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Vector invariants of permutation groups in characteristic zero

Zentrum Mathematik — M11, Technische Universität München, Boltzmannstrasse 3, 85748 Garching bei München, Germany

E-mail Address: [email protected]

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and

Müfit sezer

https://orcid.org/0000-0002-6704-5399

Department of Mathematics, Bilkent University, Cankaya, Ankara 06800, Turkey

E-mail Address: [email protected]

Corresponding author.

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https://doi.org/10.1142/S0129167X23501112Cited by:1 (Source: Crossref)

Abstract

We consider a finite permutation group acting naturally on a vector space $V$ over a field $𝕜$ . A well-known theorem of Göbel asserts that the corresponding ring of invariants $𝕜 {[V]}^{G}$ is generated by the invariants of degree at most $(\binom{dim V}{2})$ . In this paper, we show that if the characteristic of $𝕜$ is zero, then the top degree of vector coinvariants $𝕜 {[V^{m}]}_{G}$ is also bounded above by $(\binom{dim V}{2})$ , which implies the degree bound $(\binom{dim V}{2}) + 1$ for the ring of vector invariants $𝕜 {[V^{m}]}^{G}$ . So, Göbel’s bound almost holds for vector invariants in characteristic zero as well.

Communicated by Chen-Bo Zhu

Keywords:

AMSC: 13A50

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