Iwahori-Hecke algebra of the symmetric group

Definition

Let $R$ be a unital ring and $n$ be a natural number. The Iwahori-Hecke algebra of the symmetric group of degree $n$ over the ring $R$ is defined as the $R[q]$-algebra given by the presentation:

$\langle T_{1},T_{2},\dots T_{n-1}\mid (T_{i}-q)(T_{i}+1)=0,T_{i}T_{j}=T_{j}T_{i}\ \forall |i-j|>1,T_{i}T_{i+1}T_{i}=T_{i+1}T_{i}T_{i+1}\rangle$.

In other words, it is the Iwahori-Hecke algebra corresponding to the symmetric group of degree $n$, viewed as a Coxeter group, in the usual way: Symmetric group on a finite set is a Coxeter group.

For a field with $q$ elements, specializing to the value $q$ gives the Hecke algebra of the general linear group $GL_{n}(\mathbb {F} _{q})$.

Particular cases

$n$	Iwahori-Hecke algebra of the symmetric group of degree $n$
2	Iwahori-Hecke algebra of symmetric group:S2
3	Iwahori-Hecke algebra of symmetric group:S3
4	Iwahori-Hecke algebra of symmetric group:S4