Vector invariants of permutation groups in characteristic zero

2022/11/20 by Fabian Reimers, Reimers, Fabian, Müfi̇t Sezer +1
Computer Science · Mathematics · #(13P10 #13A50 #20B99) #Advanced Differential Equations and Dynamical Systems #Coding theory and cryptography #Commutative Algebra (math.AC) #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2211.11091

openalex publication_date 2022/11/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a finite permutation group acting naturally on a vector space V over a field \Bbbk. A well known theorem of Göbel asserts that the corresponding ring of invariants \Bbbk[V]^G is generated by invariants of degree at most \binomdim V2. In this note we show that if the characteristic of \Bbbk is zero then the top degree of vector coinvariants \Bbbk[V^m]_G is also bounded above by \binomdim V2, which implies the degree bound \binomdim V2+ 1 for the ring of vector invariants \Bbbk[V^m]^G. So Göbel's bound almost holds for vector invariants in characteristic zero as well.

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