Subarrangements of type A: the weak Lefschetz property of the Artinian Orlik-Terao algebra

TL;DR

This paper investigates the weak Lefschetz property of the Artinian Orlik-Terao algebra associated with graph arrangements, revealing complex behaviors and providing new algebraic insights.

Contribution

It analyzes WLP in the Artinian Orlik-Terao algebra for graph arrangements, introducing canonical elements and refining existing algebraic results.

Findings

WLP sometimes fails for chordal graphs with Koszul algebras

WLP can hold even when initial ideals do not satisfy WLP

Constructs canonical kernel elements for tensor product decompositions

Abstract

In 1994, Orlik and Terao introduced a commutative Artinian analog S/I(A) of the Orlik-Solomon algebra of a hyperplane arrangement A to answer a question of Aomoto. A central topic of investigation in the study of Artinian algebras is the Weak Lefschetz Property (WLP). We analyze WLP for the Artinian Orlik-Terao algebra of graphc arrangements. Even for chordal graphs (which give rise to Koszul algebras) WLP sometimes fails; conversely an analysis of the state polytope shows WLP can hold even when WLP fails for all possible initial ideals. More generally, for any algebra with a tensor product decomposition, we construct canonical elements in the kernel of the multiplication map, refining previous results in the literature.