SL2-modules of small homological dimension

TL;DR

This paper extends previous classifications of the homological dimension of invariant algebras of binary forms under SL2, providing new bounds and explicit generators for cases with small homological dimension.

Contribution

It generalizes Popov's classification by determining cases with larger homological dimensions and explicitly constructing generators and parameters for these algebras.

Findings

Classified cases with hdR <= 100 for p=1

Classified cases with hdR <= 15 for p>1

Provided explicit generators and homogeneous parameters

Abstract

Let Vn be the SL2-module of binary forms of degree n and let V = Vn1+...+Vnp . We consider the algebra R of polynomial functions on V invariant under the action of SL2. The measure of the intricacy of these algebras is the length of their chains of syzygies, called homological dimension hdR. Popov gave in 1983 a classification of the cases in which hdR <=10 for a single binary form (p = 1) or hdR <=3 for a system of two or more binary forms (p > 1). We extend Popov's result and determine for p = 1 the cases with hdR <= 100, and for p > 1 those with hdR <= 15. In these cases we give a set of homogeneous parameters and a set of generators for the algebra R.