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Flow Polytopes and the Space of Diagonal Harmonics

  • Ricky Ini Liu (a1), Alejandro H. Morales (a2) and Karola Mészáros (a3)

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.

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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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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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