Let $G$ be a finite group. Associated to a complex orthogonal representation $\pi$ of $G$ is a sequence of cohomological invariants $w_i(\pi)$, called the Stiefel-Whitney Classes (SWCs) of $\pi$, which live in the group cohomology $H^*(G, Z/2Z)$.

 

There are not many explicit calculations in the literature on these characteristic classes for non-abelian groups $G$. In a series of joint works with Prof. Steven Spallone, we have computed SWCs for several finite groups of Lie type in terms of character values at diagonal involutions. These calculations can answer some interesting questions, such as: What is the subalgebra of $H^*(G,Z/2Z)$ generated by SWCs of all orthogonal $\pi$? Is it the whole group cohomology ring? This talk will give an overview of some of our results for $G=Sp(2n,q)$ when $q$ is odd. Also, as a consequence, we obtain a restriction result that allows for a “universal” calculation of $w_k$ in terms of character values, valid for all $n$ sufficiently large.