On the depth of invariant rings of infinite groups

TL;DR

This paper investigates the depth properties of invariant rings under infinite reductive groups, demonstrating that their Cohen-Macaulay defect can grow arbitrarily large, especially for certain representations of SL2.

Contribution

It constructs explicit faithful representations of reductive but not linearly reductive groups showing unbounded Cohen-Macaulay defect in invariant rings.

Findings

Existence of representations with arbitrarily large Cohen-Macaulay defect

Explicit construction for groups like SL2

Refinements for specific cases

Abstract

Let K be an algebraically closed field. For a finitely generated graded K algebra R, let cmdef R := dim R - depth R denote the Cohen-Macaulay-defect of R. Let G be a linear algebraic group over K that is reductive but not linearly reductive. We show that there exists a faithful rational representation V of G (which we will give explicitly) such that cmdef K[\sum_i=1^k V]^G >= k-2 for all k. We give refinements in the case G = SL2.