Total Stiefel Whitney classes for real representations of ${\mathrm{GL}}_n$ over ${\mathbb{F}}_q,\mathbb{R}$ and $\mathbb{C}$

Abstract

We compute the total Stiefel Whitney class for a real representation $\pi$ of ${\mathrm{GL}}_n({\mathbb{F}}_q)$, where $q$ is odd in terms of character values of $\pi$ on order $2$ diagonal elements. We also compute the total Stiefel Whitney class of real representations of ${\mathrm{GL}}_n(\mathbb{R})$ and ${\mathrm{GL}}_n(\mathbb{C})$.