---
tag: 5J2P2AGY4N
kind: concept
name: Hecke algebra
msc: [20C08, 20F55]
specializes: [3CTW1P4HBP]
sources:
  - url: https://en.wikipedia.org/wiki/Hecke_algebra
same_as:
  wikidata: Q48970267
  wikipedia: Hecke algebra
---

## Summary

The name Hecke algebra covers several related algebras; the most common is the Iwahori–Hecke algebra of a [[Coxeter group|G6FND7X69J]].
The Iwahori–Hecke algebra is a deformation of the group algebra of the Coxeter group, depending on a parameter $q$.
At $q = 1$ it is the group algebra.
Let $G$ be a finite group of Lie type over a field with $q$ elements.
Iwahori showed that the Hecke algebra is the endomorphism algebra of the permutation module of $G$ on the cosets of a Borel subgroup.
Other Hecke algebras are built from double cosets of a subgroup, as for the [[Hecke operators|S4SV3QZ8B1]] of number theory, or from locally compact groups.

## Explanations

### L1 human

A Coxeter group, such as the [[symmetric group|0515A6D2Y1]], is generated by reflections $s$ with $s^2 = 1$ and some braid relations.
The Hecke algebra keeps the braid relations but replaces $s^2 = 1$ by a rule with a parameter $q$, namely $T_s^2 = (q - 1) T_s + q$.
When $q = 1$ this is again $T_s^2 = 1$, so the Hecke algebra is a one-parameter family of algebras through the group algebra.
For the group with two elements, the Hecke algebra is spanned by $1$ and $T$, with $T^2 = (q-1)T + q$.
These algebras appear in the representation theory of finite and $p$-adic groups, in knot invariants and in the combinatorics of Kazhdan–Lusztig polynomials.