Author: Bruno Kahn 
Title: Les classes de Chern modulo $p$ d'une repr\'esentation r\'eguli\`ere
(French)
Date: March 15, 1998.

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.

Status: to appear in the Proceedings of the May 1997 Fes algebra and
number theory conference, Marcel Dekker.