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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Smith equivalence of representations
for finite perfect groups


Authors: Erkki Laitinen and Krzysztof Pawalowski
Journal: Proc. Amer. Math. Soc. 127 (1999), 297-307
MSC (1991): Primary 57S17, 57S25
MathSciNet review: 1468195
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Abstract: Using smooth one-fixed-point actions on spheres and a result due to Bob Oliver on the tangent representations at fixed points for smooth group actions on disks, we obtain a similar result for perfect group actions on spheres. For a finite group $G$, we compute a certain subgroup $IO'(G)$ of the representation ring $RO(G)$. This allows us to prove that a finite perfect group $G$ has a smooth $2$-proper action on a sphere with isolated fixed points at which the tangent representations of $G$ are mutually nonisomorphic if and only if $G$ contains two or more real conjugacy classes of elements not of prime power order. Moreover, by reducing group theoretical computations to number theory, for an integer $n \ge 1$ and primes $p, q$, we prove similar results for the group $G = A_{n}$, $\operatorname{SL} _{2}(\mathbb{F} _{p})$, or ${\operatorname{PSL}} _{2}(\mathbb{F} _{q})$. In particular, $G$ has Smith equivalent representations that are not isomorphic if and only if $n \ge 8$, $p \ge 5$, $q \ge 19$.


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Additional Information

Erkki Laitinen
Affiliation: Faculty of Mathematics and Computer Science, Adam Mickiewicz University of Poznań, ul. Jana Matejki 48/49, PL–60–769 Poznań, Poland
Email: kpa@math.amu.edu.pl

Krzysztof Pawalowski
Affiliation: Faculty of Mathematics and Computer Science, Adam Mickiewicz University of Poznań, ul. Jana Matejki 48/49, PL–60–769 Poznań, Poland

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04544-X
PII: S 0002-9939(99)04544-X
Keywords: Finite perfect group, action on sphere, Smith equivalence of representations
Received by editor(s): August 30, 1996
Received by editor(s) in revised form: May 10, 1997
Communicated by: Thomas Goodwillie
Article copyright: © Copyright 1999 American Mathematical Society