Comptes Rendus
no. 6
Combinatorics
Symmetries on plabic graphs and associated polytopes
[Symétries dans les graphes plan bicolores et les polytopes associés]

Présenté par : Michele Vergne

Xin Fang1 ; Ghislain Fourier2
1 University of Cologne, Mathematical Institute, Weyertal 86–90, 50931 Cologne, Germany
2 Leibniz Universität Hannover, Institute for Algebra, Number Theory and Discrete Mathematics, Welfengarten 1, 30167 Hannover, Germany
Comptes Rendus. Mathématique, Volume 356 (2018) no. 6, pp. 581-585.

Nous expliquons, pour les variétés grasmanniennes, comment la dualité entre les polytopes de Gelfand–Tsetlin et les polytopes de Feigin–Fourier–Littelman–Vinberg émerge dans différentes structures positives.

For Grassmann varieties, we explain how the duality between the Gelfand–Tsetlin polytopes and the Feigin–Fourier–Littelmann–Vinberg polytopes arises from different positive structures.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2018.05.003
Xin Fang 1 ; Ghislain Fourier 2

1 University of Cologne, Mathematical Institute, Weyertal 86–90, 50931 Cologne, Germany
2 Leibniz Universität Hannover, Institute for Algebra, Number Theory and Discrete Mathematics, Welfengarten 1, 30167 Hannover, Germany 
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Xin Fang; Ghislain Fourier. Symmetries on plabic graphs and associated polytopes. Comptes Rendus. Mathématique, Volume 356 (2018) no. 6, pp. 581-585. doi : 10.1016/j.crma.2018.05.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.05.003/

[1] F. Ardila; T. Bliem; D. Salazar Gelfand–Tsetlin polytopes and Feigin–Fourier–Littelmann–Vinberg polytopes as marked poset polytopes, J. Comb. Theory, Ser. A, Volume 118 (2011) no. 8, pp. 2454-2462

[2] X. Fang Polytopes arising from mirror plabic graphs, Oberwolfach Rep., Volume 13 (2016) no. 1, pp. 626-628

[3] E. Feigin; G. Fourier; P. Littelmann PBW filtration and bases for irreducible modules in type An, Transform. Groups, Volume 16 (2011) no. 1, pp. 71-89

[4] I.M. Gelfand; M.L. Tsetlin Finite-dimensional representations of the group of unimodular matrices, Dokl. Akad. Nauk SSSR (N.S.), Volume 71 (1950), pp. 825-828

[5] K. Kaveh; A.G. Khovanskii Newton–Okounkov bodies, semigroups of integral points, graded algebras and intersection theory, Ann. of Math. (2), Volume 176 (2012) no. 2, pp. 925-978

[6] B. Keller On cluster theory and quantum dilogarithm identities, Representations of Algebras and Related Topics, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, Switzerland, 2011, pp. 85-116

[7] R.K. Lazarsfeld; M. Mustaţă Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér. (4), Volume 42 (2009), pp. 783-835

[8] A. Postnikov Total positivity, Grassmannians, and networks | arXiv

[9] A. Postnikov; D. Speyer; L. Williams Matching polytopes, toric geometry, and the non-negative part of the Grassmannian, J. Algebraic Comb., Volume 30 (2009) no. 2, pp. 173-191

[10] K. Rietsch; L. Williams Newton–Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians, 2017 (preprint) | arXiv

[11] R. Stanley Two poset polytopes, Discrete Comput. Geom., Volume 1 (1986) no. 1, pp. 9-23

[12] K. Talaska A formula for Plücker coordinates associated with a planar network, Int. Math. Res. Not. (2008) (rnn-081)

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