + (d − n)m211n−2 n−1 2 Set t1 = n+1 , t2 = n+1 giving n αfµ ≥ −(d − n)m 1 n−2 − 2(d − n)m n−1 − t (n + 1)(n − 1)m n−1 2 + n 2 1 1 2 1 +2(d − n)m1n−1 + (d − n)m211n−2 = 0. Notice that we chose t1 so that it canceled the m211n−2 term, i.e., the largest term left. Continuing in this way, one can prove non-negativity for all small values of m and r = 0, n. Later, in section 3.6, we will show a method of organizing the partitions so that non-negativity can be verified for families without resorting to ad-hoc methods as above. 3.5 Another Partial Order on Partitions Before we discuss the refinement of the Richards-Sahi conjecture, we will introduce a partial order ≺ that is of independent interest. Afterward, we define a relation ≺H , in fact a sub-relation of , which features in the refined conjecture. Let us first define the partial order and then show, immediately after, that it is well-defined. 66 Definition 3.4. Let P be the set of partitions. Define a partial order on P by λ µ if and only if there exists a partition ν such that λ ≤ ν and ν ⊆ µ. Proposition 3.5. The relation is a partial order on the set of partitions. Proof. Clearly, is reflexive. Moreover, if λ µ and µ λ then we have ν1, ν2 such that λ ≤ ν1 ⊆ µ µ ≤ ν2 ⊆ λ. If |λ|= 6 |µ| then this is not possible. Therefore, ν1 = µ and ν2 = λ which implies that λ ≤ µ and µ ≤ λ. The antisymmetry of ≤ implies the antisymmetry of . It remains to show transitivity. Suppose that λ µ and µ ν then we have two intermediate partitions α, β such that λ ≤ α ⊆ µ µ ≤ β ⊆ ν. If we can find γ such that α ≤ γ and γ ⊆ β (3.6) then the transitivity of ≤ and ⊆ would imply λ ν as desired. We proceed by induction on n = |ν| − |λ|, where the n = 0 case is obviously true (since then the inclusions become equalities in the expressions above). Assume for the moment that n > 1 and that it has been proved for all values smaller than n (we will address the n = 1 case shortly). If µ = β, then λ ≤ α ⊆ µ ⊆ ν and we are done. Suppose then that µ < β. We may assume that β = ν, otherwise, |β| < |ν| and by the inductive hypothesis there is a γ such that 3.6 holds, giving the result. To summarize, we have reduced it to the case that λ ⊂ µ < ν. 67 Since we have assumed that it is true for n = 1, we can find α1 such that λ ⊆ α1 ⊂: µ < ν and so α1 ν with intermediate γ1, i.e., such that α1 ≤ γ1 ⊂ ν. If |γ1| − |λ| < |ν| − |λ|, then by the induction hypothesis, there is an intermediate λ1 such that λ ≤ λ1 ⊆ γ1 which in turn implies that λ ≤ λ1 ⊂ ν. It remains to show the case n = 1, i.e., that λ ⊂: µ. Let µ0 = µ <: µ1 <: ··· <: µm = ν. We use a separate inductive argument on m to show that λ ν. The case m = 0 was done above. Suppose that m > 0 and that we have proved it for all values less than m. If we can find an intermediate partition γ such that λ ≤ γ ⊆ µ1, then by the inductive hypothesis, γ ν. This in turn implies that there is a γ1 such that λ ≤ γ ≤ γ1 ⊆ ν as desired. Therefore, we only need to prove it for m = 1. Assume that µ <: ν. There are two cases: 1. For some i, λi + 1 = µi and µi = νi + 1. 