Specht module cohomology and integral designs

TL;DR

This paper explores the connection between integral designs and the cohomology of Specht modules in symmetric groups, providing new insights into when certain cohomology groups are non-zero.

Contribution

It introduces a novel link between integral designs and Specht module cohomology, offering a new approach to construct elements satisfying specific cohomological criteria.

Findings

Discovered a surprising link between integral designs and symmetric group cohomology.

Constructed elements satisfying Hemmer's criterion for non-vanishing cohomology.

Enhanced understanding of the structure of Specht modules and their cohomology.

Abstract

We aim to construct an element satisfying Hemmer's combinatorial criterion for $H^1(\mathfrak{S}_n, S^\lambda)$ to be non-vanishing. In the process, we discover an unexpected and surprising link between the combinatorial theory of integral designs and the representation theory of the symmetric groups.