Capacity bounds on integral flows and the Kostant partition function

TL;DR

This paper investigates the asymptotic behavior of the Kostant partition function, providing improved bounds and settling existing conjectures, with implications for combinatorics and representation theory.

Contribution

It advances understanding of the Kostant partition function's asymptotics by improving bounds and confirming conjectures, using Lorentzian polynomials and flow polytope techniques.

Findings

Improved lower bounds on the Kostant partition function.

Settled conjectures of O'Neill and Yip.

Derived new two-sided bounds using flow polytope subdivisions.

Abstract

The type $A$ Kostant partition function is an important combinatorial object with various applications: it counts integer flows on the complete directed graph, computes Hilbert series of spaces of diagonal harmonics, and can be used to compute weight and tensor product multiplicities of representations. In this paper we study asymptotics of the Kostant partition function, improving on various previously known lower bounds and settling conjectures of O'Neill and Yip. Our methods build upon recent results and techniques of Br\"and\'en-Leake-Pak, who used Lorentzian polynomials and Gurvits' capacity method to bound the number of lattice points of transportation and flow polytopes. Finally, we also give new two-sided bounds using the Lidskii formulas from subdivisions of flow polytopes.