2. For some i, λi + 1 = µi and µi = νi. For (1), µj ≤ νj for j 6= i. Therefore λi = νi and λj ≤ νj for j 6= i. So λ ⊆ ν and we are done. For (2), let i0, j0, k0 be indices such that µi0 = λi0 + 1 68 νj0 = µj0 + 1 νk0 = µk0 − 1. Note that i0 6= k0, j0 < k0, µk0+1 < µk0 , and µj0 < µj0−1. We now have two subcases: i0 = j0 − 1 and i0 6= j0 − 1. When i0 6= j0 − 1, define the partition γ by γj = λj, for j∈ / {j0, k0} γj0 = λj0 + 1, γk0 = λk0 − 1. This is well defined for two reasons. First, i0 6= k0 means that λk0 = µk0 and so γk0+1 = λk0+1 ≤ µk0+1 < µk0 = λk0 = γk0 + 1. More to the point, γk0+1 ≤ γk0 . Second, λj0−1 = µj0−1 = νj0−1 > µj0 implies that γj0−1 ≥ λj0 + 1 = γj0 (unless j0 = 1 in which case it is already well defined). Now, ν ⊃ γ since γj = λj ≤ µj = νj for j∈ / {j0, k0} γj0 = λj0 + 1 ≤ µj0 + 1 = νk0 γk0 = λk0 − 1 = µk0 − 1 = νk0 . In the case that i0 = j0 − 1, define the partition γ by γj = λj, for j∈ / {i0, k0} γi0 = λi0 + 1, γk0 = λk0 − 1. To show that this is well-defined, we repeat the same argument for γk0 and for γi0 we note that λi0 + 1 = µi0 ≤ λi0−1 = µi0−1 if i0 > 1. We again have that ν ⊃ γ since γj = λj = µj ≤ νj for j∈ / {i0, k0} γi0 = λi0 + 1 = µi0 = νk0 γk0 = λk0 − 1 = µk0 − 1 = νk0 . It follows that λ ν, completing the proof. 69 The partial order is a natural fusion of the dominance and inclusion order. It has a sub-relation H , which, despite the unfortunate notation, is not a partial order (it fails transitivity), but is featured in the expansion X |λ|−|µ| sλ(x1, . . . , xn, y) = aλµy mµ(x1, . . . , xn), µH λ were aλµ are positive integers. Nonetheless, if λ H µ, then λ µ. First, let us define the relation and then later prove the above expansion. Definition 3.6. Let P be the set of partitions. Define a relation H on P by λ H µ if and only if there exists a partition ν such that λ ≤ ν, ν ⊆ µ, and µ \ ν is a horizontal strip. Proposition 3.7. Let λ ∈ P and n + 1 ≥ `(λ), then X |λ|−|µ| sλ(x1, . . . , xn, y) = aλµy mµ(x1, . . . , xn), µH λ where aλµ are positive integers. T Proof. Recall that sλ is the sum of monomials of the form x , where T ranges over all semi-standard tableaux of shape λ (where xn+1 = y), see section 3.2. Now, for any semi-standard tableaux T the n + 1 entries form a horizontal strip of λ. In the other direction, if we fixed a horizontal strip λ \ µ and label those boxes with n + 1, then we can fill the boxes in µ to form any semi-standard tableaux T 0 of shape µ, then adding the boxes containing n + 1 to T 0 forms a semi-standard tableaux T of shape λ. Therefore, using the semi-standard tableaux expansion of sλ, X |λ|−|µ| sλ(x1, . . . , xn, y) = y sµ(x1, . . . , xn) µ⊆λ,λ\µis a hor. strip X X |λ|−|µ| = Kµνy mν(x1, . . . , xn) µ⊆λ,λ\µis a hor. strip ν≤µ X |λ|−|µ| = aλµy mµ(x1, . . . , xn), µH λ where Kµν ∈ N are Kostka numbers, which are positive whenever ν ≤ µ, and aλµ are non-empty sums of Kostka numbers. 70 3.6 Certain Chains of Partitions ≤n Given n ∈ N, there is a way of decomposing the set P of partitions of weight at most n into disjoint chains with respect to inclusion. Moreover, each of the chains will have one (and only one, of course) partition of weight n. Definition 3.8. Define the following map C : λ 7→ {λ = λ0 ⊃ λ1 ⊃ · · · ⊃ λk(λ)} where k(λ) is the number of nonzero elements in the set {λ1, . . . , λ`(λ)} and, for i = 1, . . . , k(λ), λi is attained from λ by removing its ith smallest part. We will often use the notation λ∗ = λk(λ). Example 3.9. If λ = (3, 2, 1), then C(λ) = {(3, 2, 1), (3, 2, 0), (3, 1, 0), (2, 1, 0)}. Lemma 3.10. Let λ be a partition and n ≥ |λ|. If λ0 is the partition