Gr\"obner bases for the Hilbert ideal and coinvariants of the Dihedral group $D_{2p}$}

TL;DR

This paper investigates the structure of the Hilbert ideal and coinvariants for dihedral groups over fields of characteristic two, providing explicit Gr"obner bases, degree bounds, and top degree computations.

Contribution

It introduces a universal Gr"obner basis for the Hilbert ideal of $D_{2p}$ and establishes sharp degree bounds and coinvariant degrees.

Findings

Universal Gr"obner basis consisting of invariants and monomials.

Sharp bounds for degrees of basis elements.

Computed the top degree of coinvariants.

Abstract

We consider a finite dimensional representation of the dihedral group $D_{2p}$ over a field of characteristic two where $p$ is an odd prime and study the corresponding Hilbert ideal $I_H$. We show that $I_H$ has a universal Gr\" {o}bner basis consisting of invariants and monomials only. We provide sharp bounds for the degree of an element in this basis and in a minimal generating set for $I_H$. We also compute the top degree of coinvariants.