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- Title
Hessenberg varieties and hyperplane arrangements.
- Authors
Abe, Takuro; Horiguchi, Tatsuya; Masuda, Mikiya; Murai, Satoshi; Sato, Takashi
- 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 𝒜 I {\mathcal{A}_{I}} , the regular nilpotent Hessenberg variety Hess (N , I) {\operatorname{Hess}(N,I)} , and the regular semisimple Hessenberg variety Hess (S , I) {\operatorname{Hess}(S,I)}. We show that a certain graded ring derived from the logarithmic derivation module of 𝒜 I {\mathcal{A}_{I}} is isomorphic to H * (Hess (N , I)) {H^{*}(\operatorname{Hess}(N,I))} and H * (Hess (S , I)) W {H^{*}(\operatorname{Hess}(S,I))^{W}} , the invariants in H * (Hess (S , I)) {H^{*}(\operatorname{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)) {H^{*}(G/B)\to H^{*}(\operatorname{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)) {H^{*}(\operatorname{Hess}(N,I))} in types B, C, and G. Such a presentation was already known in type A and when Hess (N , I) {\operatorname{Hess}(N,I)} is the Peterson variety. Moreover, we find the volume polynomial of Hess (N , I) {\operatorname{Hess}(N,I)} and see that the hard Lefschetz property and the Hodge–Riemann relations hold for Hess (N , I) {\operatorname{Hess}(N,I)} , despite the fact that it is a singular variety in general.
- Subjects
LINEAR algebraic groups; C*-algebras; SEMISIMPLE Lie groups; WEYL groups; SURJECTIONS
- Publication
Journal für die Reine und Angewandte Mathematik, 2020, Vol 2020, Issue 764, p241
- ISSN
0075-4102
- Publication type
Article
- DOI
10.1515/crelle-2018-0039