Alexander duals of symmetric simplicial complexes and Stanley-Reisner Ideals

TL;DR

This paper investigates the Alexander duals of symmetric simplicial complexes derived from invariant monomial ideals, revealing polynomial growth patterns in their generators and faces using combinatorial and geometric methods.

Contribution

It introduces the avoidance up to symmetry tool and demonstrates polynomial growth of orbit generators and faces in Alexander duals of symmetric ideals.

Findings

Number of orbit generators of $I_n^ op$ is polynomial in $n$ for large $n$.

Number of $i$-dimensional faces of Stanley-Reisner complexes is polynomial in $n$ for large $n$.

Degree of minimal generators grows linearly with $n$.

Abstract

Given an ascending chain $(I_n)_{n\in\mathbb{N}}$ of $\Sym$-invariant squarefree monomial ideals, we study the corresponding chain of Alexander duals $(I_n^\vee)_{n\in\mathbb{N}}$. Using a novel combinatorial tool, which we call \emph{avoidance up to symmetry}, we provide an explicit description of the minimal generating set up to symmetry in terms of the original generators. Combining this result with methods from discrete geometry, this enables us to show that the number of orbit generators of $I_n^\vee$ is given by a polynomial in $n$ for sufficiently large $n$. The same is true for the number of orbit generators of minimal degree, this degree being a linear function in $n$ eventually. The former result implies that the number of $\Sym$-orbits of primary components of $I_n$ grows polynomially in $n$ for large $n$. As another application, we show that, for each $i\geq 0$, the number of $i$-dimensional faces of the associated Stanley-Reisner complexes of $I_n$ is also given by a polynomial in $n$ for large $n$.