Depth and detection in modular invariant theory

Elmer, Jonathan ORCID logoORCID: (2009) Depth and detection in modular invariant theory. Journal of Algebra, 322 (5) . pp. 1653-1666. ISSN 0021-8693 [Article] (doi:10.1016/j.jalgebra.2009.04.036)

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Let G be a finite group acting linearly on a vector space V over a field of characteristic p dividing the group order, and let R denote S(V∗). We study the R^G modules H^i(G, R), for i ≥ 0 with R^G itself as a special case. There are lower bounds for depth of (H^i(G, R)) and for depth(R^G). We show that a certain sufficient condition for their attainment (due to Fleischmann, Kemper and Shank) may be modified to give a condition which is both necessary and sufficient. We apply our main result to classify the representations of the Klein four-group for which depth(R^G) attains its lower bound. We also use our new condition to show that the if G = P × Q, with P a p-group and Q an abelian p'-group, then the depth of R G attains its lower bound if and only if the depth of R^P does so.

Item Type: Article
Additional Information: Available online 12 May 2009
Research Areas: A. > School of Science and Technology > Design Engineering and Mathematics
Item ID: 19270
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Depositing User: Jonathan Elmer
Date Deposited: 14 Apr 2016 10:56
Last Modified: 10 Jun 2022 22:20

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