Representations of Ariki–Koike Algebras and Gröbner–Shirshov Bases

Abstract

In this paper, we investigate the structure of Ariki–Koike algebras and their Specht modules using Gröbner–Shirshov basis theory and combinatorics of Young tableaux. For a multipartition λ, we find a presentation of the Specht module Sλ given by generators and relations, and determine its Gröbner–Shirshov pair. As a consequence, we obtain a linear basis of Sλ consisting of standard monomials with respect to the Gröbner–Shirshov pair. We show that this monomial basis can be canonically identified with the set of cozy tableaux of shape λ. 2000 Mathematics Subject Classification 16Gxx, 05Exx.