Resolutions of 2 and 3 Dimensional Rings of Invariants for Cyclic Groups

Abstract

Let G be the cyclic group of order n, and suppose F is a field containing a primitive nth root of unity. We consider the ring of invariants F[W] ^G of a three dimensional representation W of G where G ⊂ 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.

Key Words:

• Betti numbers
• Cyclic group
• Gröbner bases
• Invariant theory
• Minimal resolutions
• Monomial ideals

2010 Mathematics Subject Classification:

• 13A50
• 13D02

Notes

Communicated by I. Shestakov.