Stiefel-Whitney Classes of Representations of Some Finite Groups of Lie Type

Orthogonal representations $\pi$ of a finite group $G$ have invariants $w_i(\pi)$ living in the $i$th degree cohomology group $H^i(G,Z/2Z)$, called Stiefel-Whitney Classes (SWCs). The sum $w(\pi)=1+w_1(\pi)+w_2(\pi)+\dots$ is called the total SWC of $\pi$.

It seems there are not many explicit calculations in the literature of SWCs for non-abelian groups. We have computed the total SWCs for symplectic groups Sp$(4,q)$ and Sp$(6,q)$ with $q$ odd as well as special linear groups SL$(n,q)$ in the cases: (i) $n=2$ for any $q$, (ii) $n=3$ when $q$ is odd, and (iii)  $n$ is odd and $q\equiv 3$ (mod 4). We will give an overview of our results in this talk. 

This is joint work with Dr. Steven Spallone.