Comptes Rendus
no. 3
Combinatorics
Flow polytopes with Catalan volumes
[Polytopes de flot avec volumes de Catalan]

Présenté par : Michèle Vergne

Sylvie Corteel1 ; Jang Soo Kim2 ; Karola Mészáros3
1 IRIF, CNRS et Université Paris-Diderot, 75205 Paris cedex 13, France
2 Sungkyunkwan University, 2066 Seobu-ro, Jangan-gu, Suwon, Gyeonggi-do 16419, South Korea
3 Department of Mathematics, Cornell University, Ithaca NY 14853, USA
Comptes Rendus. Mathématique, Volume 355 (2017) no. 3, pp. 248-259.

Le polytope de Chan–Robbins–Yuen peut être considéré comme le polytope de flot du graphe complet avec vecteur de flot (1,0,…,0,−1). Le volume normalisé du polytope de Chan–Robbins–Yuen est égal au produit de nombres de Catalan consécutifs, mais il n'existe pas de preuve combinatoire de ce fait. Nous considérons une extension naturelle de ce polytope, à savoir le polytope de flot du graphe complet avec vecteur de flot (1,1,0,…,0,−2). Nous montrons que le volume de ce polytope est une certaine puissance de 2 fois le produit de nombres de Catalan consécutifs. Notre preuve utilise des identités de termes constants et approfondit encore le mystère combinatoire de la raison pour laquelle ces nombres apparaissent. De plus, nous introduisons deux familles de polytopes de flot dont les volumes sont donnés par des formules produits.

The Chan–Robbins–Yuen polytope can be thought of as the flow polytope of the complete graph with netflow vector (1,0,…,0,−1). The normalized volume of the Chan–Robbins–Yuen polytope equals the product of consecutive Catalan numbers, yet there is no combinatorial proof of this fact. We consider a natural generalization of this polytope, namely, the flow polytope of the complete graph with netflow vector (1,1,0,…,0,−2). We show that the volume of this polytope is a certain power of 2 times the product of consecutive Catalan numbers. Our proof uses constant-term identities and further deepens the combinatorial mystery of why these numbers appear. In addition, we introduce two more families of flow polytopes whose volumes are given by product formulas.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2017.01.007
Sylvie Corteel 1 ; Jang Soo Kim 2 ; Karola Mészáros 3

1 IRIF, CNRS et Université Paris-Diderot, 75205 Paris cedex 13, France
2 Sungkyunkwan University, 2066 Seobu-ro, Jangan-gu, Suwon, Gyeonggi-do 16419, South Korea
3 Department of Mathematics, Cornell University, Ithaca NY 14853, USA 
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     title = {Flow polytopes with {Catalan} volumes},
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     pages = {248--259},
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Sylvie Corteel; Jang Soo Kim; Karola Mészáros. Flow polytopes with Catalan volumes. Comptes Rendus. Mathématique, Volume 355 (2017) no. 3, pp. 248-259. doi : 10.1016/j.crma.2017.01.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.01.007/

[1] W. Baldoni; M. Vergne Kostant partitions functions and flow polytopes, Transform. Groups, Volume 13 (2008) no. 3–4, pp. 447-469

[2] C.S. Chan; D.P. Robbins On the volume of the polytope of doubly stochastic matrices, Exp. Math., Volume 8 (1999) no. 3, pp. 291-300

[3] C.S. Chan; D.P. Robbins; D.S. Yuen On the volume of a certain polytope, Exp. Math., Volume 9 (2000) no. 1, pp. 91-99

[4] L. Escobar; K. Mészáros Subword complexes via triangulations of root polytopes, 2015 | arXiv

[5] L. Escobar; K. Mészáros Toric matrix Schubert varieties and their polytopes, Proc. Amer. Math. Soc., Volume 144 (2016) no. 12, pp. 5081-5096

[6] L. Hille, Quivers, cones and polytopes, Linear Algebra Appl. (365), 215–237.

[7] R.I. Liu; K. Mészáros; A.H. Morales Flow polytopes and the space of diagonal harmonics, 2016 | arXiv

[8] K. Mészáros Product formulas for volumes of flow polytopes, Proc. Amer. Math. Soc. (2015) no. 3, pp. 937-954

[9] K. Mészáros Pipe dream complexes and triangulations of root polytopes belong together, SIAM J. Discrete Math., Volume 30 (2016) no. 1, pp. 100-111

[10] K. Mészáros; A.H. Morales Flow polytopes of signed graphs and the Kostant partition function, Int. Math. Res. Not. (2015) no. 3, pp. 830-871

[11] K. Mészáros; A.H. Morales; B. Rhoades The polytope of Tesler matrices, Sel. Math., Volume 23 (2017) no. 1, pp. 425-454

[12] W.G. Morris Constant Term Identities for Finite and Affine Root Systems: Conjectures and Theorems, University of Wisconsin-Madison, 1982 (PhD Thesis)

[13] D. Zeilberger Proof of a conjecture of Chan, Robbins, and Yuen, Electron. Trans. Numer. Anal., Volume 9 (1999), pp. 147-148

[14] D. Zeilberger Sketch of a proof of an intriguing conjecture of Karola Meszaros and Alejandro Morales regarding the volume of the Dn analog of the Chan–Robbins–Yuen polytope (or: the Morris–Selberg constant term identity strikes again!), 2014 | arXiv

Cité par Sources :

☆ Corteel is partially supported by the project Emergences “Combinatoire à Paris”. Kim is partially supported by National Research Foundation of Korea (NRF) grants (NRF-2016R1D1A1A09917506) and (NRF-2016R1A5A1008055). Mészáros is partially supported by a National Science Foundation Grant (DMS 1501059).

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