p-Groups and the polynomial ring of invariants question

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Abstract

Let G⊂GL(V) be a finite p-group, where p=charF and dimF⁡V is finite. Let S(V) be the symmetric algebra of V, S(V)G the subring of G-invariants, and V the dual space of V. We determine when S(V)G is a polynomial ring. Theorem A Suppose dimF⁡V=3. Then S(V)G is a polynomial ring if and only if G is generated by transvections. Theorem B Suppose dimF⁡V⩾4. Then S(V)G is a polynomial ring if and only if: (1) S(V)GU is a polynomial ring for each subspace U⊂V with dimF⁡U=2, where GU={g∈G|g(u)=u, ∀u∈U}, and (2) S(V)G is Cohen-Macaulay. Alternatively, (1) can be replaced by the equivalent condition: (3) dim⁡(non-smooth locus ofS(V)G)⩽1.

Original languageEnglish
Article number108120
JournalAdvances in Mathematics
Volume397
DOIs
StatePublished - 5 Mar 2022

Bibliographical note

Funding Information:
I thank Gregor Kemper for discussions regarding [11] and Lemma 2.4.

Publisher Copyright:
© 2021 Elsevier Inc.

Keywords

  • Invariant theory
  • p-groups
  • Polynomial rings

ASJC Scopus subject areas

  • Mathematics (all)

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