The 2-modular permutation modules on fixed point free involutions of symmetric groups

TL;DR

This paper analyzes the structure of permutation modules of symmetric groups acting on fixed point free involutions in even characteristic, identifying components, vertices, Brauer quotients, and characters.

Contribution

It provides a detailed enumeration and structural analysis of permutation modules related to fixed point free involutions in symmetric groups over even characteristic fields.

Findings

Components of permutation modules are explicitly enumerated.

Vertices and Brauer quotients for each component are determined.

Associated ordinary characters are identified.

Abstract

We enumerate over even characteristic the components of the permutation module of the symmetric group of even degree acting on the set of its fixed point free involutions. We find the vertex and Brauer quotient for each component, and the ordinary character associated with each component.