Hessenberg varieties and hyperplane arrangements

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

Research output: Contribution to journalArticle

3 Citations (Scopus)

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

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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