Hecke algebra

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 $R$-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.