Equivariant Hilbert series for hierarchical models

TL;DR

This paper introduces invariant filtrations and equivariant Hilbert series for hierarchical models, providing conditions for rationality, explicit formulas, and algorithms, with applications to toric ideals and symmetric group actions.

Contribution

It develops a framework for analyzing invariant filtrations and their Hilbert series in hierarchical models, including explicit formulas and algorithms for rational functions.

Findings

Equivariant Hilbert series are rational functions under certain conditions.

Explicit formulas for Hilbert series with coefficients in number fields.

An algorithm for computing rational functions with rational coefficients.

Abstract

Toric ideals to hierarchical models are invariant under the action of a product of symmetric groups. Taking the number of factors, say m, into account, we introduce and study invariant filtrations and their equivariant Hilbert series. We present a condition that guarantees that the equivariant Hilbert series is a rational function in m+1 variables with rational coefficients. Furthermore we give explicit formulas for the rational functions with coefficients in a number field and an algorithm for determining the rational functions with rational coefficients. A key is to construct finite automata that recognize languages corresponding to invariant filtrations.