The Benson-Symonds Invariant for Permutation Modules

TL;DR

This paper calculates the Benson-Symonds invariant for permutation modules of the symmetric group associated with two-part partitions, using advanced representation theory and combinatorics.

Contribution

It provides the first explicit computation of the Benson-Symonds invariant for a significant class of modules in symmetric group representation theory.

Findings

Explicit formulas for the invariant for two-part partition permutation modules

Enhanced understanding of module projectivity proximity measures

New combinatorial methods applied to invariant calculation

Abstract

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 permutation modules of the symmetric group corresponding to two-part partitions using tools from representation theory and combinatorics.