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Symmetric Powers of Modular Representations for Groups with a Sylow Subgroup of Prime Order

✍ Scribed by Ian Hughes; Gregor Kemper

Publisher: Elsevier Science
Year: 2001
Tongue: English
Weight: 207 KB
Volume: 241
Category: Article
ISSN: 0021-8693
DOI: 10.1006/jabr.2000.8710

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✦ Synopsis

Let V be a representation of a finite group G over a field of characteristic p. If p does not divide the group order, then Molien's formula gives the Hilbert series of the invariant ring. In this paper we find a replacement of Molien's formula which works in the case that G is divisible by p but not by p 2 . We also obtain formulas which give generating functions encoding the decompositions of all symmetric powers of V into indecomposables. Our methods can be applied to determine the depth of the invariant ring without computing any invariants. This leads to a proof of a conjecture of the second author on certain invariants of GL 2 p .

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