Solomon-Terao algebra of hyperplane arrangements

TL;DR

This paper introduces the Solomon-Terao algebra associated with hyperplane arrangements, generalizing coinvariant algebras, and explores its properties, including conditions for being a complete intersection and its relation to free arrangements.

Contribution

It defines the Solomon-Terao algebra for hyperplane arrangements, investigates its algebraic properties, and establishes criteria for when it is a complete intersection, extending previous algebraic frameworks.

Findings

Solomon-Terao algebra is Artinian for generic η.

It is a complete intersection if and only if the arrangement is free.

Provides a Hilbert polynomial factorization for free arrangements.

Abstract

We introduce a new algebra associated with a hyperplane arrangement $\mathcal{A}$, called the Solomon-Terao algebra $\mbox{ST}(\mathcal{A},\eta)$, where $\eta$ is a homogeneous polynomial. It is shown by Solomon and Terao that $\mbox{ST}(\mathcal{A},\eta)$ is Artinian when $\eta$ is generic. This algebra can be considered as a generalization of coinvariant algebras in the setting of hyperplane arrangements. The class of Solomon-Terao algebras contains cohomology rings of regular nilpotent Hessenberg varieties. We show that $\mbox{ST}(\mathcal{A},\eta)$ is a complete intersection if and only if $\mathcal{A}$ is free. We also give a factorization formula of the Hilbert polynomials when $\mathcal{A}$ is free, and pose several related questions, problems and conjectures.