Iwahori-Hecke algebra of a Coxeter group

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Definition

Let G be a Coxeter group with Coxeter presentation:

G := \langle s_1, s_2, \dots, s_n \mid (s_is_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_iT_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.