Signed Young Modules and Simple Specht Modules

TL;DR

This paper establishes the labels of simple Specht modules as signed Young modules and explores their properties, including tensoring effects, Green vertices, and cohomological aspects, advancing the understanding of symmetric group representations.

Contribution

It determines the signed Young module labels of simple Specht modules and analyzes properties of indecomposable signed Young modules, including tensoring with sign representation.

Findings

Identified labels of simple Specht modules as signed Young modules.

Determined the effect of tensoring with the sign representation on indecomposable signed Young modules.

Computed Green vertices, Green correspondents, cohomological varieties, and complexities for simple Specht modules.

Abstract

By a result of Hemmer, every simple Specht module of a finite symmetric group over a field of odd characteristic is a signed Young module. While Specht modules are parametrized by partitions, indecomposable signed Young modules are parametrized by certain pairs of partitions. The main result of this article establishes the signed Young module labels of simple Specht modules. Along the way we prove a number of results concerning indecomposable signed Young modules that are of independent interest. In particular, we determine the label of the indecomposable signed Young module obtained by tensoring a given indecomposable signed Young module with the sign representation. As consequences, we obtain the Green vertices, Green correspondents, cohomological varieties, and complexities of all simple Specht modules and a class of simple modules of symmetric groups, and extend the results of Gill on periodic Young modules to periodic indecomposable signed Young modules.