On the one dimensional representations of Ariki–Koike algebras at roots of unity

Abstract

We study the natural labeling of the one dimensional representations for Ariki–Koike algebras at roots of unity. For Hecke algebras of types A and B, some of these representations can be identified with the socle of the Steinberg representation of a finite reductive group. We here give closed formulas for them. This uses, in particular, several results concerning crystal isomorphisms and the Mullineux involution.