Comptes Rendus
Combinatoire
Refinements and Symmetries of the Morris identity for volumes of flow polytopes
Alejandro H. Morales1   ; William Shi2
1 Department of Mathematics and Statistics, UMass, Amherst, MA 01003, USA.
2 Northview High School, Johns Creek, GA 30097, USA.
Comptes Rendus. Mathématique, Volume 359 (2021) no. 7, pp. 823-851.

Flow polytopes are an important class of polytopes in combinatorics whose lattice points and volumes have interesting properties and relations. The Chan–Robbins–Yuen (CRY) polytope is a flow polytope with normalized volume equal to the product of consecutive Catalan numbers. Zeilberger proved this by evaluating the Morris constant term identity, but no combinatorial proof is known. There is a refinement of this formula that splits the largest Catalan number into Narayana numbers, which Mészáros gave an interpretation as the volume of a collection of flow polytopes. We introduce a new refinement of the Morris identity with combinatorial interpretations both in terms of lattice points and volumes of flow polytopes. Our results generalize Mészáros’s construction and a recent flow polytope interpretation of the Morris identity by Corteel–Kim–Mészáros. We prove the product formula of our refinement following the strategy of the Baldoni–Vergne proof of the Morris identity. Lastly, we study a symmetry of the Morris identity bijectively using the Danilov–Karzanov–Koshevoy triangulation of flow polytopes and a bijection of Mészáros–Morales–Striker.

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DOI : 10.5802/crmath.218
Alejandro H. Morales 1 ; William Shi 2

1 Department of Mathematics and Statistics, UMass, Amherst, MA 01003, USA.
2 Northview High School, Johns Creek, GA 30097, USA.
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Alejandro H. Morales; William Shi. Refinements and Symmetries of the Morris identity for volumes of flow polytopes. Comptes Rendus. Mathématique, Volume 359 (2021) no. 7, pp. 823-851. doi : 10.5802/crmath.218. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.218/

[1] Kazuhiko Aomoto Jacobi polynomials associated with Selberg integrals, SIAM J. Math. Anal., Volume 18 (1987) no. 2, pp. 545-549 | DOI | MR | Zbl

[2] Welleda Baldoni; Michèle Vergne Kostant partitions functions and flow polytopes, Transform. Groups, Volume 13 (2008) no. 3-4, pp. 447-469 | DOI | MR | Zbl

[3] Velleda Baldoni-Silva; Michèle Vergne Morris identities and the total residue for a system of type A r , Noncommutative harmonic analysis. In honor of Jacques Carmona (Patrick Delorme, ed.) (Progress in Mathematics), Volume 220, Springer, 2004, pp. 1-19 | DOI | MR | Zbl

[4] Matias von Bell; Rafael S. González D’Léon; Francisco A. Mayorga Cetina; Martha Yip A unifying framework for the nu-Tamari lattice and principal order ideals in Young’s lattice (2021) (https://arxiv.org/abs/2101.10425)

[5] Carolina Benedetti; Rafael S. González D’Léon; Christopher R. H. Hanusa; Pamela E. Harris; Apoorva Khare; Alejandro H. Morales; Martha Yip A combinatorial model for computing volumes of flow polytopes, Trans. Am. Math. Soc., Volume 372 (2019) no. 5, pp. 3369-3404 | DOI | MR | Zbl

[6] Clara S. Chan; David P. Robbins; David S. Yuen On the volume of a certain polytope, Exp. Math., Volume 9 (2000) no. 1, pp. 91-99 | DOI | MR | Zbl

[7] Sylvie Corteel; Jang Soo Kim; Karola Mészáros Flow polytopes with Catalan volumes, C. R. Math. Acad. Sci. Paris, Volume 355 (2017) no. 3, pp. 248-259 | DOI | MR | Zbl

[8] Vladimir I. Danilov; Alexander V. Karzanov; Gleb A. Koshevoy Coherent fans in the space of flows in framed graphs, Proceedings of the 24th international conference on formal power series and algebraic combinatorics, FPSAC 2012, Nagoya, Japan, July 30–August 3, 2012 (Discrete Mathematics & Theoretical Computer Science. Proceedings), The Association. Discrete Mathematics & Theoretical Computer Science (DMTCS), Nancy, 2012, pp. 481-490 | Zbl

[9] Jesús A. De Loera; Jörg Rambau; Francisco Santos Triangulations: structures for algorithms and applications, Algorithms and Computation in Mathematics, 25, Springer, 2010 | Zbl

[10] Laura Escobar; Karola Mészáros Subword complexes via triangulations of root polytopes, Algebr. Comb., Volume 1 (2018) no. 3, pp. 395-414 | Numdam | MR | Zbl

