The structure of rank 3 permutation modules for O_{2n}(2) and U_m(2) acting on nonsingular points

TL;DR

This paper investigates the detailed structure of permutation modules in odd characteristic for rank 3 actions of orthogonal and unitary groups over finite fields, focusing on nonsingular points.

Contribution

It provides a comprehensive analysis of the module structure for specific rank 3 actions of orthogonal and unitary groups over finite fields in odd characteristic.

Findings

Detailed module decompositions for $O_{2n}^{ ext{±}}(2)$ and $U_m(2)$

Identification of irreducible constituents in permutation modules

Insights into the representation theory of classical groups over finite fields

Abstract

We study the odd-characteristic structure of permutation modules for the rank 3 natural actions of $O_{2n}^{\pm}(2)$ ($n\geq3$) and $U_{m}(2)$ ($m\geq4$) on nonsingular points of their standard modules.