Combinatorial Representation Theory

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 17 September 2013

Appendix B

B3. Hecke algebras and “Hecke algebras” of Coxeter groups

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 $$\mathscr{H}(G,B)=\left\{\sum _{g\in G}{a}_{g}g\hspace{0.17em}\right|\hspace{0.17em}{a}_g\in ℂ,\hspace{0.17em}\text{and}\hspace{0.17em}{a}_g=a_h\hspace{0.17em}\text{if}\hspace{0.17em}BgB=BhB\}$$ 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 ${𝔽}_q$ 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 $$T_{𝔰}{T}_w=\{\begin{array}{cc}{T}_{sw},& \text{if}\hspace{0.17em}\ell \left(sw\right)>\ell \left(w\right),\\ (q-1)T_w+q{T}_{sw}& \text{if}\hspace{0.17em}\ell \left(sw\right)<\ell \left(w\right),\end{array}$$ if $s$ 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 $G=GL(n,{𝔽}_q)$ 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 §67, and [Hum1990] Chpt. 7. The article [Cur1988-2] is also very informative.

Notes and references

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..

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