obtained from λ by adding a single row of length n − |λ|, then λ ∈ C(λ0). m1 m2 mr Proof. Let λ = (α1 , α2 , . . . , αr ) written in frequency notation. Let k be the index such that αk ≥ n−|λ| and αk+1 < n−|λ| where we take α0 := ∞. Then the composition 0 m1 mk mk+1 mr λ := (α1 , . . . , αk , n − |λ|, αk+1 , . . . , αr ) is a partition. Furthermore, C(λ0) is the sequence: 0 m1 mk+1 mr−1 mr λ = (α1 , . . . , n − |λ|, αk+1 , . . . , αr−1 , αr ) 1 m1 mk+1 mr−1 mr−1 λ = (α1 , . . . , n − |λ|, αk+1 , . . . , αr−1 , αr ) 2 m1 mk+1 mr−1−1 mr λ = (α1 , . . . , n − |λ|, αk+1 , . . . , αr−1 , αr ) . . 71 r−k m1 mk+1−1 mr−1 mr λ = (α1 , . . . , n − |λ|, αk+1 , . . . , αr−1 , αr ) λr−k+1 = λ . . This completes the proof. ≤n Proposition 3.11. Let n ∈ N, P be the set of partitions of weight at most n, and n ≤n P ⊆ P be the subset of partitions of weight n, then the set {C(λ)}λ∈Pn is a partition ≤n of P into |Pn| many disjoint chains, with respect to the inclusion order. ≤n Proof. Let µ ∈ P , then by the previous lemma, there is a unique λ ∈ Pn such that µ ∈ C(λ). This proves the disjointness of the chains, the fact that the union of the ≤n chains in Q = {C(λ)}λ∈Pn equals P , and that Q contains |Pn| many chains. The sequence C(λ) can be described more easily in terms of frequency notation. Let m1 m2 mr λ = (α1 , α2 , . . . , αr ) where α1 > α2 > . . . > αr. The sequence C(λ) is obtained by subtracting 1 from the multiplicities in the following way 0 m1 m2 mr−1 mr λ = (α1 , α2 , . . . , αr−1 , αr ) 1 m1 m2 mr−1 mr−1 λ = (α1 , α2 , . . . , αr−1 , αr ) 2 m1 m2 mr−1−1 mr λ = (α1 , α2 , . . . , αr−1 , αr ) . . r−1 m1 m2−1 mr−1 mr λ = (α1 , α2 , . . . , αr−1 , αr ) r m1−1 m2 mr−1 mr λ = (α1 , α2 , . . . , αr−1 , αr ). Proposition 3.12. Let λ be a partition and C(λ) = {λ0 ⊃ λ1 ⊃ · · · ⊃ λk}, then for each j = 0, . . . , k − 1, the skew diagram λj \ λj+1 is a horizontal strip. 72 Proof. The prove that any skew diagram λ \ µ is a horizontal strip, it suffices to show: if µj < λj, then λj ≤ µj−1, for all j. This is so because if λ \ µ contains two boxes (a, j), (b, j), a < b, then µa < j ≤ λa and µb < j ≤ λb, which implies that µb−1 < λb. Now, let j ∈ {0, . . . , k − 1} and λj (λj+1, resp.) be attained from λ by removing the i1th row (i2th row, resp.) of size n1 (n2, resp.). Note that n1 < n2 and i2 < i1. Therefore λj and λj+1 are equal in rows i < i and i ≥ i . Moreover, λj+1 < λj and 2 1 i2 i2 λj+1 = λj+1 = λj for i ≤ i < i −1. It follows that λj \λj+1 is a horizontal strip. i2 i i+1 2 1 The chains have the following curious ‘universal’ property. If we ever have the situation described by the diagram below, where µ0 \ µ is a horizontal strip and λ, λ0 are as above, then this diagram “commutes”, i.e., λ0 ≤ µ0. λ0 − − − µ0 ∪ ∪ (hor. strip) λ ≤ µ Proposition 3.13. Let λ, λ0 be partitions such that λ ∈ C(λ0). For any partitions µ, µ0 such that λ ≤ µ, µ0 \ µ is a horizontal strip, and |µ0| = |λ0|, then λ0 ≤ µ0. 0 Proof. First, let’s describe λ more precisely. If k is the index of λ such that λk ≥ 0 0 0 |λ | − |λ| and λk+1 < |λ | − |λ|, then the parts of λ are λj if j ≤ k 0 0 λj = |λ | − |λ| if j = k + 1 . λj−1 if j > k + 1 Second, note that λ ≤ µ and µ ⊆ µ0 implies i i i X X X 0 λj ≤ µj ≤ µj j=1 j=1 j=1 for all i ≥ 1. 