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<x\mbox{~for~}s\in S\\T_xT_y&=&T_{xy}&\mbox{if~}\ell(x)+\ell(y)=\ell(xy)\end{array}

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.

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