Flow polytopes of partitions

TL;DR

This paper introduces a new family of flow polytopes associated with partitions, demonstrating that their limiting forms are products of scaled simplices and establishing their consistent combinatorial types, diverging from traditional analytic proof methods.

Contribution

It defines a novel family of flow polytopes for partitions and proves their limiting forms are products of scaled simplices, revealing their combinatorial invariance.

Findings

Limiting flow polytopes are products of scaled simplices.

All polytopes in a fixed family share the same combinatorial type.

Special case recovers known Tesler polytope results.

Abstract

Recent progress on flow polytopes indicates many interesting families with product formulas for their volume. These product formulas are all proved using analytic techniques. Our work breaks from this pattern. We define a family of closely related flow polytopes $\mathcal{F}_{(\lambda, {\bf a})}$ for each partition shape $\lambda$ and netflow vector ${\bf a}\in \mathbb{Z}^n_{> 0}$. In each such family, we prove that there is a polytope (the limiting one in a sense) which is a product of scaled simplices, explaining their product volumes. We also show that the combinatorial type of all polytopes in a fixed family $\mathcal{F}_{(\lambda, {\bf a})}$ is the same. When $\lambda$ is a staircase shape and ${\bf a}$ is the all ones vector the latter result specializes to a theorem of the first author with Morales and Rhoades, which shows that the combinatorial type of the Tesler polytope is a product of simplices.