Moving vectors and core blocks of Ariki-Koike algebras

TL;DR

This paper classifies core blocks of Ariki-Koike algebras using moving vectors, establishing conditions for equivalence and linking their simple modules and decomposition numbers to classical combinatorial objects.

Contribution

It introduces a classification of core blocks via moving vectors and connects their properties to known algebraic and combinatorial structures.

Findings

Classification of core blocks by moving vectors

Necessary and sufficient conditions for Scopes equivalence

Relation of graded decomposition numbers to Iwahori-Hecke algebras

Abstract

We classify the core blocks of Ariki-Koike algebras by their moving vectors. Using this classification, we obtain a necessary and sufficient condition for Scopes equivalence between two core blocks, and express the number of simple modules lying in a core block as a classical Kostka number. Under certain conditions on the multicharge and moving vector, we further relate the graded decomposition numbers of these blocks in characteristic zero with the graded decomposition numbers of the Iwahori-Hecke algebras of type $A$.