A combinatorial approach to Specht module cohomology

TL;DR

This paper introduces a combinatorial criterion to determine the nonvanishing of the first cohomology of Specht modules over symmetric groups in odd characteristic, extending known results and enabling new computations.

Contribution

It provides a new combinatorial criterion for Specht module cohomology in degree one, generalizing James' degree zero result, and offers explicit computations and conjectures.

Findings

A combinatorial criterion for nonvanishing H^1

Explicit descriptions of cohomology modules

New conjectures related to Specht module cohomology

Abstract

For a Specht module S^\lambda for the symmetric group \Sigma_d, the cohomology H^i(\Sigma_d, S^\lambda) is known only in degree i=0. We give a combinatorial criterion equivalent to the nonvanishing of the degree i=1 cohomology, valid in odd characteristic. Our condition generalizes James' solution in degree zero. We apply this combinatorial description to give some computations of Specht module cohomology, together with an explicit description of the corresponding modules. Finally we suggest some general conjectures that might be particularly amenable to proof using this description.