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This article is cited in 4 scientific papers (total in 4 papers)
The Volume Polynomial of Regular Semisimple Hessenberg Varieties and the Gelfand–Zetlin Polytope
Megumi Haradaa, Tatsuya Horiguchib, Mikiya Masudac, Seonjeong Parkd a Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, Ontario L8S4K1, Canada
b Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, 1-5 Yamadaoka, Suita, Osaka 565-0871, Japan
c Department of Mathematics, Osaka City University, 3-3-138, Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan
d Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology (KAIST), 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea
Abstract:
Regular semisimple Hessenberg varieties are subvarieties of the flag variety $\mathrm {Flag}(\mathbb C^n)$ arising naturally at the intersection of geometry, representation theory, and combinatorics. Recent results of Abe, Horiguchi, Masuda, Murai, and Sato as well as of Abe, DeDieu, Galetto, and Harada relate the volume polynomials of regular semisimple Hessenberg varieties to the volume polynomial of the Gelfand–Zetlin polytope $\mathrm {GZ}(\lambda )$ for $\lambda =(\lambda _1,\lambda _2,\dots ,\lambda _n)$. In the main results of this paper we use and generalize tools developed by Anderson and Tymoczko, by Kiritchenko, Smirnov, and Timorin, and by Postnikov in order to derive an explicit formula for the volume polynomials of regular semisimple Hessenberg varieties in terms of the volumes of certain faces of the Gelfand–Zetlin polytope, and also exhibit a manifestly positive, combinatorial formula for their coefficients with respect to the basis of monomials in the $\alpha _i := \lambda _i-\lambda _{i+1}$. In addition, motivated by these considerations, we carefully analyze the special case of the permutohedral variety, which is also known as the toric variety associated to Weyl chambers. In this case, we obtain an explicit decomposition of the permutohedron (the moment map image of the permutohedral variety) into combinatorial $(n-1)$-cubes, and also give a geometric interpretation of this decomposition by expressing the cohomology class of the permutohedral variety in $\mathrm {Flag}(\mathbb C^n)$ as a sum of the cohomology classes of a certain set of Richardson varieties.
Keywords:
Hessenberg variety, flag variety, Schubert variety, Richardson variety, permutohedral variety, volume polynomials, Gelfand–Zetlin polytope, Young tableaux.
Received: December 25, 2018 Revised: January 10, 2019 Accepted: March 28, 2019
Citation:
Megumi Harada, Tatsuya Horiguchi, Mikiya Masuda, Seonjeong Park, “The Volume Polynomial of Regular Semisimple Hessenberg Varieties and the Gelfand–Zetlin Polytope”, Algebraic topology, combinatorics, and mathematical physics, Collected papers. Dedicated to Victor Matveevich Buchstaber, Corresponding Member of the Russian Academy of Sciences, on the occasion of his 75th birthday, Trudy Mat. Inst. Steklova, 305, Steklov Math. Inst. RAS, Moscow, 2019, 344–373; Proc. Steklov Inst. Math., 305 (2019), 318–344
Linking options:
https://www.mathnet.ru/eng/tm4014https://doi.org/10.4213/tm4014 https://www.mathnet.ru/eng/tm/v305/p344
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