0 0 Now, suppose that λ 6≤ µ . This means that there is some i0 so that i0 X 0 0 µj − λj < 0. j=1 73 By the inequalities in the second paragraph, it must be the case that i0 ≥ k + 1. The condition that µ0 \ µ is a horizontal strip implies that i2 X 0 0 µj − µj ≤ µi1 − µi2 . j=i1 In particular, X 0 0 µj − µj ≤ µi . j≥i Using these facts we see that, i0 i0−1 i0 X 0 0 X 0 X 0 λj = |λ | − |λ| + λj > µj j=1 j=1 j=1 which implies i0−1 0 X 0 X 0 0 |λ | − |λ| − µj − λj = µj − λj > µi0 , j=1 j≥i0 a contradiction. The proposition follows. 3.7 Refinement of Conjecture In this section, we give a refined version of the conjecture. First, we need to have an understanding of the decomposition of these identities in terms of the monomial basis (see sec. 3.2). Let Pµ be the set of partitions λ satisfying λ H µ. It follows that r X r αµ = aµλmλ λ∈Pµ r for some coefficients aµλ ∈ Q. The set Mµ ⊆ Pµ of maximal partitions is Mµ = {λ ∈ Pµ|λ ∈ C(ν) iff λ = ν}. Proposition 3.13 gives us a characterization of the set Mµ. The proof of the following lemma is straightforward using the arguments in the proofs in 3.6. Lemma 3.14. The set {C(λ)}λ∈Mµ is a partition of the set Pµ. 74 Proposition 3.15. Let µ be a partition, then Mµ = {λ : λ ≤ µ}. Proof. If λ ≤ ν and µ \ ν is a horizontal strip, then, using the previous two lemmas, we can find a λ0 such that λ ∈ C(λ0) and λ0 ≤ µ. 3.7.1 Positive Functions γµλ The most important component of the refined conjecture involves functions that are non-negative on the set 1 ≥ x1 ≥ x2 ≥ · · · ≥ xm ≥ 0. Let λ, µ be partitions such that mµ(1) µ ⊇ λ. Define cµλ := and the function mλ(1) γµλ := cµλmλ − mµ. Also, consider the set K := {(x1, . . . , xm) : 1 ≥ x1 ≥ x2 ≥ · · · ≥ xm ≥ 0}. Proposition 3.16. If x ∈ K and µ ⊇ λ, then γµλ(x) ≥ 0. Proof. If λ = µ then γµλ = 0 and we are done. Suppose that λ ⊂ µ. If x1 = ··· = xm = 0, then γµλ = 0. Suppose that x = (x1, . . . , xm) ∈ K and xm > 0. The inequality γµλ ≥ 0 is equivalent to mµ(1)mλ(x) ≥ mλ(1)mµ(x). Consider the expressions Mν,ρ(x, y) = mν(y)mρ(x) for partitions ν, ρ, then the non- negativity of γµλ is equivalent to the non-negativity of Mν,ρ(x, 1) − Mν,ρ(1, x). If ν ⊂ ρ then X σ·ν τ·ρ τ·ν τ·ρ c(Mν,ρ(x, y) − Mν,ρ(y, x)) = y x − x y σ,τ∈Sm X = xτ·νyσ·ν(y(σ·ρ)\(σ·ν) − x(τ·ρ)\(τ·ν)) σ,τ∈Sm where c is the product of the sizes of the stabilizers of ν and ρ in Sm. Substituting α m y1 = ··· = ym = 1 and noting that 1 − x ≥ 0 for all α ∈ N gives the result. 75 µ,r Definition 3.17. For any partition µ and Pieri root r, we define the values bλ,i for r λ ∈ Mµ to be the coefficients in the expansion of αµ in γλi−1λi , i.e., |C(λ)|−1 r X X µ,r X αµ = bλ,i γλi−1λi + fµλ∗ mλ∗ λ∈Mµ i=1 λ∈Mµ where C(λ) = {λ = λ0 ⊃ λ1 ⊃ · · · ⊃ λ∗}. Conjecture 3.18. For any partition µ and Pieri root r, |C(λ)|−1 r X X µ,r αµ = bλ,i γλi−1λi . λ∈Mµ i=1 µ,r r Furthermore, for all λ ∈ Mµ and i ≥ 1, bλ,i ≥ 0. In particular, αµ(x) ≥ 0 for x ∈ K. µ,r There is a recurrence relation between the bλ,i involving the monomial expansion coefficients r X r αµ = aµλmλ λ∈Pµ and the weights involved in the definition of the γµλ, γµλ := cµλmλ − mµ. r µ,r r Let bµλi := bi in the expression of Γµλ for λ ∈ Mµ, then r r r bµλi = cλi−1λi bµλi−1 − aµλi−1 (3.7) Write mk mk−1 m1 m0 λ = (α1 , α2 , . . . , αk , 0 ). Another way of writing the above recursion fully in terms of the aµλ is i r X r bµλi = − cλj λi aµλj . j=0 r Multiplying out by mλi (1) we deduce that, bµλi ≥ 0 if and