Iwahori-Hecke algebra of a Coxeter group

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

$$G := \langle s_1, s_2, \dots, s_n \mid (s_is_j)^{m_{ij}} \rangle$$

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 $$q$$.