Stuttering blocks of Ariki-Koike algebras

TL;DR

This paper investigates the combinatorial structure of multipartitions under a shift action, reducing the problem to a convex optimization and applying it to representation theory of complex reflection groups.

Contribution

It introduces a new approach to analyze orbit sizes of multipartitions using abaci and convex optimization, with applications to Hecke algebra representations.

Findings

Minimal orbit size depends only on the multiset orbit size.

Reduced the problem to a convex integer optimization problem.

Provided applications to the representation theory of complex reflection groups.

Abstract

We study a shift action defined on multipartitions and on residue multisets of their Young diagrams. We prove that the minimal orbit cardinality among all multipartitions associated to a given multiset depends only on the orbit cardinality of the multiset. Using abaci, this problem reduces to a convex optimisation problem over the integers with linear constraints. We solve it by proving an existence theorem for binary matrices with prescribed row, column and block sums. Finally, we give some applications to the representation theory of the Hecke algebra of the complex reflection group $G(r,p,n)$.