only if i X r aµλj mλj (1) ≤ 0. j=0 76 Now, the coefficients fµλ∗ are explicitly k−1 r r r X r fµλ∗ = aµλ∗ − bµλ∗ = aµλ∗ + cλj λk−1 aµλj , j=0 k = k(λ), which has the same sign as k X r mλk−1 (1)fµλ∗ = aµλj mλj (1). j=0 In fact, since the value of mλ(1) is a multinomial coefficient, after clearing denominators the right hand side becomes k k X r X X r aµλj mj = mjhµνKνλj j=0 j=0 ν:λj ≤ν where, r gµ(ν, −µ1) gµ(ν, r) hµν = − . (3.8) (d + µ1)gµ(µ, −µ1) (d − r)gµ(µ, r) The following conjecture is therefore a much stronger version of the assertion in con- jecture 3.18 that fµλ∗ = 0. Conjecture 3.19. For any r ∈ Rµ, k X X gµ(ν, r) mjKνλj = Kµλ. (d − r)gµ(µ, r) j=0 ν:λj ≤ν This conjecture has been verified empirically. We will show data at the end of this dissertation. Suppose that aµλ ≥ 0 for all λ such that |λ| < |µ|, then we have the following observation. r Proposition 3.20. Let µ be a partition of weight n. If aµλ ≥ 0 for all λ with |λ| < n r r and for all ν, |ν| = n we have that fµν∗ = 0, then for all i ∈ {1, . . . , k(ν)} bµνi ≥ 0. X r aµλj = kµνKνλj ν:λj ≤ν r where kµν = 0 if µ \ ν is not a horizontal strip. 77 3.8 Kostka-Content Identities After some algebraic manipulation, conjecture 3.19 can be put into a polynomial form. This allows for an inductive proof for certain families of partitions. This new form does µ,r not address the positivity of the bλ,i coefficients. Let σ, τ be a partitions and recall that Gσ = {i : σi > σi+1} Y gσ(τ, t) = (t + τj − j). j∈Gσ Also recall that p (x) = Q x − r. Enumerate the elements in R and G by µ r∈Rµ µ µ Rµ = {r0, r1, . . . , rN } Gµ = {g1, . . . , gN } − such that r0 = r , r0 < r1 < ··· < rN , and g1 < g2 < ··· < gN . Then we have that µgi = gi−1 − ri−1 where g0 = 0. Using this notation, we can write the polynomial pµ as N Y pµ(x) = (x + (gi+1 − gi) + µgi+1 − gi+1) i=0 which resembles the polynomials gµ(λ, t). In fact, using the frequency notation µ = m1 m2 mN (n1 n2 ··· nN ), we have that gi − gi−1 = ti. The last factor is x − rN = x − `. Suppose that N = 1 then can expand pµ and put it in the form pµ = (x + m1 + µg1 − g1)(x + µg2 − g2) = (x + m1 + µg1 − g1)(x + µ` − `) = (x + µg1 − g1)(x + µ` − `) + m1(x + µ` − `). It follows that the expansion holds in the case that µ = λ and N = 1. 78 Throughout this section, by abuse of notation, the values Kσ,τ refer to the coef- ficients of the monomial symmetric functions in the expansion of sµ(x1, . . . , xn) for a specified n. For the first part, we may assume that n >> 0 and so Kσ,τ are the Kostka numbers. The following conjecture is equivalent to conjecture 3.19. Conjecture 3.21 (Kostka-Content Identity). For any partitions µ, λ such that λ ≤ µ, mk mk−1 m1 where λ = (n1 , n2 , . . . , nk ) ni > 0, k Y X X Kµλ t − r = (t − `(λ))gµ(µ, t)Kµλ + mjgµ(ν, t)Kνλj j r∈Rµ j=1 ν∈Hµ:|ν|=|λ | j mk mj −1 m1 where λ = (n1 , . . . , nk−j+1, . . . , nk ). The proof of the equivalence of the two conjectures is a straightforward algebraic manipulation that involves clearing denominators and then checking the degree of the polynomial on both sides. This polynomial form gives us the means to more easily check the vanishing part of conjecture 3.18. 