Journal of the Mathematical Society of Japan

Solomon–Terao algebra of hyperplane arrangements

Takuro ABE, Toshiaki MAENO, Satoshi MURAI, and Yasuhide NUMATA

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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, Volume 71, Number 4 (2019), 1027-1047.

Dates
Received: 24 February 2018
First available in Project Euclid: 17 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1563350426

Digital Object Identifier
doi:10.2969/jmsj/79957995

Mathematical Reviews number (MathSciNet)
MR4023295

Zentralblatt MATH identifier
07174394

Subjects
Primary: 32S22: Relations with arrangements of hyperplanes [See also 52C35]
Secondary: 13E10: Artinian rings and modules, finite-dimensional algebras

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

Citation

ABE, Takuro; MAENO, Toshiaki; MURAI, Satoshi; NUMATA, Yasuhide. Solomon–Terao algebra of hyperplane arrangements. J. Math. Soc. Japan 71 (2019), no. 4, 1027--1047. doi:10.2969/jmsj/79957995. https://projecteuclid.org/euclid.jmsj/1563350426


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