Vector invariants of permutation groups in characteristic zero

TL;DR

This paper extends G"obel's degree bound for invariants of permutation groups to the setting of vector coinvariants in characteristic zero, showing the bounds are nearly optimal.

Contribution

It demonstrates that the degree bounds for invariants extend to vector coinvariants in characteristic zero, nearly matching G"obel's original bounds.

Findings

Top degree of vector coinvariants is bounded by inom{ ext{dim} V}{2}

Degree bound for vector invariants is inom{ ext{dim} V}{2} + 1

G"obel's bound nearly applies in characteristic zero

Abstract

We consider a finite permutation group acting naturally on a vector space $V$ over a field $\Bbbk$. A well known theorem of G\"obel asserts that the corresponding ring of invariants $\Bbbk[V]^G$ is generated by invariants of degree at most $\binom{\dim V}{2}$. 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 $\binom{\dim V}{2}$, which implies the degree bound $\binom{\dim V}{2}+ 1$ for the ring of vector invariants $\Bbbk[V^m]^G$. So G\"obel's bound almost holds for vector invariants in characteristic zero as well.