3.8.1 An Inductive Argument In certain cases, we can use induction to prove conjecture 3.21. To do this, note the following relationship between the g and p polynomials: gµ+m(µ, t) = pµ(t − 1) = and pµ+m(t) = (t + m)pµ(t − 1), where m > µ1. If m = µ1, then the analogous identities are pµ(t) gµ+µ1 (µ, t) = and t + µ1 − 1 (t + µ1) pµ+µ1 (t) = pµ(t − 1). (t + µ1 − 1) Using these identities we can prove the following. 79 Proposition 3.22. Let µ, λ be partitions such that λ ≤ µ and conjecture 3.19 holds, then for any m ≥ µ1 conjecture 3.19 holds for µ + m, λ + m. Proof. Recall that conjecture 3.19 states that, in the same setting as in the proposition statement, `(λ) X X Kµλpµ(t) = (t − `(λ))gµ(µ, t)Kµλ + gµ(ν, t)Kν,λ−λj . j=1 ν∈Hµ:|ν|=|λ|−λj Letν ˆ = ν + m for any partition ν. Suppose that m > µ1. Using the identities above, a straightforward calculation yields the result: `(λˆ) X X (t − `(λˆ))g (ˆµ, t)K + g (ν, t)K µˆ µˆλˆ µˆ ν,λˆ−λˆj j=1 ˆ ˆ ν∈Hµˆ:|ν|=|λ|−λj `(λ)+1 X X = (t − `(λ) − 1)g (ˆµ, t)K + g (ν, t)K µˆ µˆλˆ µˆ ν,λˆ−λˆj j=1 ˆ ˆ ν∈Hµˆ:|ν|=|λ|−λj X = (t + m − 1) (t − `(λ) − 1)gµ(µ, t − 1)Kµλ + gµ(ν, t − 1)Kν,λ−λj 1≤j≤`(λ),ν∈Hµ +gµ+m(µ, t)Kµλ = (t + m − 1)pµ(t − 1)Kµλ + gµ+m(µ, t)Kµλ = (t + m)pµ(t − 1)Kµλ = pµ+m(t)Kµˆλˆ. In the case that m = µ1, the last 4 lines do not hold. Instead we have, = m1gµ+m(µ, t)Kµλ + ((t − `(λ) − 1)gµ(µ, t − 1)Kµλ X + gµ(ν, t − 1)Kν,λ−λj 1≤j≤`(λ),ν∈Hµ = pµ(t − 1)Kµλ + gµ+m(µ, t)Kµλ pµ(t − 1) = pµ(t − 1)Kµλ + Kµλ t + µ1 − 1 pµ(t − 1) = (t + µ1) Kµλ t + µ1 − 1 = pµ+m(t)Kµˆλˆ. 80 Note that the set Hµ+m can be described in terms of Hµ as G Hµ+m = {k + ν : ν ∈ Hµ, µ1 ≤ k ≤ m} = (k + Hµ). µ1≤k≤m This leads us to the following corollary to proposition 3.22. µ,r Corollary 3.23. Let µ, λ be partitions such that λ ≤ µ. Fix i and suppose that bλ,i is µ+m,r decreasing as a function of r where r ∈ Rµ, then for any m ≥ µ1, bλ+m,i is decreasing as a function of r where r ∈ Rµ+m. 3.8.2 Case λ = µ In this subsection, we will prove the Kostka-Content identity for the case when λ = µ using the conventions in 3.21. Proposition 3.24. Let µ and λ be partitions, then Kostka-Content identity holds when µ = λ. mk mk−1 m1 Proof. Let µ = n1 n2 ··· nk and suppose that λ = µ then the right hand side of the Kostka-Content identity is k X X (t − `(µ))gµ(µ, t)Kµ,µ + mjgµ(ν, t)Kνµj . j j=1 ν∈Hµ:|ν|=|λ | Since ν ≥ µj implies that ν = µj we have that the above polynomial is k X j (t − `(µ))gµ(µ, t) + mjgµ(µ , t). j=1 Furthermore, if we let Rµ = {r1, . . . , rk, rk+1} where −µ1 = r1 < r2 < ··· < rk+1 = `(µ) then one can directly verify that k−j k j Y Y gµ(µ , t) = (t − rp − mk−p+1) (t − rp − mk−p+1 + µip − µip+1 ) p=1 p=k−j+1 k−j k Y Y = (t − rp − mk−p+1) (t − rp+1). p=1 p=k−j+1 81 The crucial observation is that k k k−j k Y X Y Y (t − `(µ)) (t − rp − mk−p+1) + mj (t − rp − mk−p+1) (t − rp+1) p=1 j=1 p=1 p=k−j+1 k−1 Y = (t − `(µ))(t − rk) (t − rp − mk−p+1) p=1 k k−j k X Y Y + mj (t − rp − mk−p+1) (t − rp+1) j=2 p=1 p=k−j+1 k−2 Y = (t − `(µ))(t − rk)(t − rk−1) (t − rp − mk−p+1) p=1 k k−j k X Y Y + mj (t − rp − mk−p+1) (t − rp+1) j=3 p=1 p=k−j+1 . . = pµ(x). 