Comptes Rendus
no. 8
Combinatorics
Newton polytopes and symmetric Grothendieck polynomials
[Polytopes de Newton et polynômes symétriques de Grothendieck]

Présenté par : the Editorial Board

Laura Escobar1 ; Alexander Yong1
1 Department of Mathematics, University of Illinois at Urbana–Champaign, 1409 W. Green Street, Urbana, IL 61801, USA
Comptes Rendus. Mathématique, Volume 355 (2017) no. 8, pp. 831-834.

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2017.07.003
Laura Escobar 1 ; Alexander Yong 1

1 Department of Mathematics, University of Illinois at Urbana–Champaign, 1409 W. Green Street, Urbana, IL 61801, USA 
@article{CRMATH_2017__355_8_831_0,
     author = {Laura Escobar and Alexander Yong},
     title = {Newton polytopes and symmetric {Grothendieck} polynomials},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {831--834},
     publisher = {Elsevier},
     volume = {355},
     number = {8},
     year = {2017},
     doi = {10.1016/j.crma.2017.07.003},
     language = {en},
}
TY  - JOUR
AU  - Laura Escobar
AU  - Alexander Yong
TI  - Newton polytopes and symmetric Grothendieck polynomials
JO  - Comptes Rendus. Mathématique
PY  - 2017
SP  - 831
EP  - 834
VL  - 355
IS  - 8
PB  - Elsevier
DO  - 10.1016/j.crma.2017.07.003
LA  - en
ID  - CRMATH_2017__355_8_831_0
ER  - 
%0 Journal Article
%A Laura Escobar
%A Alexander Yong
%T Newton polytopes and symmetric Grothendieck polynomials
%J Comptes Rendus. Mathématique
%D 2017
%P 831-834
%V 355
%N 8
%I Elsevier
%R 10.1016/j.crma.2017.07.003
%G en
%F CRMATH_2017__355_8_831_0
Laura Escobar; Alexander Yong. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.07.003/

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

[2] A. Lascoux; M.-P. Schützenberger 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] C. Lenart Combinatorial aspects of the K-theory of Grassmannians, Ann. Comb., Volume 4 (2000) no. 1, pp. 67-82

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

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

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

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

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

Cité par Sources :

Commentaires - Politique