# Hecke Algebra of a Coxeter Group

Let $G$ be a Coxeter group with a generating set $S.$

Definition 1.  The Hecke algebra of $G,$ denoted $H(G),$ is generated by elements $\{T_g\}_{g\in G}$ over the ring $\mathbb{Z}[q^{1/2},q^{-1/2}]$ such that multiplication satisfies

$\begin{array}{lcll}T_sT_x&=&qT_{sx}+(q-1)T_x&\mbox{if~}sx

where the order is Bruhat.  It is trivial to verify that the Hecke algebra is unital with unit $T_1.$

Proposition 2.  If $s\in S,$ then $T_s$ is invertible with inverse

$T_{s}^{-1}=(q^{-1}-1)T_1+q^{-1}T_s.$

Proof.

$\begin{array}{lcl}T_sT_{s}^{-1}&=&T_s\left((q^{-1}-1)T_1+q^{-1}T_s\right)\\&=&T_s(q^{-1}T_1-T_1+q^{-1}T_s)\\&=&q^{-1}T_sT_1-T_sT_1+q^{-1}T_sT_s\\&=&(q^{-1}-1)T_sT_1+q^{-1}T_sT_s\\&=&(q^{-1}-1)T_s+T_1+q^{-1}(q-1)T_s\\&=&T_1.\end{array}$

It follows that all elements of the form $T_g$ with $g\in G$ are invertible, so the Hecke algebra is a division algebra.  Moreover it is a *-algebra with involution defined by

$\displaystyle \overline{\sum_{g\in G}p_g(q^{1/2})T_g}=\sum_{g\in G}p_g(q^{1/2})T_{g^{-1}}^{-1}.$

It turns out that there is a basis $\{x_g\}_{g\in G}$ of $H(G)$ where

$\displaystyle x_g=q^{\ell(g)/2}\sum_{h\leq g}(-1)^{\ell(h,g)}q^{-\ell(h)}P_{h,g}T_h$

with $\ell(h,g)=|\ell(h)-\ell(g)|$ and $P_{h,g}\in\mathbb{Z}[q],$ which can be proven to uniquely exist for elements $h,g\in G,$ which are called the Kazhdan-Lusztig polynomials.

[1]  Björner, Anders and Francesco Brenti.  Combinatorics of Coxeter Groups.  Vol. 231.  Graduate Texts in Mathematics.  Springer.  2005.