3.8.3 Jack Hook-Length Recursion Formula Conjecture 3.21 for µ ` n, λ = 1n can be written as Y X fµ t − r = (t − n)gµ(µ, t)fµ + n gµ(ν, t)fν r∈Rµ ν⊂:µ where fρ = Kρ,1n for any partition ρ. Note that every ν such that ν ⊂: µ is of the i form µ − (i, µi), for i ∈ Gµ. In light of this fact, let µ denote the partition µ − (i, µi). Evaluating both sides at t = r where r is a Pieri root and dividing out by gµ(µ, r) (which is never 0 at Pieri roots) yields X r + µi − i − 1 0 = (r − n)fµ + n fµi . r + µi − i i∈Gµ Further manipulation yields the more manageable X fµi −rfµ = n . i − r − µi i∈Gµ 82 We will now give a proof of the following version of conjecture 1 for λ = 1n. It uses the hook length formula for fµ crucially. The proof uses the technique (of interpolation polynomials) found in the proof of the hook formula in [Ba]. i Proposition 3.25. For µ ` n, r ∈ Pµ, and µ = µ − (i, µi) for i ∈ Gµ, X fµi rfµ = n . r + µi − i i∈Gµ Proof. Recall that for any partition λ, |λ|! f = λ H(λ) Q where H(λ) = s∈λ hs(λ) and h(i,j)(λ) = λi + λj − i − j + 1 is the hook length at the box (i, j). Now fix a Pieri root r0. We will prove the identity for r = r0. Solving for r0 on the left hand side of the purported identity, we get Q X 1 s∈µ hs(µ) r0 = Q i . r0 + µi − i s∈µi hs(µ ) i∈Gµ Q s∈µ hs(µ) In the quotient Q i , there are many cancellations. s∈µi hs(µ ) Lemma 3.26. Q Q (c(µ \ µi) + r) s∈µ hs(µ) r∈Rµ Q i = −Q i j i h (µ ) (c(µ \ µ ) − c(µ \ µ )) s∈µ s j∈Gµ,j6=i where c(s) is the content of s. The lemma can be proved directly by carefully keeping track of all of the cancel- i lations. A crucial observation is that r + µi − i = r + c(µ \ µ ) and so we must prove that Q (c(µ \ µi) + r) X r∈Rµ\{r0} −r0 = Q i j . j∈G ,j6=i(c(µ \ µ ) − c(µ \ µ )) i∈Gµ µ Now, consider the polynomials Q (c(µ \ µi) + r) X r∈Rµ\{r0} Y j P (t) = Q i j (t − c(µ \ µ )) and j∈G ,j6=i(c(µ \ µ ) − c(µ \ µ )) i∈Gµ µ j∈Gµ,j6=i 83 Y Q(t) = (t + r) r∈Rµ\{r0} then, by degree considerations and vanishing at c(µ \ µi), we see that Y Q(t) − P (t) = (t − c(µ \ µi)). i∈Gµ Solving for P (t) and computing the coefficient of t|Gµ|−1, we get that Q (c(µ \ µi) + r) X r∈Rµ\{r0} X X = r + c(µ \ µi) = −r . Q (c(µ \ µi) − c(µ \ µj)) 0 j∈Gµ,j6=i i∈Gµ r∈Rµ\{r0} i∈Gµ (This last equality follows from the fact that P r + P c(µ \ µi) = 0). r∈Rµ i∈Gµ Corollary 3.27. Let µ be a partition of n for n ≥ 1, then 1 X fµ−(i,j) fµ = µ1 µ1 − h(1,j)(µ) (i,j)∈µ where fµ−s = 0 if µ − s is not a partition and hs(µ) is the hook length of s in µ. Here is an analogous Jack version that uses the fact that the Pieri roots of µ are the negative contents of the inner corners of µ. An inner corner of µ is the box (i, µi+1) rδ where i ∈ Gµ ∪ {0}. For this, we will need the concept of interpolation polynomials Pλ defined in [KP]. Proposition 3.28. Let µ be a partition, xi be the outer corners of µ, and yi be the i inner corners of µ, and µ = µ − xi, then Q X j(cα(xi) − cα(yj)) X rδ α|µ| = − Q = α Pµi (µ + rδ) j,j6=i(cα(xi) − cα(xj)) i∈Gµ µi where cα(i, j) = αj − i. Call the values cα(s), for s a box, the α-content of s. Let xi = (ai, bi) for i = 1, . . . , m, then y0 = (0, b1), y1 = (a1, b2), . . . , yn = (an, 0). The proof of the above proposition follows along the same lines as the proof of 3.25. 