Unipotent Hecke algebras: the structure, representation theory, and combinatorics

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 26 March 2015

This is an excerpt of the PhD thesis Unipotent Hecke algebras: the structure, representation theory, and combinatorics by F. Nathaniel Edgar Thiem.

Introduction

Combinatorial representation theory takes abstract algebraic structures and attempts to give explicit tools for working with them. While the central questions may take an abstract form, the answers from this perspective use known combinatorial objects in computations and constructive descriptions. This thesis examines a family of Hecke algebras, giving algorithms that elucidate some of the structural and representation theoretic results classically developed from an algebraic and geometric point of view. Thus, a classical description of a natural basis becomes an actual construction of this basis, and an efficient algorithm for computing products in this basis replaces a classical formula.

A large body of literature in combinatorial representation theory is dedicated to the study of the symmetric group $S_n,$ giving rise to a wealth of combinatorial ideas. Some of these ideas have been generalized to Weyl groups, a larger class of groups whose structure is essential in the study of Lie groups and Lie algebras. Even more generally, the Iwahori-Hecke algebra gives a “quantum” version of the Weyl group situation.

Iwahori [Iwa1964] and Iwahori-Matsumoto [IMa1965] introduced the Iwahori-Hecke algebra as a first step in classifying the irreducible representations of finite Chevalley groups and reductive $p\text{-adic}$ Lie groups. Subsequent work (e.g. [CSh1999] [KLu1979] [LVo1983]) has established Hecke algebras as fundamental tools in the representation theory of Lie groups and Lie algebras, and advances on subfactors and quantum groups by Jones [Jon1983], Jimbo [Jim1986], and Drinfeld [Dri1987] gave Hecke algebras a central role in knot theory [Jon1987], statistical mechanics [Jon1989], mathematical physics, and operator algebras.

Unipotent Hecke algebras are a family of algebras that generalize the Iwahori-Hecke algebra. Another classical example of a unipotent Hecke algebra is the Gelfand-Graev Hecke algebra, which has connections with unipotent orbits [Kaw1975], Kloosterman sums [CSh1999], and in the $p\text{-adic}$ case, Hecke operators (see [Bum1998, 4.6,4.8]). Interpolating between these two classical examples will be fundamental to my analysis.

There are three ingredients in the construction of a Hecke algebra: a group $G,$ a subgroup $U$ and a character $\chi :U\to ℂ\text{.}$ The Hecke algebra associated with the triple $(G,U,\chi )$ is the centralizer algebra $$\mathscr{H}(G,U,\chi )={\text{End}}_{ℂG}\left({\text{Ind}}_U^G\left(\chi \right)\right)\text{.}$$ It has a natural basis indexed by a subset $N_{\chi }$ of $U\text{-double}$ coset representatives in $G\text{.}$

A unipotent Hecke algebra is $${\mathscr{H}}_{\mu }=\mathscr{H}(G,U,{\psi }_{\mu })={\text{End}}_{ℂG}\left({\text{Ind}}_U^G\left({\psi }_{\mu }\right)\right),$$ where $U$ is a maximal unipotent subgroup of a finite Chevalley $G$ (such as ${GL}_n\left({𝔽}_q\right)\text{),}$ and ${\psi }_{\mu }$ is an arbitrary linear character. In this context, the classical algebras are $$\begin{array}{ccc}\text{the Yokonuma Hecke algebra [Yok1969-2]}& ⟷& {\psi }_{\mu }\hspace{0.17em}\text{trivial,}\\ \text{the Gelfand-Graev Hecke algebra [GGr1962]}& ⟷& {\psi }_{\mu }\hspace{0.17em}\text{in general position.}\end{array}$$

After reviewing some of the underlying mathematics in Chapter 2, Chapter 3 analyzes unipotent Hecke algebras of general type. The main results are (a) a generalization to unipotent Hecke algebras of the braid calculus used in Iwahori-Hecke algebra multiplication. I use two approaches: the first gives “local” relations similar to braid relations, and the second gives a more global algorithm for computing products. In particular, this provides another solution to the difficult problem of multiplication in the Gelfand-Graev Hecke algebra (see [Cur1988] for a geometric approach to this problem). (b) the construction of a large commutative subalgebra ${ℒ}_{\mu }$ in ${\mathscr{H}}_{\mu },$ which may be used as a Cartan subalgebra to analyze the representations of ${\mathscr{H}}_{\mu }$ via weight space decompositions;

Chapters 4, 5 and part of 6 consider the case when $G={GL}_n\left({𝔽}_q\right),$ the general linear group over a finite field with $q$ elements. These chapters can all be read independently of Chapter 3, and I would recommend that readers unfamiliar with Chevalley groups begin with Chapter 4.

Chapter 4 describes the structure of the unipotent Hecke algebras when $G={GL}_n\left({𝔽}_q\right)\text{;}$ in this case, unipotent Hecke algebras are indexed by compositions $\mu$ of $n\text{.}$ The main results are (c) an explicit basis. Suppose ${\mathscr{H}}_{\mu }$ corresponds to the composition $\mu ,$ and $N_{\mu }$ indexes the basis of ${\mathscr{H}}_{\mu }\text{.}$ There is an explicit bijection from $N_{\mu }$ to the set of polynomial matrices over ${𝔽}_q$ with row-degree sums and column-degree sums equal to $\mu$ (see Chapter 4 for details). In particular, this gives an explicit construction of the matrices in $N_{\mu }\text{.}$ (d) a pictorial approach to multiplying the natural basis elements, inspired by (a) above.

Chapter 5 proceeds to develop the representation theory of unipotent Hecke algebras in the case $G={GL}_n\left(𝔽\right)\text{.}$ This chapter relies heavily on a decomposition of ${\text{Ind}}_U^G\left({\psi }_{\mu }\right)$ given by Zelevinsky in [Zel1981-2]. The main results are (e) a generalization of the RSK correspondence which provides a combinatorial proof of the identity $$\text{dim}\left({\mathscr{H}}_{\mu }\right)=\sum _{\underset{{\mathscr{H}}_{\mu }\text{-modules}\hspace{0.17em}V}{\text{Irreducible}}}\text{dim}{\left(V\right)}^2\text{;}$$ (f) a computation of the weight space decomposition of an ${\mathscr{H}}_{\mu }\text{-module}$ with respect to a commutative subalgebra ${ℒ}_{\mu }$ (see (b) above).

Chapter 6 concludes by studying the representation theory of Yokonuma Hecke algebras (the special case when ${\psi }_{\mu }$ is trivial). The main results are (g) a reduction in general type from the study of irreducible Yokonuma Hecke algebra modules to the study of irreducible modules for a family of “sub”algebras; (h) an explicit construction of the irreducible modules of the Yokonuma algebra in the case $G={GL}_n\left({𝔽}_q\right)\text{.}$

Notes and References

This is an excerpt of the PhD thesis Unipotent Hecke algebras: the structure, representation theory, and combinatorics by F. Nathaniel Edgar Thiem.

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