Resolutions of 2 and 3 dimensional rings of invariants for cyclic groups

TL;DR

This paper characterizes the structure of invariant rings for cyclic groups acting on 2D and 3D representations, providing explicit generators, relations, and Betti numbers, with a focus on quadratic relations and combinatorial simplicity.

Contribution

It offers a detailed description of minimal generators, relations, and free resolutions for invariant rings of cyclic groups in 2D and 3D, including Gr"obner bases and Betti numbers.

Findings

Minimal generators and quadratic relations identified.

Gr"obner basis with simple combinatorial structure established.

Explicit graded Betti numbers for free resolutions provided.

Abstract

Let $G$ be the cyclic group of order $n$ and suppose ${\bf F}$ is a field containing a primitive $n^\text{th}$ root of unity. We consider the ring of invariants ${\bf F}[W]^G$ of a three dimensional representation $W$ of $G$ where $G \subset \text{SL}(W)$. We describe minimal generators and relations for this ring and prove that the lead terms of the relations are quadratic. These minimal generators for the relations form a Gr\"obner basis with a surprisingly simple combinatorial structure. We describe the graded Betti numbers for a minimal free resolution of $F[W]^G$. The case where $W$ is any two dimensional representation of $G$ is also handled.