A filtration on the cohomology rings of regular nilpotent Hessenberg varieties
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Mathematische Zeitschrift
A filtration on the cohomology rings of regular nilpotent Hessenberg varieties Megumi Harada1 · Tatsuya Horiguchi2 · Satoshi Murai3 · Martha Precup4 · Julianna Tymoczko5 Received: 4 July 2020 / Accepted: 23 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Let n be a positive integer. The main result of this manuscript is a construction of a filtration on the cohomology ring of a regular nilpotent Hessenberg variety in G L(n, C)/B such that its associated graded ring has graded pieces (i.e., homogeneous components) isomorphic to rings which are related to the cohomology rings of Hessenberg varieties in G L(n − 1, C)/B, showing the inductive nature of these rings. In previous work, the first two authors, together with Abe and Masuda, gave an explicit presentation of these cohomology rings in terms of generators and relations. We introduce a new set of polynomials which are closely related to the relations in the above presentation and obtain a sequence of equivalence relations they satisfy; this allows us to derive our filtration. In addition, we obtain the following three corollaries. First, we give an inductive formula for the Poincaré polynomial of these varieties. Second, we give an explicit monomial basis for the cohomology rings of regular nilpotent Hessenberg varieties with respect to the presentation mentioned above. Third, we derive a basis of the set of linear relations satisfied by the images of the Schubert classes in the cohomology rings of regular nilpotent Hessenberg varieties. Finally, our methods and results suggest many directions for future work; in particular, we propose a definition of “Hessenberg Schubert polynomials” in the context of regular nilpotent Hessenberg varieties, which generalize the classical Schubert polynomials. We also outline several open questions pertaining to them.
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Statement of the main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Proof of main theorem, part 1: a filtration of R/Ih . . . . . . . . . . . . . . . . . 5 Proof of main theorem, part 2: the quotient ring . . . . . . . . . . . . . . . . . . . 6 An inductive formula for Poincaré polynomials . . . . . . . . . . . . . . . . . . . 7 A monomial basis for H ∗ (Hess(N, h)) . . . . . . . . . . . . . . . . . . . . . . . 8 Linear relations on Schubert classes in H ∗ (Hess(N, h)) . . . . . . . . . . . . . . 9 Further directions: a proposal for a definition of Hessenberg Schubert polynomials References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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