Hessenberg varieties and hyperplane arrangements

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

Research output: Contribution to journalArticle

2 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 - Jan 1 2019

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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

All Science Journal Classification (ASJC) codes

  • 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.
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