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

  • Karola Mészáros
  • Alejandro H. MoralesEmail author
  • Jessica Striker
Article

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.

Keywords

Flow polytopes Alternating sign matrices Chan–Robbins–Yuen polytope Order polytopes Triangulations 

Mathematics Subject Classification

05A15 05A19 05A20 52B05 52A38 (Primary) 05C20 05C21 52B11 52B22 (Secondary) 

Notes

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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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsCornell UniversityIthacaUSA
  2. 2.School of MathematicsInstitute for Advanced StudyPrincetonUSA
  3. 3.Department of Mathematics and StatisticsUMassAmherstUSA
  4. 4.Department of MathematicsNorth Dakota State UniversityFargoUSA

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