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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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.


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

J. Math. Soc. Japan, Volume 71, Number 4 (2019), 1027-1047.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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  • [1] T. Abe, Divisionally free arrangements of hyperplanes, Invent. Math., 204 (2016), 317–346.
  • [2] T. Abe, M. Barakat, M. Cuntz, T. Hoge and H. Terao, The freeness of ideal subarrangements of Weyl arrangements, J. European Math. Soc., 18 (2016), 1339–1348.
  • [3] T. Abe, T. Horiguchi, M. Masuda, S. Murai and T. Sato, Hessenberg varieties and hyperplane arrangements, J. Reine Angew. Math., to appear. DOI:
  • [4] T. Abe and L. Kühne, Heavy hyperplane in multiarrangements and their freeness, J. Algebraic Combin., 48 (2018), 581–606.
  • [5] A. Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogénes de groupes de Lie compacts, Ann. of Math. (2), 57 (1953), 115–207.
  • [6] C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math., 77 (1955), 778–782.
  • [7] F. De Mari, C. Procesi and M. A. Shayman, Hessenberg varieties, Trans. Amer. Math. Soc., 332 (1992), 529–534.
  • [8] R. Epure and M. Schulze, A Saito criterion for holonomic divisors, Manuscr. Math., to appear. arXiv:1711:10259.
  • [9] M. Harada and J. Tymoczko, Poset pinball, GKM-compatible subspaces, and Hessenberg varieties, J. Math. Soc. Japan, 69 (2017), 945–994.
  • [10] T. Harima, T. Maeno, H. Morita, Y. Numata, A. Wachi and J. Watanabe, The Lefschetz Properties, Lecture Notes in Math, 2080, Springer, Heidelberg, 2013.
  • [11] P. Orlik and L. Solomon, Combinatorics and topology of complements of hyperplane arrangements, Invent. Math., 56 (1980), 167–189.
  • [12] P. Orlik and H. Terao, Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften, 300, Springer-Verlag, Berlin, 1992.
  • [13] P. Orlik and H. Terao, Arrangements and Milnor fibers, Math. Ann., 301 (1995), 211–235.
  • [14] K. Saito, On the uniformization of complements of discriminant loci, AMS Summer Institute, Williams college, 1975, RIMS Kōkyūroku, 287 (1977), 117–137.
  • [15] W. Slofstra, Rationally smooth Schubert varieties and inversion hyperplane arrangements, Adv. Math., 285 (2015), 709–736.
  • [16] L. Smith, Polynomial invariants of finite groups, Res. Notes Math., A K Peters, Wellesley, MA, 1995.
  • [17] L. Solomon and H. Terao, A formula for the characteristic polynomial of an arrangement, Adv. Math., 64 (1987), 305–325.
  • [18] E. Sommers and J. Tymoczko, Exponents of $B$-stable ideals, Trans. Amer. Math. Soc., 358 (2006), 3493–3509.
  • [19] H. Terao, Arrangements of hyperplanes and their freeness I, II, J. Fac. Sci. Univ. Tokyo, 27 (1980), 293–320.
  • [20] H. Terao, Generalized exponents of a free arrangement of hyperplanes and Shepherd–Todd–Brieskorn formula, Invent. Math., 63 (1981), 159–179.