Geodesic
The Chern classes modulo $p$ of a regular representation
Documenta mathematica, Tome 4 (1999), pp. 167-178
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Let G be a finite group and ρ a complex linear representation of G. In 1961, Atiyah and Venkov independently defined Chern classes ci​(ρ) with values in the integral or mod p cohomology of G. We consider here the mod p Chern classes of the regular representation rG​ of G. Venkov claimed that ci​(rG​)=0 for i−pn−1, where pn 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 ci​(rG​) are p-powers of cohomology classes invariant by Aut(G) as soon as G is a non-abelian p-group.
DOI : 10.4171/dm/57
Classification : 20C15, 20J06
Mots-clés : finite groups, Chern classes, regular representation 
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     author = {Bruno Kahn},
     title = {The {Chern} classes modulo $p$ of a regular representation},
     journal = {Documenta mathematica},
     pages = {167--178},
     year = {1999},
     volume = {4},
     doi = {10.4171/dm/57},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/57/}
}
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Bruno Kahn. The Chern classes modulo $p$ of a regular representation. Documenta mathematica, Tome 4 (1999), pp. 167-178. doi: 10.4171/dm/57

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