Combinatorics
Newton polytopes and symmetric Grothendieck polynomials
Comptes Rendus. Mathématique, Volume 355 (2017) no. 8, pp. 831-834.

Symmetric Grothendieck polynomials are inhomogeneous versions of Schur polynomials that arise in combinatorial K-theory. A polynomial has saturated Newton polytope (SNP) if every lattice point in the polytope is an exponent vector. We show that the Newton polytopes of these Grothendieck polynomials and their homogeneous components have SNP. Moreover, the Newton polytope of each homogeneous component is a permutahedron. This addresses recent conjectures of C. Monical–N. Tokcan–A. Yong and of A. Fink–K. Mészáros–A. St. Dizier in this special case.

Les polynômes symétriques de Grothendieck sont des versions inhomogènes des polynômes de Schur qui apparaissent dans la K-théorie combinatoire. Un polynôme a un polytope de Newton saturé (SNP) si chaque point entier dans le polytope est un vecteur d'exposant. Nous montrons que les polytopes de Newton de ces polynômes de Grothendieck et leurs composants homogènes ont un SNP. En outre, le polytope de Newton de chaque composant homogène est un permutoèdre. Cela concerne les récentes conjectures de C. Monical–N. Tokcan–A. Yong et de A. Fink–K. Mészáros–A. St. Dizier dans ce cas spécial.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2017.07.003
Escobar, Laura 1; Yong, Alexander 1

1 Department of Mathematics, University of Illinois at Urbana–Champaign, 1409 W. Green Street, Urbana, IL 61801, USA
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Escobar, Laura; Yong, Alexander. Newton polytopes and symmetric Grothendieck polynomials. Comptes Rendus. Mathématique, Volume 355 (2017) no. 8, pp. 831-834. doi : 10.1016/j.crma.2017.07.003. http://www.numdam.org/articles/10.1016/j.crma.2017.07.003/

[1] Buch, A. A Littlewood–Richardson rule for the K-theory of Grassmannians, Acta Math., Volume 189 (2002) no. 1, pp. 37-78

[2] Lascoux, A.; Schützenberger, M.-P. Structure de Hopf de l'anneau de cohomologie et de l'anneau de Grothendieck d'une variété de drapeaux, C. R. Acad. Sci. Paris, Sér. I, Volume 295 (1982) no. 11, pp. 629-633

[3] Lenart, C. Combinatorial aspects of the K-theory of Grassmannians, Ann. Comb., Volume 4 (2000) no. 1, pp. 67-82

[4] Marshall, A.W.; Olkin, I.; Arnold, B.C. Inequalities: Theory of Majorization and Its Applications, Springer Series in Statistics, Springer, New York, 2011

[5] Mészáros, K.; Dizier, A. St. Generalized permutahedra to Grothendieck polynomials via flow polytopes (preprint) | arXiv

[6] Monical, C. Set-valued skyline fillings (preprint) | arXiv

[7] Monical, C.; Tokcan, N.; Yong, A. Newton polytopes in algebraic combinatorics (preprint) | arXiv

[8] Rado, R. An inequality, J. Lond. Math. Soc., Volume 27 (1952), pp. 1-6

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