On syzygies for rings of invariants of abelian groups

TL;DR

This paper explores the relationship between zero-sum sequences and rings of invariants of abelian groups, providing new bounds on presentations and syzygies by reversing traditional transfer directions.

Contribution

It introduces a method to derive presentations of invariant rings from multiplicity free cases, linking zero-sum theory with invariant theory in a novel way.

Findings

Provides bounds on the presentation of block monoids

Establishes a connection between syzygies and binomial ideals

Uses catenary degree to analyze monoid congruences

Abstract

It is well known that results on zero-sum sequences over a finitely generated abelian group can be translated to statements on generators of rings of invariants of the dual group. Here the direction of the transfer of information between zero-sum theory and invariant theory is reversed. First it is shown how a presentation by generators and relations of the ring of invariants of an abelian group acting linearly on a finite dimensional vector space can be obtained from a presentation of the ring of invariants for the corresponding multiplicity free representation. This combined with a known degree bound for syzygies of rings of invariants, yields bounds on the presentation of a block monoid associated to a finite sequence of elements in an abelian group. The results have an equivalent formulation in terms of binomial ideals, but here the language of monoid congruences and the notion of catenary degree is used.