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On Flow Polytopes, Order Polytopes, and Certain Faces of the Alternating Sign Matrix Polytope

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Abstract

We study an alternating sign matrix analogue of the Chan–Robbins–Yuen polytope, which we call the ASM-CRY polytope. We show that this polytope has Catalan many vertices and its volume is equal to the number of standard Young tableaus of staircase shape; we also determine its Ehrhart polynomial. We achieve the previous by proving that the members of a family of faces of the alternating sign matrix polytope which includes ASM-CRY are both order and flow polytopes. Inspired by the above results, we relate three established triangulations of order and flow polytopes, namely Stanley’s triangulation of order polytopes, the Postnikov–Stanley triangulation of flow polytopes and the Danilov–Karzanov–Koshevoy triangulation of flow polytopes. We show that when a graph G is a planar graph, in which case the flow polytope \({{\mathcal {F}}}_G\) is also an order polytope, Stanley’s triangulation of this order polytope is one of the Danilov–Karzanov–Koshevoy triangulations of \({{\mathcal {F}}}_G\). Moreover, for a general graph G we show that the set of Danilov–Karzanov–Koshevoy triangulations of \({{\mathcal {F}}}_G\) equals the set of framed Postnikov–Stanley triangulations of \({{\mathcal {F}}}_G\). We also describe explicit bijections between the combinatorial objects labeling the simplices in the above triangulations.

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Notes

  1. See Definition 3.1 and the discussion in Sect. 3.1 for the relation of nonnegative integer flows with a given netflow vector to Kostant partition functions as well as Theorem 3.4.

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Acknowledgements

The authors are grateful to Alexander Postnikov for generously sharing his insights and questions. The authors are also grateful to the anonymus referee for numerous helpful comments and suggestions. AHM and JS would like to thank ICERM and the organizers of its Spring 2013 program in Automorphic Forms during which part of this work was done. The authors also thank the SageMath community [26] for developing and sharing their code by which some of this research was conducted.

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Authors and Affiliations

  1. Department of Mathematics, Cornell University, Ithaca, NY, 14853, USA

    Karola Mészáros

  2. School of Mathematics, Institute for Advanced Study, Princeton, NJ, 08540, USA

    Karola Mészáros

  3. Department of Mathematics and Statistics, UMass, Amherst, MA, 01003, USA

    Alejandro H. Morales

  4. Department of Mathematics, North Dakota State University, Fargo, ND, 58102, USA

    Jessica Striker

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  1. Karola Mészáros
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  2. Alejandro H. Morales
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  3. Jessica Striker
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Correspondence to Alejandro H. Morales.

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KM was partially supported by a National Science Foundation Grant (DMS 1501059) as well as by a von Neumann Fellowship at the IAS funded by the Fund for Mathematics and the Friends of the Institute for Advanced Study.

AHM was partially supported by a CRM-ISM Postdoctoral Fellowship and an AMS-Simons travel grant.

JS was partially supported by a National Security Agency Grant (H98230-15-1-0041), the North Dakota EPSCoR National Science Foundation Grant (IIA-1355466), the NDSU Advance FORWARD program sponsored by National Science Foundation grant (HRD-0811239), and a grant from the Simons Foundation/SFARI (527204, JS).

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Mészáros, K., Morales, A.H. & Striker, J. On Flow Polytopes, Order Polytopes, and Certain Faces of the Alternating Sign Matrix Polytope. Discrete Comput Geom 62, 128–163 (2019). https://doi.org/10.1007/s00454-019-00073-2

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