Flow Polytopes and the Space of Diagonal Harmonics

Liu, Ricky Ini ; Morales, Alejandro H. ; Mészáros, Karola

Canadian journal of mathematics, Tome 71 (2019) no. 6, pp. 1495-1521

Résumé

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.

Détail

DOI : 10.4153/CJM-2018-007-3

Mots-clés : flow polytope, threshold graph, diagonal harmonic, Tesler matrix

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