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

TL;DR

This paper provides explicit formulas for one-dimensional representations of Ariki-Koike algebras at roots of unity, linking them to representations of finite reductive groups and utilizing crystal isomorphisms and Mullineux involution.

Contribution

It introduces closed formulas for these representations, enhancing understanding of their structure and connections to other algebraic objects.

Findings

Explicit formulas for one-dimensional representations at roots of unity

Identification with socle of Steinberg representations in types A and B

Use of crystal isomorphisms and Mullineux involution in derivations

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.