---
tag: B0587TVWZY
kind: definition
name: Hecke algebra
msc: [20C08, 20F55]
defines: 5J2P2AGY4N
uses: [G6FND7X69J, 3CTW1P4HBP, 0310REJJY8, 0DZ9DKAKZR]
sources:
  - url: https://en.wikipedia.org/wiki/Hecke_algebra
---

## Definition

This is the Iwahori–Hecke algebra.
Let $(W, S)$ be a [[Coxeter system|G6FND7X69J]], with Coxeter matrix $(m_{st})_{s,t \in S}$, so that $m_{st}$ is the order of $st$ in $W$.
Let $R$ be a [[commutative ring|0310REJJY8]] with identity and let $q \in R$.
The *Hecke algebra* $H_q(W)$ is the [[associative algebra|3CTW1P4HBP]] over $R$ generated by elements $T_s$ for $s \in S$, subject to the relations:

1. $(T_s - q)(T_s + 1) = 0$ for every $s \in S$.
2. $T_s T_t T_s \cdots = T_t T_s T_t \cdots$, with $m_{st}$ factors on each side, for all $s \ne t$ in $S$ with $m_{st} < \infty$.

## Remarks

For $w \in W$ with a reduced expression $w = s_1 \cdots s_k$, put $T_w = T_{s_1} \cdots T_{s_k}$.
By the braid relations and Matsumoto's theorem $T_w$ does not depend on the reduced expression, and the $T_w$ for $w \in W$ form a basis of $H_q(W)$ as a [[free|0DZ9DKAKZR]] $R$-module.
The multiplication is $T_s T_w = T_{sw}$ if $\ell(sw) > \ell(w)$ and $T_s T_w = (q - 1) T_w + q T_{sw}$ if $\ell(sw) < \ell(w)$, where $\ell$ is the length function.
At $q = 1$ the relations become $T_s^2 = 1$ and the braid relations, which present the group algebra $R[W]$.
Example: $W = \set{1, s}$ of order $2$ gives $H_q(W) = R[T]/((T - q)(T + 1))$, free of rank $2$ with basis $1, T$.
Non-example: for $W = S_3$ with $S = \set{s, t}$ and $m_{st} = 3$, dropping relation 2 changes the algebra.
At $q = 1$ the result is the group algebra of the infinite dihedral group $\langle s, t \mid s^2 = t^2 = 1 \rangle$, which has infinite rank, while $H_1(S_3) = R[S_3]$ has rank $6$.
Many authors use the parameter $q^{1/2}$ or $v$ with $q = v^2$ and rescale the generators.