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Tr. Mat. Inst. Steklova, 2019, Volume 305, Pages 344–373 (Mi tm4014)  

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 $\lambda =(\lambda _1,\lambda _2,…,\lambda _n)$. 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.

Funding Agency Grant Number
Natural Sciences and Engineering Research Council of Canada (NSERC)
Japan Society for the Promotion of Science 17J04330
16K05152
National Research Foundation of Korea NRF-2018R1A6A3A11047606
Canada Research Chair
The first author is supported in part by an NSERC Discovery Grant and a Canada Research Chair (Tier 2) Award. She is also grateful to the Osaka City University Advanced Mathematics Institute for support and hospitality during her fruitful visit in Fall 2017, when much of this work was conducted. The second author is partially supported by JSPS Grant-in-Aid for JSPS Fellows 17J04330. The third author is supported in part by JSPS Grant-in-Aid for Scientific Research 16K05152. The fourth author acknowledges the support of the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (NRF-2018R1A6A3A11047606).


DOI: https://doi.org/10.4213/tm4014

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

Bibliographic databases:

UDC: 512.734
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, Tr. Mat. Inst. Steklova, 305, Steklov Math. Inst. RAS, Moscow, 2019, 344–373; Proc. Steklov Inst. Math., 305 (2019), 318–344

Citation in format AMSBIB
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\by Megumi~Harada, Tatsuya~Horiguchi, Mikiya~Masuda, Seonjeong~Park
\paper The Volume Polynomial of Regular Semisimple Hessenberg Varieties and the Gelfand--Zetlin Polytope
\inbook Algebraic topology, combinatorics, and mathematical physics
\bookinfo Collected papers. Dedicated to Victor Matveevich Buchstaber, Corresponding Member of the Russian Academy of Sciences, on the occasion of his 75th birthday
\serial Tr. Mat. Inst. Steklova
\yr 2019
\vol 305
\pages 344--373
\publ Steklov Math. Inst. RAS
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm4014}
\crossref{https://doi.org/10.4213/tm4014}
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\jour Proc. Steklov Inst. Math.
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