Hessenberg varieties and hyperplane arrangements

Takuro Abe, Tatsuya Horiguchi, Mikiya Masuda, Satoshi Murai, Takashi Sato

    Research output: Contribution to journalArticle

    Abstract

    Given a semisimple complex linear algebraic group G {{G}} and a lower ideal I in positive roots of G, three objects arise: the ideal arrangement AI, the regular nilpotent Hessenberg variety Hess (N, I), and the regular semisimple Hessenberg variety Hess (S, I). We show that a certain graded ring derived from the logarithmic derivation module of AI is isomorphic to H ∗ (Hess (N, I)) and H ∗ (Hess (S, I)) W, the invariants in H ∗ (Hess (S, I)) under an action of the Weyl group W of G. This isomorphism is shown for general Lie type, and generalizes Borel's celebrated theorem showing that the coinvariant algebra of W is isomorphic to the cohomology ring of the flag variety G / B {G/B}. This surprising connection between Hessenberg varieties and hyperplane arrangements enables us to produce a number of interesting consequences. For instance, the surjectivity of the restriction map H ∗ (G / B) → H ∗ (Hess (N, I)) announced by Dale Peterson and an affirmative answer to a conjecture of Sommers and Tymoczko are immediate consequences. We also give an explicit ring presentation of H ∗ (Hess (N, I)) in types B, C, and G. Such a presentation was already known in type A and when Hess (N, I) is the Peterson variety. Moreover, we find the volume polynomial of Hess (N, I) and see that the hard Lefschetz property and the Hodge-Riemann relations hold for Hess (N, I), despite the fact that it is a singular variety in general.

    Original languageEnglish
    JournalJournal fur die Reine und Angewandte Mathematik
    DOIs
    Publication statusAccepted/In press - 2019 Jan 1

    Fingerprint

    Hyperplane Arrangement
    Si
    Algebra
    Polynomials
    Semisimple
    Isomorphic
    Linear Algebraic Groups
    Surjectivity
    Flag Variety
    Graded Ring
    Cohomology Ring
    Weyl Group
    Arrangement
    Isomorphism
    Logarithmic
    Roots
    Restriction
    Ring
    Module
    Generalise

    ASJC Scopus subject areas

    • Mathematics(all)
    • Applied Mathematics

    Cite this

    Hessenberg varieties and hyperplane arrangements. / Abe, Takuro; Horiguchi, Tatsuya; Masuda, Mikiya; Murai, Satoshi; Sato, Takashi.

    In: Journal fur die Reine und Angewandte Mathematik, 01.01.2019.

    Research output: Contribution to journalArticle

    Abe, Takuro ; Horiguchi, Tatsuya ; Masuda, Mikiya ; Murai, Satoshi ; Sato, Takashi. / Hessenberg varieties and hyperplane arrangements. In: Journal fur die Reine und Angewandte Mathematik. 2019.
    @article{fdb8c2e6abdf4a6ba83111d312d8df98,
    title = "Hessenberg varieties and hyperplane arrangements",
    abstract = "Given a semisimple complex linear algebraic group G {{G}} and a lower ideal I in positive roots of G, three objects arise: the ideal arrangement AI, the regular nilpotent Hessenberg variety Hess (N, I), and the regular semisimple Hessenberg variety Hess (S, I). We show that a certain graded ring derived from the logarithmic derivation module of AI is isomorphic to H ∗ (Hess (N, I)) and H ∗ (Hess (S, I)) W, the invariants in H ∗ (Hess (S, I)) under an action of the Weyl group W of G. This isomorphism is shown for general Lie type, and generalizes Borel's celebrated theorem showing that the coinvariant algebra of W is isomorphic to the cohomology ring of the flag variety G / B {G/B}. This surprising connection between Hessenberg varieties and hyperplane arrangements enables us to produce a number of interesting consequences. For instance, the surjectivity of the restriction map H ∗ (G / B) → H ∗ (Hess (N, I)) announced by Dale Peterson and an affirmative answer to a conjecture of Sommers and Tymoczko are immediate consequences. We also give an explicit ring presentation of H ∗ (Hess (N, I)) in types B, C, and G. Such a presentation was already known in type A and when Hess (N, I) is the Peterson variety. Moreover, we find the volume polynomial of Hess (N, I) and see that the hard Lefschetz property and the Hodge-Riemann relations hold for Hess (N, I), despite the fact that it is a singular variety in general.",
    author = "Takuro Abe and Tatsuya Horiguchi and Mikiya Masuda and Satoshi Murai and Takashi Sato",
    year = "2019",
    month = "1",
    day = "1",
    doi = "10.1515/crelle-2018-0039",
    language = "English",
    journal = "Journal fur die Reine und Angewandte Mathematik",
    issn = "0075-4102",
    publisher = "Walter de Gruyter GmbH & Co. KG",

