# The Chern classes modulo $p$ of a regular representation

### Bruno Kahn

Institut de Mathematiques de Jussieu Universite Paris 7 Case 7012 75251 Paris Cedex 05 France

## Abstract

Let $G$ be a finite group and $ρ$ a complex linear representation of $G$. In 1961, Atiyah and Venkov independently defined Chern classes $c_{i}(ρ)$ 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.

## Cite this article

Bruno Kahn, The Chern classes modulo $p$ of a regular representation. Doc. Math. 4 (1999), pp. 167–178

DOI 10.4171/DM/57