Cores and weights of multipartitions and blocks of Ariki-Koike algebras

TL;DR

This paper introduces a new way to define cores and weights of multipartitions related to Ariki-Koike algebras, leading to a classification of blocks and generalizations of known algebraic concepts.

Contribution

It defines $e$-core and $e$-weight for multipartitions and blocks of Ariki-Koike algebras without restrictions on multicharge, extending existing theories and classifications.

Findings

Defined $e$-core and $e$-weight for multipartitions and blocks.

Established an analogue of Nakayama's Conjecture for Ariki-Koike algebras.

Generalized $[w:k]$-pair concept and provided conditions for Scopes equivalence.

Abstract

Let $e$ be an integer at least two. We define the $e$-core and the $e$-weight of a multipartition associated with a multicharge as the $e$-core and the $e$-weight of its image under the Uglov map. We do not place any restriction on the multicharge for these definitions. We show how these definitions lead to the definition of the $e$-core and the $e$-weight of a block of an Ariki-Koike algebra with quantum parameter $e$, and an analogue of Nakayama's `Conjecture' that classifies these blocks. Our definition of $e$-weight of such a block coincides with that first defined by Fayers. We further generalise the notion of a $[w:k]$-pair for Iwahori-Hecke algebra of type $A$ to the Ariki-Koike algebras, and obtain a sufficient condition for such a pair to be Scopes equivalent.