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 $\det \pi =1$.