TY - JOUR

T1 - A filtration on the cohomology rings of regular nilpotent Hessenberg varieties

AU - Harada, Megumi

AU - Horiguchi, Tatsuya

AU - Murai, Satoshi

AU - Precup, Martha

AU - Tymoczko, Julianna

N1 - Funding Information:
We are grateful to the hospitality of the Mathematical Sciences Research Institute in Berkeley, California, and the Osaka City University Advanced Mathematical Institute in Osaka, Japan, where parts of this research was conducted. Some of the material contained in this paper are based upon work supported by the National Security Agency under Grant No. H98230-19-1-0119, The Lyda Hill Foundation, The McGovern Foundation, and Microsoft Research, while the first, fourth, and fifth authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the summer of 2019. We are also grateful for a crucial idea from Claudia Miller and for helpful conversations with Adam Van Tuyl. The first author is supported by a Natural Science and Engineering Research Council Discovery Grant and a Canada Research Chair (Tier 2) from the Government of Canada. The second author is supported by JSPS Grant-in-Aid for JSPS Research Fellow: 17J04330 and by JSPS Grant-in-Aid for Young Scientists: 19K14508. The third author is partially supported by Kakenhi 16J04761 and Waseda University Grant Research Base Creation 2019C-134. The fourth author is supported in part by NSF DMS-1954001. The fifth author is supported in part by NSF DMS-1800773.

PY - 2020

Y1 - 2020

N2 - 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 GL(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 GL(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.

AB - 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 GL(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 GL(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.

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U2 - 10.1007/s00209-020-02646-x

DO - 10.1007/s00209-020-02646-x

M3 - Article

AN - SCOPUS:85096012748

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

ER -