Flow polytopes and the space of diagonal harmonics

TL;DR

This paper explores the $(q,t)$-Ehrhart functions of flow polytopes for threshold graphs, generalizing known results and proving a conjecture related to the Hilbert series of diagonal harmonics.

Contribution

It extends the study of $(q,t)$-Ehrhart functions to threshold graph flow polytopes with arbitrary netflow vectors, generalizing previous special cases and proving a key conjecture.

Findings

Generalized $(q,t)$-Ehrhart functions for threshold graph flow polytopes.

Connected these functions to the Hilbert series of diagonal harmonics.

Proved a conjecture about the $(q, q^{-1})$-Ehrhart function for complete graphs.

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.