A remark on a conjecture of Derksen

TL;DR

This paper discusses a bound on the degrees of pth syzygies in invariant rings of complex representations of finite groups, improving previous conjectured bounds using twisted commutative algebra techniques.

Contribution

It provides a new, simpler bound of pg^3 on the degrees of syzygies, refining Derksen's conjecture through algebraic methods.

Findings

Established a bound of pg^3 for syzygy degrees

Applied twisted commutative algebra to invariant theory

Simplified the approach to bounding syzygy degrees

Abstract

Let V be a complex representation of a finite group G of order g. Derksen conjectured that the pth syzygies of the invariant ring Sym(V)^G are generated in degrees at most (p+1)g. We point out that a simple application of the theory of twisted commutative algebras -- using an idea due to Weyl -- gives the bound pg^3.