Kronecker positivity and 2-modular representation theory

TL;DR

This paper proves semisimplicity of certain Specht modules in 2-modular representation theory and applies these results to verify Saxl's conjecture for a broad class of partitions, advancing understanding of Kronecker coefficients.

Contribution

It introduces new semisimplicity results for Specht modules labeled by 2-separated partitions and applies these to confirm Saxl's conjecture in new cases.

Findings

Specht modules labeled by 2-separated partitions are semisimple

Complete decomposition of these modules into graded simple modules

Verification of Saxl's conjecture for a large class of partitions

Abstract

This paper consists of two prongs. Firstly, we prove that any Specht module labelled by a 2-separated partition is semisimple and we completely determine its decomposition as a direct sum of graded simple modules. Secondly, we apply these results and other modular representation theoretic techniques on the study of Kronecker coefficients and hence verify Saxl's conjecture for a large new class of partitions.