# 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:

$\langle T_1, T_2, \dots T_{n-1} \mid (T_i - q)(T_i + 1) = 0, T_iT_j = T_jT_i \ \forall |i - j| > 1, T_iT_{i+1}T_i = T_{i+1}T_iT_{i+1} \rangle$.

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