Constructing all irreducible Specht modules in a block of the symmetric group

TL;DR

This paper develops a method to construct and count all p-irreducible Specht modules within a block of the symmetric group, based on a recent characterization of their corresponding partitions.

Contribution

It introduces a decomposition technique for partitions associated with p-irreducible modules, enabling explicit construction and enumeration within blocks.

Findings

Constructed all p-irreducible Specht modules in a block.

Provided a counting method for these modules.

Validated the characterization of p-irreducible modules.

Abstract

For any prime p, we construct, and simultaneously count, all of the complex Specht modules in a given p-block of the symmetric group which remain irreducible when reduced modulo p. We call the Specht modules with this property p-irreducible modules. Recently Fayers has proven a conjecture of James and Mathas that provides a characterization of the partitions that correspond to the p-irreducible modules. In this paper we present a method for decomposing the partitions corresponding to p-irreducible modules, and we use this decomposition to construct and count all of the partitions corresponding to p-irreducible Specht modules in a given block.