Rationality of Equivariant Hilbert Series and Asymptotic Properties

TL;DR

This paper studies the asymptotic behavior of invariants of modules over polynomial algebras with symmetric group actions, introducing an equivariant Hilbert series that reveals growth patterns and structural properties.

Contribution

It introduces a rational equivariant Hilbert series for finitely generated FI- or OI-modules, linking it to growth rates of invariants and providing an algorithm for computation.

Findings

The equivariant Hilbert series is rational for finitely generated modules.

Krull dimension grows linearly, multiplicity grows exponentially in n.

Degree j component dimensions grow polynomially in n.

Abstract

An FI- or an OI-module $\mathbf{M}$ over a corresponding noetherian polynomial algebra $\mathbf{P}$ may be thought of as a sequence of compatible modules $\mathbf{M}_n$ over a polynomial ring $\mathbf{P}_n$ whose number of variables depends linearly on $n$. In order to study invariants of the modules $\mathbf{M}_n$ in dependence of $n$, an equivariant Hilbert series is introduced if $\mathbf{M}$ is graded. If $\mathbf{M}$ is also finitely generated, it is shown that this series is a rational function. Moreover, if this function is written in reduced form rather precise information about the irreducible factors of the denominator is obtained. This is key for applications. It follows that the Krull dimension of the modules $\mathbf{M}_n$ grows eventually linearly in $n$, whereas the multiplicity of $\mathbf{M}_n$ grows eventually exponentially in $n$. Moreover, for any fixed degree $j$, the vector space dimensions of the degree $j$ components of $\mathbf{M}_n$ grow eventually polynomially in $n$. As a consequence, any graded Betti number of $\mathbf{M}_n$ in a fixed homological degree and a fixed internal degree grows eventually polynomially in $n$. Furthermore, evidence is obtained to support a conjecture that the Castelnuovo-Mumford regularity and the projective dimension of $\mathbf{M}_n$ both grow eventually linearly in $n$. It is also shown that modules $\mathbf{M}$ whose width $n$ components $\mathbf{M}_n$ are eventually Artinian can be characterized by their equivariant Hilbert series. Using regular languages and finite automata, an algorithm for computing equivariant Hilbert series is presented.