Volumes and Ehrhart polynomials of flow polytopes

TL;DR

This paper provides a geometric and combinatorial proof of formulas relating the volume and Ehrhart polynomials of flow polytopes, generalizing Lidskii's result to arbitrary graphs and revealing their underlying structure.

Contribution

It constructs canonical polytopal subdivisions of flow polytopes to prove generalized formulas for volume and Ehrhart polynomials, offering new geometric insights.

Findings

Canonical subdivisions reveal geometric structure of flow polytopes

Generalized Lidskii formulas for arbitrary graphs

Enumerative properties of flow polytopes derived from subdivisions

Abstract

The Lidskii formula for the type $A_n$ root system expresses the volume and Ehrhart polynomial of the flow polytope of the complete graph with nonnegative integer netflows in terms of Kostant partition functions. For every integer polytope the volume is the leading coefficient of the Ehrhart polynomial. The beauty of the Lidskii formula is the revelation that for these polytopes its Ehrhart polynomial function can be deduced from its volume function! Baldoni and Vergne generalized Lidskii's result for flow polytopes of arbitrary graphs $G$ and nonnegative integer netflows. While their formulas are combinatorial in nature, their proofs are based on residue computations. In this paper we construct canonical polytopal subdivisions of flow polytopes which we use to prove the Baldoni-Vergne-Lidskii formulas. In contrast with the original computational proof of these formulas, our proof reveal their geometry and combinatorics. We conclude by exhibiting enumerative properties of the Lidskii formulas via our canonical polytopal subdivisions.