Let $G$ be a finite group. Under the identification of the representation ring $R_{\mathbb{C}}(G)$ with the equivariant K-theory $KU^0_G(\ast)$ of the point, followed by Atiyah-Segal completion-projection to the K-theory $KU^0(B G)$ of the classifying space

$$ \widehat{(-)} \;:\; R_{\mathbb{C}}(G) \simeq KU_G^0(\ast) \longrightarrow KU^0(B G) $$

every linear representation $V \in R_{\mathbb{C}}(G)$ induces a K-theory class $\widehat V \in KU^0(B G)$.

The former has a character $\chi_{V}$, the latter has a Chern character $\mathrm{ch}(\widehat V)$, or rather a sequence $c_n(\widehat V)$ of Chern classes.

Is there an algebraic expression of the Chern classes $\mathrm{c}_n(\widehat V)$ in terms of the character $\chi_V$?

Atiyah discussed this question way back in

  • Michael Atiyah, "Characters and cohomology of finite groups", Publications Mathématiques de l’IHÉS, Volume 9 (1961) , p. 23-64 (numdam)

in the appendix. There he gave an explicit formula for $c_1(\widehat{V})$ in terms of $\chi_V$, but about the other Chern classes, he wrote:

It would be highly desirable to have a direct algebraic definition of them, but [...] this is still unsolved.

Is this still the case?

  • 2
    $\begingroup$ Doesn't Symonds' splitting principle for group representations solve the problem? If not, what would be missing? $\endgroup$ Jul 8, 2019 at 18:51


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