The Benson-Symonds Invariant for Permutation Modules

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.