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Comptes Rendus Mathématique
Volume 355, n° 8
pages 831-834 (août 2017)
Doi : 10.1016/j.crma.2017.07.003
Received : 25 May 2017 ;  accepted : 5 July 2017
Newton polytopes and symmetric Grothendieck polynomials
Polytopes de Newton et polynômes symétriques de Grothendieck
 

Laura Escobar , Alexander Yong
 Department of Mathematics, University of Illinois at Urbana–Champaign, 1409 W. Green Street, Urbana, IL 61801, USA 

Abstract

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.

The full text of this article is available in PDF format.
Résumé

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.

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