TY - JOUR

T1 - Solomon–Terao algebra of hyperplane arrangements

AU - Abe, Takuro

AU - Maeno, Toshiaki

AU - Murai, Satoshi

AU - Numata, Yasuhide

N1 - Funding Information:
2010 Mathematics Subject Classification. Primary 32S22; Secondary 13E10. Key Words and Phrases. hyperplane arrangements, logarithmic derivation modules, free arrangements, Solomon–Terao formula, complete intersection ring, Artinian ring. 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.
Publisher Copyright:
© 2019 The Mathematical Society of Japan.

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

KW - Artinian ring

KW - Complete intersection ring

KW - Free arrangements

KW - Hyperplane arrangements

KW - Logarithmic derivation modules

KW - Solomon–Terao formula

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U2 - 10.2969/jmsj/79957995

DO - 10.2969/jmsj/79957995

M3 - Article

AN - SCOPUS:85075518432

VL - 71

SP - 1027

EP - 1047

JO - Journal of the Mathematical Society of Japan

JF - Journal of the Mathematical Society of Japan

SN - 0025-5645

IS - 4

ER -