An integral Riemann-Roch formula for induced representations of finite groups

Evens, Leonard; Kahn, Daniel S.

doi:10.1090/S0002-9947-1978-0511413-4

An integral Riemann-Roch formula for induced representations of finite groups
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by Leonard Evens and Daniel S. Kahn PDF

Trans. Amer. Math. Soc. 245 (1978), 331-347 Request permission

Abstract:

Let H be a subgroup of the finite group G, $\xi$ a finite dimensional complex representation of H and $\rho$ the induced representation of G. If ${s_k}(\rho ) \in {H^{2k}}(G,\textbf {Z})$, $k \geqslant 1$ denote the characteristic classes bearing the same relation to power sums that Chern classes bear to elementary symmetric functions, then we prove the following, \begin{equation} \bar N (k)( {{s_k}(\rho ) - {\text {T}}{{\text {r}}_{H \to G}}({s_k}(\xi ))}) = 0, \end{equation} where \begin{equation} \bar N(k) = {\prod _{\begin {array}{*{20}{c}} {p|N(k)} \\ {p{\text {prime}}} \\ \end{array}}}p \end{equation} and \begin{equation} N(k) = \left ( {\begin {array}{*{20}{c}} {\prod \limits _{p{\text {prime}}} {{p^{[k/p - 1]}}}} \end{array} } \right )/k!. \end{equation} (Tr denotes transfer.) Moreover, $\bar N (k)$ is the least integer with this property. This settles a question originally raised in a paper of Knopfmacher in which it was conjectured that the required bound was N(k).

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