Stiefel-Whitney Classes for Finite Symplectic Groups

TL;DR

This paper derives explicit formulas for Stiefel-Whitney classes of orthogonal representations of finite symplectic groups, linking them to character values and providing universal formulas for specific classes.

Contribution

It introduces new formulas for Stiefel-Whitney classes of finite symplectic groups and computes the subring generated by these classes for a specific case.

Findings

Formulas for total Stiefel-Whitney classes in terms of character values.

Universal formulas for the 4th and 8th Stiefel-Whitney classes.

Computation of the subring generated by SWCs for n=2.

Abstract

Let $q$ be an odd prime power, and $G=\text{Sp}(2n,q)$ the finite symplectic group. We give an expression for the total Stiefel-Whitney Classes (SWCs) for orthogonal representations $\pi$ of $G$, in terms of character values of $\pi$ at elements of order $2$. We give "universal formulas'' for the fourth and eighth SWCs. For $n=2$, we compute the subring of the mod $2$ cohomology generated by the SWCs $w_k(\pi)$.