This article is cited in 5 papers

The Volume Polynomial of Regular Semisimple Hessenberg Varieties and the Gelfand–Zetlin Polytope

Megumi Harada^a, Tatsuya Horiguchi^b, Mikiya Masuda^c, Seonjeong Park<sup>d

a</sup> 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.

UDC: 512.734

Received: December 25, 2018
Revised: January 10, 2019
Accepted: March 28, 2019

DOI: 10.4213/tm4014

 English version: Proceedings of the Steklov Institute of Mathematics, 2019, 305, 318–344

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