## Abstract

Let G⊂GL(V) be a finite p-group, where p=charF and dim_{F}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 dim_{F}V=3. Then S(V)^{G} is a polynomial ring if and only if G is generated by transvections. Theorem B Suppose dim_{F}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 dim_{F}U=2, where G_{U}={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 language | English |
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Article number | 108120 |

Journal | Advances in Mathematics |

Volume | 397 |

DOIs | |

State | Published - 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)