Coinvariant stresses, Lefschetz properties and random complexes

TL;DR

This paper explores the algebraic and geometric properties of simplicial complexes, connecting Lefschetz properties, inverse systems, and rigidity theory, and demonstrates that most homology spheres fail the weak Lefschetz property in certain algebraic reductions.

Contribution

It extends the understanding of Lefschetz properties in monomial ideals and establishes probabilistic predictions for their failure in random complexes.

Findings

Most homology spheres have monomial artinian reductions failing the WLP.

A probabilistic model predicts intervals where Lefschetz property failure is expected.

Connections between algebraic invariants and geometric rigidity are further elucidated.

Abstract

Lefschetz properties and inverse systems have played key roles in understanding the $h$-vector of simplicial spheres. In 1996, Lee established connections between these two algebraic tools and rigidity theory, an area often used in the study of motions of geometric complexes. One of the key ideas, is to translate geometric information about a complex, coming from vertex coordinates, to the algebraic notion of a linear system of parameters. In this paper, we explore similar connections in the nonlinear case, by using recent results of Herzog and Moradi (2021) where they prove that a subset of the elementary symmetric polynomials is always a system of parameters for the Stanley-Reisner ideal of a complex. We investigate connections to the study of Lefschetz properties of monomial ideals. Using this perspective, we recover and extend the well known result of Migliore, Mir\'o-Roig and Nagel on the failure of the WLP of monomial almost complete intersections, by showing that, with one simple exception, every homology sphere has a monomial artinian reduction failing the weak Lefschetz property. Finally, we state probabilistic consequences of our results under a model introduced by Linial and Meshulam. We prove that there exists an open interval for the probability parameter where failure of Lefschetz properties of monomial ideals should be expected.