Hilbert Functions of Finite Group Orbits: Abelian and Metacyclic Groups

This thesis is concerned with Hilbert functions of ideals of finite group orbits, focusing in particular on abelian and metacyclic groups. Conjecture 16 in Section 4 characterizes the Hilbert functions that arise from two-dimensional representations of certain abelian groups, and a proof of the necessity of its conditions is provided. In Section 6, Proposition 18 gives a bound for the Hilbert function of the orbit ideal of an induced representation of a metacyclic group. Proposition 21 improves this result in the twodimensional case, providing a description of the actual Hilbert function. Conjecture 24 characterizes the three-dimensional representations that fail to achieve the bound in Proposition 18, while Proposition 25 reformulates this characterization in terms of a simple number theoretic condition.