Comptes Rendus
Combinatorics
Newton polytope of good symmetric polynomials
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 767-775.

We introduce a general class of symmetric polynomials that have saturated Newton polytope and their Newton polytope has integer decomposition property. The class covers numerous previously studied symmetric polynomials.

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DOI: 10.5802/crmath.430
Classification: 52B20, 05E05

Duc-Khanh Nguyen 1; Giao Nguyen Thi Ngoc 2; Hiep Dang Tuan 3; Thuy Do Le Hai 4

1 Department of Mathematics and Statistics, University at Albany, Albany, NY 12222, USA
2 Faculty of Advanced Science and Technology, University of Science and Technology - The University of Da Nang, 54 Nguyen Luong Bang, Da Nang, Vietnam
3 Department of Mathematics, Dalat University, 1 Phu Dong Thien Vuong, Ward 8, Dalat City, Lam Dong, Vietnam
4 Institute of Mathematics, Vietnam academy of science and technology, 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Duc-Khanh Nguyen; Giao Nguyen Thi Ngoc; Hiep Dang Tuan; Thuy Do Le Hai. Newton polytope of good symmetric polynomials. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 767-775. doi : 10.5802/crmath.430. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.430/

[1] Alexander Barvinok Lattice points and lattice polytopes, Handbook of discrete and computational geometry, CRC Press, 2017, pp. 185-210

[2] Margaret Bayer; Bennet Goeckner; Su Ji Hong; Tyrrell McAllister; McCabe Olsen; Casey Pinckney; Julianne Vega; Martha Yip Lattice polytopes from Schur and symmetric Grothendieck polynomials, Electron. J. Comb., Volume 28 (2021) no. 2, P2.45, 36 pages | MR | Zbl

[3] Winfried Bruns; Tim Römer h-vectors of Gorenstein polytopes, J. Comb. Theory, Ser. A, Volume 114 (2007) no. 1, pp. 65-76 | DOI | MR | Zbl

[4] David A. Cox; John B. Little; Henry K. Schenck Toric Varieties, Graduate Studies in Mathematics, 124, American Mathematical Society, 2011

[5] Laura Escobar; Alexander Yong Newton polytopes and symmetric Grothendieck polynomials, C. R. Math. Acad. Sci. Paris, Volume 355 (2017) no. 8, pp. 831-834 | DOI | Numdam | MR | Zbl

[6] Ehrhart Eugène Sur les polyèdres rationnels homothétiques à n dimensions, C. R. Math. Acad. Sci. Paris, Volume 254 (1962), pp. 616-618 | Zbl

[7] William Fulton Introduction to Toric Varieties, Annals of Mathematics Studies, 131, Princeton University Press, 2016

[8] Vesselin Gasharov Incomparability graphs of (3+1)-free posets are s-positive, Discrete Math., Volume 157 (1996) no. 1-3, pp. 193-197 | DOI | MR | Zbl

[9] Boris Ya Kazarnovskii; Askold G. Khovanskii; Alexander I. Esterov Newton polytopes and tropical geometry, Russ. Math. Surv., Volume 76 (2021) no. 1, pp. 91-175 | DOI | MR | Zbl

[10] Thomas Lam; Pavlo Pylyavskyy Combinatorial Hopf algebras and K-homology of Grassmanians, Int. Math. Res. Not., Volume 2007 (2007) no. 9, RMN125, 48 pages | Zbl

[11] Jacob P. Matherne; Alejandro H. Morales; Jesse Selover The Newton polytope and Lorentzian property of chromatic symmetric functions (2022) | arXiv

[12] Cara Monical; Neriman Tokcan; Alexander Yong Newton polytopes in algebraic combinatorics, Sel. Math., New Ser., Volume 25 (2019) no. 5, 66, 37 pages | MR | Zbl

[13] Hidefumi Ohsugi; Takayuki Hibi Special simplices and Gorenstein toric rings, J. Comb. Theory, Ser. A, Volume 113 (2006) no. 4, pp. 718-725 | DOI | MR | Zbl

[14] Richard Rado An inequality, J. Lond. Math. Soc., Volume 27 (1952) no. 1, pp. 1-6 | DOI

[15] Jan Schepers; Leen Van Langenhoven Unimodality questions for integrally closed lattice polytopes, Ann. Comb., Volume 17 (2013) no. 3, pp. 571-589 | DOI | MR | Zbl

[16] Richard P. Stanley On the number of reduced decompositions of elements of Coxeter groups, Eur. J. Comb., Volume 5 (1984) no. 4, pp. 359-372 | DOI | MR | Zbl

[17] Richard P. Stanley A symmetric function generalization of the chromatic polynomial of a graph, Adv. Math., Volume 111 (1995) no. 1, pp. 166-194 | DOI | MR | Zbl

[18] Richard P. Stanley Enumerative Combinatorics. Volume 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, 1999 | DOI

[19] Richard P. Stanley; John R. Stembridge On immanants of Jacobi–Trudi matrices and permutations with restricted position, J. Comb. Theory, Ser. A, Volume 62 (1993) no. 2, pp. 261-279 | DOI | MR | Zbl

[20] John R. Stembridge Shifted tableaux and the projective representations of symmetric groups, Adv. Math., Volume 74 (1989) no. 1, pp. 87-134 | DOI | MR | Zbl

[21] John R. Stembridge Immanants of totally positive matrices are nonnegative, Bull. Lond. Math. Soc., Volume 23 (1991) no. 5, p. 442-428 | MR | Zbl

[22] Bernd Sturmfels Grobner bases and convex polytopes, University Lecture Series, 8, American Mathematical Society, 1996

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