Flow Polytopes and the Space of Diagonal Harmonics

Abstract

AbstractA 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.