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arXiv · 1110.5067

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

John C. Harris·David L. Wehlau
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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 Gröbner 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.

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John C. Harris, David L. Wehlau. 2012-05-14. Resolutions of 2 and 3 dimensional rings of invariants for cyclic groups. https://arxiv.org/abs/1110.5067

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