# Iwahori-Hecke algebra of the symmetric group

## Definition

Let be a unital ring and be a natural number. The **Iwahori-Hecke algebra** of the symmetric group of degree over the ring is defined as the -algebra given by the presentation:

.

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

For a field with elements, *specializing* to the value gives the Hecke algebra of the general linear group .

## Particular cases

Iwahori-Hecke algebra of the symmetric group of degree | |
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2 | Iwahori-Hecke algebra of symmetric group:S2 |

3 | Iwahori-Hecke algebra of symmetric group:S3 |

4 | Iwahori-Hecke algebra of symmetric group:S4 |