Last update: 17 September 2013
Let be a finite group and let be a subgroup of The Hecke algebra of the pair is the subalgebra of the group algebra of The elements as runs over a set of representatives of the double cosets form a basis of
Let be a finite Chevalley group over the field with elements and fix a Borel subgroup of The pair determines a pair where is the Weyl group of and is a set of simple reflections in (with respect to The Iwahori-Hecke algebra corresponding to is the Hecke algebra In this case the basis elements are indexed by the elements of the Weyl group corresponding to the pair and the multiplication is given by if is a simple reflection in In this formula is the length of i.e. the minimum number of factors needed to write as a product of simple reflections.
A particular example of the Iwahori-Hecke algebra occurs when and is the subgroup of upper triangular matrices. Then the Weyl group is the symmetric group and the simple reflections in the set are the transpositions In this case the algebra is the Iwahori-Hecke algebra of type and (as we will see later) can be presented by generators and relations 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 just by defining it to be the algebra with basis and multiplication if These algebras are not true Hecke algebras except when is a Weyl group.
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..