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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: Leonard Evens and Daniel S. Kahn
Journal: Trans. Amer. Math. Soc. 245 (1978), 331-347
MSC: Primary 55R40; Secondary 20C99
MathSciNet review: 511413
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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,

$\displaystyle \bar N (k)( {{s_k}(\rho ) - {\text{T}}{{\text{r}}_{H \to G}}({s_k}(\xi ))}) = 0,$ (1)

where

$\displaystyle \bar N(k) = {\prod _{\begin{array}{*{20}{c}} {p\vert N(k)} \\ {p{\text{prime}}} \\ \end{array}}}p$ (2)

and

$\displaystyle N(k) = \left( {\begin{array}{*{20}{c}} {\prod\limits_{p{\text{prime}}} {{p^{[k/p - 1]}}}} \end{array} } \right)/k!.$ (3)

(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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1978-0511413-4
PII: S 0002-9947(1978)0511413-4
Keywords: Characteristic class, Chern class, Knopfmacher, Riemann-Roch formula, induced representation, group, transfer, wreath product
Article copyright: © Copyright 1978 American Mathematical Society