Some algebras with the weak Lefschetz property

TL;DR

This paper links lozenge tilings to algebraic properties, providing a simple criterion to verify the weak Lefschetz property in monomial ideals and showing these ideals have semistable syzygy bundles.

Contribution

It introduces a new, easily checkable criterion connecting lozenge tilings to the weak Lefschetz property in monomial ideals and proves their syzygy bundles are semistable.

Findings

Established a criterion for the weak Lefschetz property using lozenge tilings.

Proved that monomial ideals with this property have semistable syzygy bundles.

Connected combinatorial tiling models to algebraic properties.

Abstract

Using a connection to lozenge tilings of triangular regions, we establish an easily checkable criterion that guarantees the weak Lefschetz property of a quotient by a monomial ideal. It is also shown that each such ideal also has a semistable syzygy bundle.