# Iwahori-Hecke algebra of a Coxeter group

## Definition

Let be a Coxeter group with Coxeter presentation:

where and . The Iwahori-Hecke algebra of over a ring is defined as the -algebra (for an indeterminate ) generated by with the following relations:

and the Artin braid relations:

,

where the length of both sides is . If is even, the left side ends with and the right side ends with . Otherwise, the left side ends with and the right side ends with .

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

## For the Weyl group of a Chevalley group

If is the Weyl group of a Chevalley group, then 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 , taken over the ring is isomorphic to the Iwahori-Hecke algebra described above, specialized at .

### The symmetric group and general linear groups

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