Solomon–Terao algebra of hyperplane arrangements

Abstract

We introduce a new algebra associated with a hyperplane arrangement 𝒜, called the Solomon–Terao algebra 𝑆𝑇(𝒜, 𝜂), where 𝜂 is a homogeneous polynomial. It is shown by Solomon and Terao that 𝑆𝑇(𝒜, 𝜂) is Artinian when 𝜂 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 𝑆𝑇(𝒜, 𝜂) is a complete intersection if and only if 𝒜 is free. We also give a factorization formula of the Hilbert polynomials of 𝑆𝑇(𝒜, 𝜂) when 𝒜 is free, and pose several related questions, problems and conjectures.