Last update: 17 September 2013

Let $G$ be a finite group and let $B$ be a subgroup of $G\text{.}$ The
*Hecke algebra* of the pair $(G,B)$ is the subalgebra
$$\mathscr{H}(G,B)=\left\{\sum _{g\in G}{a}_{g}g\hspace{0.17em}\right|\hspace{0.17em}{a}_{g}\in \u2102,\hspace{0.17em}\text{and}\hspace{0.17em}{a}_{g}={a}_{h}\hspace{0.17em}\text{if}\hspace{0.17em}BgB=BhB\}$$
of the group algebra of $G\text{.}$ The elements
$${T}_{w}=\frac{1}{\left|B\right|}\sum _{g\in BwB}g,$$
as $w$ runs over a set of representatives of the double cosets $B\backslash G/B,$
form a basis of $\mathscr{H}(G,B)\text{.}$

Let $G$ be a finite Chevalley group over the field ${\mathbb{F}}_{q}$ with $q$ elements
and fix a Borel subgroup $B$ of $G\text{.}$ The pair $(G,B)$
determines a pair $(W,S)$ where $W$ is the Weyl group of $G$
and $S$ is a set of simple reflections in $W$ (with respect to $B\text{).}$
The *Iwahori-Hecke algebra* corresponding to $G$ is the Hecke algebra
$\mathscr{H}(G,B)\text{.}$ In this case the basis elements
${T}_{w}$ are indexed by the elements $w$ of the Weyl group $W$ corresponding to the pair
$(G,B)$ and the multiplication is given by
$${T}_{\U0001d530}{T}_{w}=\{\begin{array}{cc}{T}_{sw},& \text{if}\hspace{0.17em}\ell \left(sw\right)>\ell \left(w\right),\\ (q-1){T}_{w}+q{T}_{sw}& \text{if}\hspace{0.17em}\ell \left(sw\right)<\ell \left(w\right),\end{array}$$
if $s$ is a simple reflection in $W\text{.}$ In this formula
$\ell \left(w\right)$ is the *length* of $w,$
i.e. the minimum number of factors needed to write $w$ as a product of simple reflections.

A particular example of the Iwahori-Hecke algebra occurs when $G=GL(n,{\mathbb{F}}_{q})$
and $B$ is the subgroup of upper triangular matrices. Then the Weyl group $W,$ is the symmetric group
${S}_{n},$ and the simple reflections in the set $S$ are the transpositions
${s}_{i}=(i,i+1),$
$1\le i\le n-1\text{.}$ In this case the algebra
$\mathscr{H}(G,B)$ is the *Iwahori-Hecke algebra of type*
${A}_{n-1}$ and (as we will see later) can be presented by generators
${T}_{1},\dots ,{T}_{n-1}$
and relations
$$\begin{array}{cc}{T}_{i}{T}_{j}={T}_{j}{T}_{i},& \text{for}\hspace{0.17em}|i-j|>1,\\ {T}_{i}{T}_{i+1}{T}_{i}={T}_{i+1}{T}_{i}{T}_{i+1},& \text{for}\hspace{0.17em}1\le i\le n-2,\\ {T}_{i}^{2}=(q-1){T}_{i}+q,& \text{for}\hspace{0.17em}2\le i\le n\text{.}\end{array}$$
See Section B5 for more facts about the Iwahori-Hecke algebras of type A. In particu- lar, these Iwahori-Hecke algebras also appear as tensor power centralizer algebras, see Theorem B5.3. This is some kind of miracle: the Iwahori-Hecke algebras of type A are the only Iwahori-Hecke algebras which arise naturally as tensor power centralizers.

In view of the multiplication rules for the Iwahori-Hecke algebras of Weyl groups it is easy to define a “Hecke algebra” for all Coxeter groups $(W,S),$ just by defining it to be the algebra with basis ${T}_{w},$ $w\in W,$ and multiplication $${T}_{s}{T}_{w}=\{\begin{array}{cc}{T}_{sw},& \text{if}\hspace{0.17em}\ell \left(sw\right)>\ell \left(w\right),\\ (q-1){T}_{w}+q{T}_{sw},& \text{if}\hspace{0.17em}\ell \left(sw\right)<\ell \left(w\right),\end{array}$$ if $s\in S\text{.}$ These algebras are not true Hecke algebras except when $W$ is a Weyl group.

**References**

For references on Hecke algebras see [CRe1987] (Vol I, Section 11). For references on Iwahori-Hecke algebras see [Bou1968] Chpt. IV §2 Ex. 23-25, [CRe1987]
Vol. II §

This is the survey paper *Combinatorial Representation Theory*, written by Hélène Barcelo and Arun Ram.

*Key words and phrases.* Algebraic combinatorics, representations.

Barcelo was supported in part by National Science Foundation grant DMS-9510655.

Ram was supported in part by National Science Foundation grant DMS-9622985.

This paper was written while both authors were in residence at MSRI. We are grateful for the hospitality and financial support of MSRI..