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 $\mathrm{GL}_n(\mathbb{C})$.