The Chern classes modulo $p$ of a regular representation

Abstract

Let $G$ be a finite group and $\rho$ a complex linear representation of $G$. In 1961, Atiyah and Venkov independently defined Chern classes $c_i(\rho )$ with values in the integral or mod $p$ cohomology of $G$. We consider here the mod $p$ Chern classes of the regular representation $r_G$ of $G$. Venkov claimed that $c_i(r_G)=0$ for $i<p^n-p^{n-1}$, where $p^n$ is the highest power of $p$ dividing $|G|$; however his proof is only valid for $G$ elementary abelian. In this note, we show Venkov's assertion is valid for any $G$. The proof also shows that the $c_i(r_G)$ are $p$-powers of cohomology classes invariant by $Aut(G)$ as soon as $G$ is a non-abelian $p$-group.