Equivariant Hilbert Series of Monomial Orbits

TL;DR

This paper calculates the equivariant Hilbert series for ideals generated by monomial orbits under increasing function actions, revealing their dimension and degree, and comments on the efficiency of existing descriptions.

Contribution

It determines the equivariant Hilbert series for monomial orbit ideals and analyzes their algebraic properties, enhancing understanding of $ ext{Inc}( )$-invariant ideals.

Findings

Explicit formulas for Hilbert series of monomial orbit ideals

Dimension and degree of these ideals are derived

Supports the efficiency of Nagel and R"omer's denominator description

Abstract

The equivariant Hilbert series of an ideal generated by an orbit of a monomial under the action of the monoid $\mbox{Inc}(\mathbb{N})$ of strictly increasing functions is determined. This is used to find the dimension and degree of such an ideal. The result also suggests that the description of the denominator of an equivariant Hilbert series of an arbitrary $\mbox{Inc}(\mathbb{N})$-invariant ideal as given by Nagel and R\"omer is rather efficient.