The Benson -- Symonds Invariant for Ordinary and Signed Permutation Modules

TL;DR

This paper computes the Benson--Symonds invariant for all signed permutation modules of the symmetric group, advancing understanding of how these modules relate to projectivity using representation theory and combinatorics.

Contribution

It provides a complete determination of the Benson--Symonds invariant for signed permutation modules, a significant generalization in the study of symmetric group modules.

Findings

Invariant computed explicitly for all signed permutation modules

Results connect the invariant to combinatorial properties of modules

Advances understanding of module projectivity in symmetric groups

Abstract

The signed permutation modules are a simultaneous generalization of the ordinary permutation modules and the twisted permutation modules of the symmetric group. In a recent paper Dave Benson and Peter Symonds defined a new invariant $\gamma_G(M)$ for a finite dimensional module $M$ of a finite group $G$ which attempts to quantify how close a module is to being projective. In this paper, we determine this invariant for all the signed permutation modules of the symmetric group using tools from representation theory and combinatorics.