On the classification of simple modules for cyclotomic Hecke algebras of type \(G(m,1,n)\) and Kleshchev multipartitions (Q5956130)

This paper studies the modular representation theory of the cyclotomic Hecke algebras of type \(G(m,1,n)\), also known as Ariki-Koike algebras. These algebras were proved to be cellular by \textit{J. J. Graham} and \textit{G. I. Lehrer} [Invent. Math. 123, No. 1, 1-34 (1996; Zbl 0853.20029)], which means that they come equipped with a set of cell modules (called ``Specht modules'' in the paper under review). The theory of cellular algebras shows that each Specht module \(S^{\underline\lambda}\) has a natural bilinear form with radical \(\text{rad }S^{\underline\lambda}\), and the nonzero quotients \(D^{\underline\lambda}=S^{\underline\lambda}/\text{rad }S^{\underline\lambda}\) form a complete and irredundantly described set of simple modules over a field. Mathas conjectured, in previous joint work with the author, that the quotient modules \(D^{\underline\lambda}\) should be nonzero if and only if \(\underline\lambda\) is a Kleshchev multipartition.NEWLINENEWLINENEWLINEThe main result of this paper is a proof of Mathas' conjecture. The technique used to achieve this is to transform the conjecture into a problem about canonical bases in Fock spaces, and then to use methods from quantum groups.