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arXiv · 1610.08370

Flow polytopes and the space of diagonal harmonics

Ricky Ini Liu·Karola Mészáros·Alejandro H. Morales
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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 $(-n, 1, \dots, 1)$. 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.

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Ricky Ini Liu, Karola Mészáros, Alejandro H. Morales. 2016-10-26. Flow polytopes and the space of diagonal harmonics. https://doi.org/10.4153/cjm-2018-007-3

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