Iwahori-Hecke algebra of a Coxeter group

Definition

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

$G : = ⟨ s_{1}, s_{2}, \dots, s_{n} ∣ (s_{i} s_{j})^{m_{i j}} ⟩$

where $m_{i j} = m_{j i}$ and $m_{i i} = 2$ . 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 $m_{i j}$ . If $m_{i j}$ 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 $q$ .