Generalised regularisation maps on partitions

TL;DR

This paper generalizes Gordon James's regularisation maps on integer partitions, providing a new algorithm for these maps in the abacus representation, advancing the understanding of modular representation theory.

Contribution

It introduces a broader class of regularisation maps on partitions and offers a formal algorithm for their computation in the abacus.

Findings

Generalisation of James's regularisation maps

Development of an explicit abacus algorithm

Enhanced understanding of partition decomposition

Abstract

In a 1976 landmark paper, Gordon James defined the regularisation maps on integer partition, yielding certain decomposition numbers for modular representations of $\mathfrak{S}_n$. We describe a generalisation of James's regularisation map and give with proof an algorithm for such maps in the abacus.