Kleshchev multipartitions and extended Young diagrams

TL;DR

This paper introduces a new simple characterization of Kleshchev and Uglov multipartitions, which are crucial in representation theory, leading to proofs of conjectures and generalizations of algorithms.

Contribution

It provides a novel, simplified description of multipartitions and extends existing algorithms and conjectures in the field.

Findings

New characterization of Kleshchev multipartitions

Proof of a generalized conjecture by Dipper, James, and Murphy

Extension of the LLT algorithm to arbitrary level

Abstract

We give a new simple characterization of the set of Kleshchev multipartitions, and more generally of the set of Uglov multipartitions. These combinatorial objects play an important role in various areas of representation theory of quantum groups, Hecke algebras or finite reductive groups. As a consequence, we obtain a proof of a generalization of a conjecture by Dipper, James and Murphy and a generalization of the LLT algorithm for arbitrary level.