Capacity bounds on integral flows and the Kostant partition function
Abstract
The type 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.
Cite
@article{arxiv.2406.07838,
title = {Capacity bounds on integral flows and the Kostant partition function},
author = {Jonathan Leake and Alejandro H. Morales},
journal= {arXiv preprint arXiv:2406.07838},
year = {2024}
}
Comments
34 pages, 3 figures, and an appendix, v2. fixed typos and added a footnote about an announced improvement to Theorem 1.1 by Balashov, Bulavenko, and Molybog that we learned about after v1. was posted. v3. added a reference to the preprint of the aforementioned authors