[11] Peter J. Forrester; S. Ole Warnaar The importance of the selberg integral, Bull. Am. Math. Soc., Volume 45 (2008) no. 4, pp. 489-534 | DOI | MR | Zbl

[12] Pavel Galashin; Gleb Nenashev; Alexander Postnikov Trianguloids and triangulations of root polytopes (https://arxiv.org/abs/1803.06239) (2018)

[13] Alexander Haupt Combinatorial proof of Selberg’s integral formula (2020) (https://arxiv.org/abs/2005.08731)

[14] Lutz Hille Quivers, cones and polytopes, Linear Algebra Appl., Volume 365 (2003), pp. 215-237 | DOI | MR | Zbl

[15] James E. Humphreys Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, 9, Springer, 1980 | Zbl

[16] Jihyeug Jang; Jang Soo Kim Volumes of flow polytopes related to caracol graphs, Electron. J. Comb., Volume 27 (2020) no. 4, P4.21 | MR | Zbl

[17] Kabir Kapoor; Karola Mészáros; Linus Setiabrata Counting integer points of flow polytopes (2019) (https://arxiv.org/abs/1906.05592) | Zbl

[18] Ricky I. Liu; Karola Mészáros; Avery St. Dizier Gelfand–Tsetlin polytopes: a story of flow and order polytopes, SIAM J. Discrete Math., Volume 33 (2019) no. 4, pp. 2394-2415 | MR | Zbl

[19] Karola Mészáros Product formulas for volumes of flow polytopes, Proc. Am. Math. Soc., Volume 143 (2015) no. 3, pp. 937-954 | DOI | MR | Zbl

[20] Karola Mészáros; Alejandro H. Morales Flow polytopes of signed graphs and the Kostant partition function, Int. Math. Res. Not., Volume 2015 (2015) no. 3, pp. 830-871 | DOI | MR | Zbl

[21] Karola Mészáros; Alejandro H. Morales; Brendon Rhoades The polytope of Tesler matrices, Sel. Math., New Ser., Volume 23 (2017) no. 1, p. 425-414 | DOI | MR | Zbl

[22] Karola Mészáros; Avery St. Dizier From generalized permutahedra to Grothendieck polynomials via flow polytopes, Algebr. Comb., Volume 3 (2020) no. 5, pp. 1197-1229 | MR | Zbl

[23] Karola Mészáros; Jessica Striker On flow polytopes, order polytopes, and certain faces of the alternating sign matrix polytope, Discrete Comput. Geom., Volume 62 (2019) no. 1, pp. 128-163 | DOI | MR | Zbl

[24] Alejandro H. Morales; Igor Pak; Greta Panova Asymptotics of principal evaluations of Schubert polynomials for layered permutations, Proc. Am. Math. Soc., Volume 147 (2019) no. 4, pp. 1377-1389 | DOI | MR | Zbl

[25] Alejandro H. Morales; Igor Pak; Greta Panova Hook formulas for skew shapes. III: multivariate and product formulas, Algebr. Comb., Volume 2 (2019) no. 5, pp. 815-861 | Numdam | MR | Zbl

[26] Walter G. Morris Constant term identities for nite and affine root systems: conjectures and theorems, Ph. D. Thesis, University of Wisconsin-Madison, USA (1982)

[27] Alexander Postnikov Permutohedra, associahedra, and beyond, Int. Math. Res. Not., Volume 2009 (2009) no. 6, pp. 1026-1106 | DOI | MR | Zbl

[28] Robert A. Proctor New symmetric plane partition identities from invariant theory work of De Concini and Procesi, Eur. J. Comb., Volume 11 (1990) no. 3, pp. 289-300 | DOI | MR | Zbl

[29] Alexander Schrijver Combinatorial optimization: polyhedra and efficiency (3 volumes), Algorithms and Combinatorics, 24, Springer, 2003 | Zbl

[30] Richard P. Stanley Acyclic flow polytopes and Kostant’s partition function (2000) (Slides in http://math.mit.edu/~rstan/transparencies/kostant.pdf)

[31] Richard P. Stanley Catalan numbers, Cambridge University Press, 2015 | Zbl

[32] Martha Yip A Fuss–Catalan variation of the caracol flow polytope (2019) (https://arxiv.org/abs/1910.10060)

[33] Doron Zeilberger Email to Richard Stanley and David Robbins, 1999 (https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/ToRobbins)

[34] Doron Zeilberger Proof of a conjecture of Chan, Robbins, and Yuen, Electron. Trans. Numer. Anal., Volume 9 (1999), pp. 147-148 | MR | Zbl

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