Last update: 17 September 2013
Let and be indeterminates. Let be the algebra over the field given by generators and relations
Now suppose that and are positive integers such that Let be indeterminates, let be a primitive root of unity and specialize the variables according to the relation where the subscripts on the are taken mod The “Hecke algebra” corresponding to the group is the subalgebra of generated by the elements Upon specializing where is a primitive root of unity, becomes the group algebra Thus is a of the group algebra of the group
The algebras were first constructed by Ariki and Koike [AKo1994], and they were classified as cyclotomic Hecke algebras of type Bn by Broué and Malle [BMa1993] and the representation theory of was studied by Ariki [Ari1995]. See [HRa1998] for information about the characters of these algebras.
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..