Stiefel-Whitney Classes for Finite Special Linear Groups of Even Rank

TL;DR

This paper computes the total Stiefel-Whitney classes for orthogonal representations of special linear groups of odd dimension over finite fields, expressing them via character values and providing explicit calculations for specific classes.

Contribution

It introduces explicit formulas for Stiefel-Whitney classes of $ ext{SL}(n,q)$ with odd $n$ and $q$, linking them to character evaluations at elements of order 2.

Findings

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

Calculation of the 4th and 8th Stiefel-Whitney classes for these groups.

Several consequences and applications derived from the computed classes.

Abstract

We compute the total Stiefel-Whitney Classes (SWCs) for orthogonal representations of special linear groups $\text{SL}(n,q)$ when $n$ and $q$ are odd. These classes are expressed in terms of character values at diagonal elements of order $2$. We give several consequences, and work out the $4$th SWC explicitly, and the $8$th SWC when the $4$th vanishes.