## Journal of the Mathematical Society of Japan

### Solomon–Terao algebra of hyperplane arrangements

#### Abstract

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.

#### Note

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.

#### Article information

Source
J. Math. Soc. Japan, Advance publication (2019), 21 pages.

Dates
First available in Project Euclid: 17 July 2019

https://projecteuclid.org/euclid.jmsj/1563350426

Digital Object Identifier
doi:10.2969/jmsj/79957995

#### Citation

ABE, Takuro; MAENO, Toshiaki; MURAI, Satoshi; NUMATA, Yasuhide. Solomon–Terao algebra of hyperplane arrangements. J. Math. Soc. Japan, advance publication, 17 July 2019. doi:10.2969/jmsj/79957995. https://projecteuclid.org/euclid.jmsj/1563350426

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