Some new decomposable Specht modules

TL;DR

This paper introduces a new family of decomposable Specht modules for the symmetric group in characteristic 2, identified by specific partitions, and provides explicit homomorphisms to demonstrate their structure.

Contribution

It presents the first new examples of decomposable Specht modules in thirty years, expanding understanding of module decomposability in characteristic 2.

Findings

New decomposable Specht modules for partitions of form (a,3,1^b)

Explicit homomorphisms constructed between Specht modules

First such examples found in three decades

Abstract

We present (with proof) a new family of decomposable Specht modules for the symmetric group in characteristic 2. These Specht modules are labelled by partitions of the form $(a,3,1^b)$, and are the first new examples found for thirty years. Our method of proof is to exhibit summands isomorphic to irreducible Specht modules, by constructing explicit homomorphisms between Specht modules.