Stiefel-Whitney Classes of Representations of $\text{SL}(2,q)$

TL;DR

This paper computes the Stiefel-Whitney classes of orthogonal representations of SL(2,q), linking them to character values, and explores their algebraic structure and Euler classes.

Contribution

It provides explicit descriptions of SWCs for representations of SL(2,q) and analyzes their algebraic and topological properties.

Findings

Determined the subalgebra generated by SWCs in cohomology.

Identified which representations have nontrivial mod 2 Euler class.

Connected character values to topological invariants.

Abstract

We describe the Stiefel-Whitney classes (SWCs) of orthogonal representations $\pi$ of the finite special linear groups $G=\text{SL}(2,\mathbb F_q)$, in terms of character values of $\pi$. From this calculation, we can answer interesting questions about SWCs of $\pi$. For instance, we determine the subalgebra of $H^*(G,\mathbb Z/2\mathbb Z)$ generated by the SWCs of orthogonal $\pi$, and we also determine which $\pi$ have nontrivial mod $2$ Euler class.