Stiefel Whitney Classes for Real representations of $\mathrm{GL}_2(\mathbb{F}_q)$

Abstract

We compute the total Stiefel Whitney class for a real representation $\pi$ of $\mathrm{GL}_2(\mathbb{F}_q)$, where $q$ is odd. The obstruction class of $\pi$ is defined to be the Stiefel Whitney class of lowest positive degree that does not vanish. We provide an expression for the obstruction class of $\pi$ in terms of its character values if $\det\pi=1$.