Incidence Modules for Symplectic Spaces in Characteristic Two

TL;DR

This paper investigates the action of finite symplectic groups in characteristic two on certain subspaces, deriving a formula for the 2-rank of an incidence matrix related to isotropic subspaces.

Contribution

It provides a general formula for the 2-rank of incidence matrices involving isotropic subspaces in symplectic spaces over characteristic two fields.

Findings

Derived a formula for the 2-rank of the incidence matrix.

Analyzed the permutation action of symplectic groups on subspaces.

Contributed to understanding the combinatorial structure of symplectic spaces.

Abstract

We study the permutation action of a finite symplectic group of characteristic 2 on the set of subspaces of its standard module which are either totally isotropic or else complementary to totally isotropic subspaces with respect to the alternating form. A general formula is obtained for the 2-rank of the incidence matrix for the inclusion of one-dimensional subspaces in the distinguished subspaces of a fixed dimension.