A combinatorial approach to first degree cohomology of Specht modules

TL;DR

This paper develops combinatorial methods to compute the first degree cohomology of Specht modules for two-part partitions over fields of characteristic p≥3, providing explicit descriptions of extensions and bounds for more complex partitions.

Contribution

It introduces a purely combinatorial approach to cohomology of Specht modules and extends results to partitions with more than two parts, including a new proof of existing bounds.

Findings

Explicit combinatorial formulas for first cohomology

Description of all non-split extensions with trivial modules

A combinatorial proof of cohomology dimension bounds

Abstract

Using purely combinatorial methods we calculate the first degree cohomology of Specht modules indexed by two part partitions over fields of characteristic $p\ge 3$. These combinatorial methods also allow us to obtain an explicit description of all of the non-split extensions of the Specht module, $S^\lambda$, by the trivial module. Applying this work to partitions with more than two parts we are able to give an entirely combinatorial proof of the bound on the dimension of the first degree cohomology given by work of Donkin and Geranios. We also obtain as a corollary a result of Weber giving a far reaching condition determining partitions for which the first cohomology of the Specht module is trivial.