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Lannes T-functor and invariants of pointwise stabilizers

Lannes T-functor and invariants of pointwise stabilizers Abstract. Let r X G D3 GLnY F be a representation of a ®nite group G over the ®eld F. The group G acts on the algebra of polynomial functions FV on V via r and the subalgebra of polynomials invariant under this action is denoted by FV G . If U t V F n is a linear subspace then the pointwise stabilizer of U is denoted by GU fg e G j gu u iu e Ug. In this note we examine the relation between FV G and FV GU when F Fq is a Galois ®eld with q elements using the T-functor introduced by J. Lannes [13]. We show that a wide variety of properties of FV G are inherited by FV GU . For example, among other things: (1) we reprove a result of R. Steinberg [26] and H. Nakajima [17] that FV GU is a polynomial algebra when FV G is; (2) we show that the Cohen-Macaulay property is inherited by FV GU from FV G ; (3) and when FV G is a complete intersection, then so is FV GU . We apply the T-functor to study degree bounds for generators of http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Lannes T-functor and invariants of pointwise stabilizers

Smith, Larry
Forum Mathematicum , Volume 12 (4) – May 29, 2000
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References (19)

Publisher
de Gruyter
Copyright
Copyright © 2000 by Walter de Gruyter GmbH & Co. KG
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.2000.015
Publisher site
See Article on Publisher Site

Abstract

Abstract. Let r X G D3 GLnY F be a representation of a ®nite group G over the ®eld F. The group G acts on the algebra of polynomial functions FV on V via r and the subalgebra of polynomials invariant under this action is denoted by FV G . If U t V F n is a linear subspace then the pointwise stabilizer of U is denoted by GU fg e G j gu u iu e Ug. In this note we examine the relation between FV G and FV GU when F Fq is a Galois ®eld with q elements using the T-functor introduced by J. Lannes [13]. We show that a wide variety of properties of FV G are inherited by FV GU . For example, among other things: (1) we reprove a result of R. Steinberg [26] and H. Nakajima [17] that FV GU is a polynomial algebra when FV G is; (2) we show that the Cohen-Macaulay property is inherited by FV GU from FV G ; (3) and when FV G is a complete intersection, then so is FV GU . We apply the T-functor to study degree bounds for generators of

Journal

Forum Mathematicum – de Gruyter

Published: May 29, 2000

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