Degree bounds for invariant skew polynomials

TL;DR

This paper extends a classical degree bound for invariant polynomials under finite group actions from polynomial rings to the exterior algebra, using representation theory and subspace arrangement techniques.

Contribution

It establishes that the degree bound for invariant skew polynomials in the exterior algebra matches the group order, adapting Derksen's approach to the exterior algebra setting.

Findings

The maximal degree of minimal invariant skew polynomials is bounded by the group order.

The proof uses a functor translating resolutions from polynomial to exterior algebra.

A regularity bound on intersections of linear ideals in the exterior algebra is established.

Abstract

When we consider the action of a finite group on a polynomial ring, a polynomial unchanged by the action is called an invariant polynomial. A famous result of Noether states that in characteristic zero the maximal degree of a minimal invariant polynomial is bounded above by the order of the group. Our work establishes that the same bound holds for invariant skew polynomials in the exterior algebra. Our approach to the problem relies on a theorem of Derksen that connects invariant theory to the study of ideals of subspace arrangements. We adapt his proof over the polynomial ring to the exterior algebra, reducing the question to establishing a bound on the Castelnuovo-Mumford regularity of intersections of linear ideals in the exterior algebra. We prove the required regularity bound using tools from representation theory. In particular, the proof relies on the existence of a functor on the category of polynomial functors that translates resolutions of ideals of subspace arrangements over the polynomial ring to resolutions of ideals of subspace arrangements over the exterior algebra.