Open Access
Translator Disclaimer
October, 2019 Solomon–Terao algebra of hyperplane arrangements
Takuro ABE, Toshiaki MAENO, Satoshi MURAI, Yasuhide NUMATA
J. Math. Soc. Japan 71(4): 1027-1047 (October, 2019). DOI: 10.2969/jmsj/79957995


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

Funding Statement

The authors are partially supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (B) 16H03924. The second author is supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (C) 16K05083.


Download Citation

Takuro ABE. Toshiaki MAENO. Satoshi MURAI. Yasuhide NUMATA. "Solomon–Terao algebra of hyperplane arrangements." J. Math. Soc. Japan 71 (4) 1027 - 1047, October, 2019.


Received: 24 February 2018; Published: October, 2019
First available in Project Euclid: 17 July 2019

zbMATH: 07174394
MathSciNet: MR4023295
Digital Object Identifier: 10.2969/jmsj/79957995

Primary: 32S22
Secondary: 13E10

Keywords: Artinian ring , complete intersection ring , free arrangements , hyperplane arrangements , logarithmic derivation modules , Solomon–Terao formula

Rights: Copyright © 2019 Mathematical Society of Japan


Vol.71 • No. 4 • October, 2019
Back to Top