84 Proof. Let Qm X j=0(cα(xi) − cα(yj)) Y P (t) = − Q (t − cα(xk)) (cα(xi) − cα(xj)) µi 1≤j≤m,j6=i 1≤k≤m;k6=i Y Q(t) = (t − cα(yi)), 0≤i≤m then Y P (t) + Q(t) = (t + β) (t − cα(xi)) 1≤i≤m where β is a constant, possibly depending on α. Solving for P (t) we see that the coefficient of tm is m m X X β − cα(xi) + cα(yi) = 0 i=1 i=0 and since m X cα(xi) = (αb1 − a1) + (αb2 − a2) + ··· + (αbn − an) i=1 (αb1 − 0) + (αb2 − a1) + ··· + (αbn − an−1) + (α · 0 − an) m X = cα(yi) i=0 it follows that β = 0. Therefore, the coefficient of tm−1 of Y P (t) = −Q(t) + t (t − cα(xi)) 1≤i≤m is X X cα(xi)cα(xj) − cα(yi)cα(yj) 1≤i m !2 m !2 m 1 X 1 X 1 X = c (x ) − c (y ) − (c (x )2 − c (y )2) 2 α i 2 α i 2 α i α i i=1 i=0 i=0 m 1 X = − (c (x )2 − c (y )2) 2 α i α i i=0 where we take x0 = (0, 0) = xm+1 for convenience. Finally, shifting the sum slightly, note that m 1 X − (c (x ) − c (y ))(c (x ) + c (y )) 2 α i+1 α i α i+1 α i i=0 m X = αbi(ai − ai−1) i=1 85 = α|µ|. To prove that X rδ |µ| = Pµi (µ + rδ) µi note that X 1 n P rδ(x) = x + ··· + x − r . λ k! 1 d 2 |λ|=k k Therefore, X |µ| P rδ(µ + rδ) = . λ k |λ|=k Define the upper and lower hook lengths for the box s = (i, j) to be 0 hλ(s; α) = α(λi − j) + λj − i + 1 0 0 hλ(s; α) = α(λi − j + 1) + λj − i respectively, where λ0 is the transpose of λ. Notice the following important fact and its immediate corollary. Lemma 3.29. Using the notation above and from 3.28, for µ :⊃ µi and R (resp. C) i are the row (resp. column) of the box xi = µ/µ , 0 Q Y hµ(s; α) Y hµ(s; α) j(cα(xi) − cα(yj)) α 0 = −Q h i (s; α) h i (s; α) (cα(xi) − cα(xj)) s∈C µ s∈R µ j,j6=i Define, for any partitions µ, λ such that λ ⊂ µ, the set Y (µ/λ) of boxes (i, j) ∈ µ 0 0 such that λi < µi and λj = µj and the rational expression 0 Y hµ(s; α)/hµ(s; α) ψµ/λ(α) = 0 . h (s; α)/hλ(s; α) s∈Y (µ/λ) λ 86 Remark 4. Note that in the 1996 IMRN paper [KS], the authors defined 0 0 Y hµ(s; α)/hµ(s; α) ψµ/λ(α) = 0 hλ(s; α)/h (s; α) s∈X(µ/λ) λ 0 0 where X(µ/λ) is the set of boxes (i, j) ∈ µ such that µi = λi and λj < µj, in order to describe the Pieri rule for Jack polynomials: (α) X 0 (α) ekPλ = ψµ/λ(α)Pµ , µ where µ runs over partitions such that µ/λ is a vertical k-strip. Also define the quantity |λ|! fλ(α) = Q , s∈λ hλ(s; α) which is the coefficient the monomial m1λ in the expansion of the Jack polynomial (α) (specifically, the normalized version Pλ ) with parameter α. Notice the striking simi- (α) larity with the Pieri rule for the case e1Pλ . Corollary 3.30. For µ any partition, X fµ(α) = ψµ/λ(α)fλ(α). λ⊂:µ The analogue of 3.25 for the Jack case can be formulated as follows. Let Pµ be the set of inner corners of µ and Rµ be their α-contents. An element of Rµ will be denoted rα to emphasize the dependence on the parameter α. i Proposition 3.31. For µ ` n, r ∈ Pµ, and µ = µ − (i, µi) where i ∈ Gµ, X fµi (α) rαfµ(α) = n . rα + αµi − i i∈Gµ Equivalently, X rα + αµi − i − 1 (n − rα)fµ(α) = n fµi (α). rα + αµi − i i∈Gµ 87 Lemma 3.32. For any partition µ and rα ∈ Rµ, then rδ X Pµi (µ + rδ) rα = . rα + αµi − i i∈Gµ Equivalently, Y X Y rδ (n − rα) (rα − n + αµj) = (rα − n + ανj))Pν (µ). j≥1 ν⊂:µ j≥1 Y X Y rδ −t (t + αµj) = (t + ανj))Pν (µ). j≥1 ν⊂:µ j≥1 The proofs of these results is almost identical to the proofs in the Schur case, ex- cept that the contents c and Pieri roots r are replaced by their Jack versions rα and cα respectively. Similar replacements can be made throughout the paper in order to produce Jack versions of different results for Schur polynomials. We hope to address the conjectures in the more general Jack setting at a future date. 88 References [B] A. Beauville. Conformal Blocks, Fusion Rules, and the Verlinde Formula, Proceed- ings of the Hirzebruch 65 Conference on Algebraic Geometry, (Ramat Gan), 75-96, 1996. 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