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Hecke algebras and the Kazhdan-Lusztig polynomials

The Hecke algebra attached to a Coxeter system $(W, S)$ is a deformation of the group algebra of $W$ defined as follows. Take the free $\mathbb{Z}[q^{ \frac{1}{2} }, q^{ - \frac{1}{2} }]$-module $\mathcal{H}_W$ with basis $T_w, w \in W$, and impose the multiplicative relations

$T_w T_s = T_{ws}$

if $\ell(sw) > \ell(w)$, and

$T_w T_s = q T_{ws} + (q - 1) T_w$

otherwise. (For now, ignore the square root of $q$.) Humphreys proves that these relations describe a unique associative algebra structure on $\mathcal{H}_W$ with $T_e$ as the identity, but the proof is somewhat unenlightening, so I will skip it. (Actually, the only purpose of this post is to motivate the definition of the Kazhdan-Lusztig polynomials, so I’ll be referencing the proofs in Humphreys rather than giving them.)

The motivation behind this definition is a somewhat long story. When $W$ is the Weyl group of an algebraic group $G$ with Borel subgroup $B$, the above relations describe the algebra of functions on $G(\mathbb{F}_q)$ which are bi-invariant with respect to the left and right actions of $B(\mathbb{F}_q)$ under a convolution product. The representation theory of the Hecke algebra is an important tool in understanding the representation theory of the group $G$, and more general Hecke algebras play a similar role; see, for example MO question #4547 and this Secret Blogging Seminar post. For example, replacing $G$ and $B$ with $\text{SL}_2(\mathbb{Q})$ and $\text{SL}_2(\mathbb{Z})$ gives the Hecke operators in the theory of modular forms.

Chevalley-Bruhat order

Before we define Bruhat order, I’d like to say a few things by way of motivation. Warning: I know nothing about algebraic groups, so take everything I say with a grain of salt.

A (maximal) flag in a vector space $V$ of dimension $n$ is a chain $V_0 \subset V_1 \subset ... \subset V_n$ of subspaces such that $\dim V_i = i$. The flag variety of $G = \text{SL}(V) = \text{SL}_n$ is, for our purposes, the “space” of all maximal flags. $\text{SL}_n$ acts on the flag variety in the obvious way, and the stabilizer of any particular flag is a Borel subgroup $B$. If $e_1, ... e_n$ denotes a choice of ordered basis, one can define the standard flag $0, \text{span}(e_1), \text{span}(e_1, e_2), ...$, whose stabilizer is the space of upper triangular matrices of determinant $1$ with respect to the basis $e_i$. This is the standard Borel, and all other Borel subgroups are conjugate to it. Indeed, it’s not hard to see that $\text{SL}_n$ acts transitively on the flag variety, so the flag variety can be identified with the homogeneous space $G/B$.

The strong exchange condition

It’s nice that Weyl groups are Coxeter groups and all, but the definition of a Coxeter group as a group with a particular kind of representation doesn’t immediately tell us why this is the appropriate level of generalization (although the faithfulness of the geometric representation is a good sign). It turns out there is a structural property, the strong exchange condition, which completely characterizes Coxeter groups among groups generated by involutions. Today we will prove this property.

Coxeter groups

At SPUR this summer I’ll be working on the Kazhdan-Lusztig polynomials, although my mentor and I haven’t quite pinned down what problem I’m working on. I thought I’d take the chance to share some interesting mathematics and also to write up some background for my own benefit. I’ll mostly be following the second half of Humphreys.

A Coxeter system $(W, S)$ is a group $W$ together with a generating set $S$ and presentation of the form

$\langle s_1, ... s_n | (s_i s_j)^{m(i, j)} = 1 \rangle$

where $m(i, j) = m(j, i), m(i, i) = 1$, and $m(i, j) \ge 2, i \neq j$. (When there is no relation between $s_i, s_j$, we write this as $m(i, j) = \infty$.) The group $W$ is a Coxeter group, and is usually understood to come with a preferred presentation, so we will often abuse terminology and use “group” and “system” interchangeably. $S$ is also referred to as the set of simple reflections in $W$, and $n$ the rank. (We will only consider finitely-generated Coxeter groups.)

Historically, Coxeter groups arose as symmetry groups of regular polytopes and as Weyl groups associated to root systems, which in turn are associated to Lie groups, Lie algebras, and/or algebraic groups; the former are very important in understanding the latter. John Armstrong over at the Unapologetic Mathematician has a series on root systems. In addition, for a non-technical overview of Coxeter groups and $q$-analogues, I recommend John Baez’s week184 through week187. The slogan you should remember is that Weyl groups are “simple algebraic groups over $\mathbb{F}_1$.”