    }

    TY - JOUR

    T1 - Hessenberg varieties and hyperplane arrangements

    AU - Abe, Takuro

    AU - Horiguchi, Tatsuya

    AU - Masuda, Mikiya

    AU - Murai, Satoshi

    AU - Sato, Takashi

    PY - 2019/1/1

    Y1 - 2019/1/1

    N2 - Given a semisimple complex linear algebraic group G {{G}} and a lower ideal I in positive roots of G, three objects arise: the ideal arrangement AI, the regular nilpotent Hessenberg variety Hess (N, I), and the regular semisimple Hessenberg variety Hess (S, I). We show that a certain graded ring derived from the logarithmic derivation module of AI is isomorphic to H ∗ (Hess (N, I)) and H ∗ (Hess (S, I)) W, the invariants in H ∗ (Hess (S, I)) under an action of the Weyl group W of G. This isomorphism is shown for general Lie type, and generalizes Borel's celebrated theorem showing that the coinvariant algebra of W is isomorphic to the cohomology ring of the flag variety G / B {G/B}. This surprising connection between Hessenberg varieties and hyperplane arrangements enables us to produce a number of interesting consequences. For instance, the surjectivity of the restriction map H ∗ (G / B) → H ∗ (Hess (N, I)) announced by Dale Peterson and an affirmative answer to a conjecture of Sommers and Tymoczko are immediate consequences. We also give an explicit ring presentation of H ∗ (Hess (N, I)) in types B, C, and G. Such a presentation was already known in type A and when Hess (N, I) is the Peterson variety. Moreover, we find the volume polynomial of Hess (N, I) and see that the hard Lefschetz property and the Hodge-Riemann relations hold for Hess (N, I), despite the fact that it is a singular variety in general.

    AB - Given a semisimple complex linear algebraic group G {{G}} and a lower ideal I in positive roots of G, three objects arise: the ideal arrangement AI, the regular nilpotent Hessenberg variety Hess (N, I), and the regular semisimple Hessenberg variety Hess (S, I). We show that a certain graded ring derived from the logarithmic derivation module of AI is isomorphic to H ∗ (Hess (N, I)) and H ∗ (Hess (S, I)) W, the invariants in H ∗ (Hess (S, I)) under an action of the Weyl group W of G. This isomorphism is shown for general Lie type, and generalizes Borel's celebrated theorem showing that the coinvariant algebra of W is isomorphic to the cohomology ring of the flag variety G / B {G/B}. This surprising connection between Hessenberg varieties and hyperplane arrangements enables us to produce a number of interesting consequences. For instance, the surjectivity of the restriction map H ∗ (G / B) → H ∗ (Hess (N, I)) announced by Dale Peterson and an affirmative answer to a conjecture of Sommers and Tymoczko are immediate consequences. We also give an explicit ring presentation of H ∗ (Hess (N, I)) in types B, C, and G. Such a presentation was already known in type A and when Hess (N, I) is the Peterson variety. Moreover, we find the volume polynomial of Hess (N, I) and see that the hard Lefschetz property and the Hodge-Riemann relations hold for Hess (N, I), despite the fact that it is a singular variety in general.

    UR - http://www.scopus.com/inward/record.url?scp=85060703145&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=85060703145&partnerID=8YFLogxK

    U2 - 10.1515/crelle-2018-0039

    DO - 10.1515/crelle-2018-0039

    M3 - Article

    JO - Journal fur die Reine und Angewandte Mathematik

    JF - Journal fur die Reine und Angewandte Mathematik

    SN - 0075-4102

    ER -