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

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

2011-10-23

arxiv.org/abs/1110.5067v1

Mathematics

Commutative Algebra

14 pages

Scientific paper

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 denote by $W_b$ the one dimensional representation of $G$ associated to the character $-b$ where $1 \leq b \leq n$. We consider the ring of invariants ${\bf F}[W]^G$ of the three dimensional representation $W=W_b \oplus W_c \oplus W_d$ of $G$ where $G \subset \text{SL}(W)$. We describe minimal generators and relations for this ring of invariants and prove that the lead terms of the relations are quadratic with respect to a carefully chosen term order. With this term order these minimal generators for the relations form a Groebner basis and the lead terms have a surprisingly simple combinatorial structure. Exploiting this structure, we describe the graded Betti numbers for a minimal free resolution of these rings of invariants. The case where $W=W_b\oplus W_c$ is any two dimensional representation of $G$ is also handled.

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