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Flow Polytopes and the Space of Diagonal Harmonics

Part of: Graph theory Algebraic combinatorics Polytopes and polyhedra

Published online by Cambridge University Press:  07 January 2019

Ricky Ini Liu ,
Alejandro H. Morales  and
Karola Mészáros
Ricky Ini Liu
Affiliation:
Department of Mathematics, North Carolina State University, Box 8205, Raleigh, NC 27695, USA Email: riliu@ncsu.edu
Alejandro H. Morales
Affiliation:
Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003, USA Email: ahmorales@math.umass.edu
Karola Mészáros
Affiliation:
Department of Mathematics, Cornell University, 212 Garden Ave., Ithaca, NY 14853, USA Email: karola@math.cornell.edu
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Abstract

A result of Haglund implies that the $(q,t)$-bigraded Hilbert series of the space of diagonal harmonics is a $(q,t)$-Ehrhart function of the flow polytope of a complete graph with netflow vector $(-n,1,\ldots ,1)$. We study the $(q,t)$-Ehrhart functions of flow polytopes of threshold graphs with arbitrary netflow vectors. Our results generalize previously known specializations of the mentioned bigraded Hilbert series at $t=1$, $0$, and $q^{-1}$. As a corollary to our results, we obtain a proof of a conjecture of Armstrong, Garsia, Haglund, Rhoades, and Sagan about the $(q,q^{-1})$-Ehrhart function of the flow polytope of a complete graph with an arbitrary netflow vector.

Keywords

flow polytopethreshold graphdiagonal harmonicTesler matrix

MSC classification

Primary: 05E10: Combinatorial aspects of representation theory
Secondary: 05C21: Flows in graphs 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry)
Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 6 , December 2019 , pp. 1495 - 1521
Copyright
© Canadian Mathematical Society 2018 

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Footnotes

Mészáros was partially supported by a National Science Foundation Grant (DMS 1501059). Morales was partially supported by an AMS-Simons travel grant.

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