Iwahori-Hecke algebra of a Coxeter group

Definition

Let $G$ be a Coxeter group with Coxeter presentation:

$G:=\langle s_{1},s_{2},\dots ,s_{n}\mid (s_{i}s_{j})^{m_{ij}}\rangle$

where $m_{ij}=m_{ji}$ and $m_{ii}=2$. The Iwahori-Hecke algebra of $G$ over a ring $R$ is defined as the $R[q]$-algebra (for an indeterminate $q$) generated by $T_{1},T_{2},\dots ,T_{n}$ with the following relations:

$(T_{i}-q)(T_{i}+1)=0$

and the Artin braid relations:

$T_{i}T_{j}\dots =T_{j}T_{i}\dots$,

where the length of both sides is $m_{ij}$. If $m_{ij}$ is even, the left side ends with $T_{j}$ and the right side ends with $T_{i}$. Otherwise, the left side ends with $T_{i}$ and the right side ends with $T_{j}$.

For specific choices of $q\in R$, we get a $R$-algebra. Thus, the $R[q]$-algebra can be viewed as a one-parameter family of $R$-algebras. When $q=1$, we get the group ring of $G$ over $R$. To distinguish itself from the algebras obtained by setting particular values of $q$, the Iwahori-Hecke algebra is also sometimes termed the generic Hecke algebra. Choosing a particular value of $q$ is specialization.

For the Weyl group of a Chevalley group

If $W$ is the Weyl group of a Chevalley group, then $W$ has a natural choice of Coxeter presentation. For this choice of Coxeter presentation, we can define the Iwahori-Hecke algebra as above. It turns out that the Hecke algebra of the Chevalley group realized over a field of size $q$, taken over the ring $R$ is isomorphic to the Iwahori-Hecke algebra described above, specialized at $q$.

The symmetric group and general linear groups

The Weyl group of the general linear group of order $n$ over any field is the symmetric group of degree $n$. The Iwahori-Hecke algebra of the symmetric group has the property that when specialized to a particular value of $q$, it gives the Hecke algebra of a general linear group of order $n$ over a field of size $q$.