Solomon–Terao algebra of hyperplane arrangements

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.