no. G4
Combinatoire
Newton polytope of good symmetric polynomials
Nguyen, Duc-Khanh1  ; Nguyen Thi Ngoc, Giao2 ; Dang Tuan, Hiep3 ; Do Le Hai, Thuy4
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
Comptes Rendus. Mathématique, Tome 361 (2023) no. G4, 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

Nguyen, Duc-Khanh 1 ; Nguyen Thi Ngoc, Giao 2 ; Dang Tuan, Hiep 3 ; Do Le Hai, Thuy 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
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Newton polytope of good symmetric polynomials},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {767--775},
     year = {2023},
     publisher = {Acad\'emie des sciences, Paris},
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     url = {https://www.numdam.org/articles/10.5802/crmath.430/}
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Nguyen, Duc-Khanh; Nguyen Thi Ngoc, Giao; Dang Tuan, Hiep; Do Le Hai, Thuy. Newton polytope of good symmetric polynomials. Comptes Rendus. Mathématique, Tome 361 (2023) no. G4, pp. 767-775. doi: 10.5802/crmath.430

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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