Flow polytopes of signed graphs and the Kostant partition function

TL;DR

This paper explores the connection between flow polytope volumes of signed graphs and the Kostant partition function, offering combinatorial proofs and studying special cases like Chan-Robbins-Yuen polytopes with conjectures on their properties.

Contribution

It provides a combinatorial approach to relate flow polytope volumes of signed graphs to the Kostant partition function, extending known results to new types and conjecturing properties of related polytopes.

Findings

Established combinatorial proofs for the relationship between flow polytope volumes and the Kostant partition function.

Studied the Chan-Robbins-Yuen polytopes for types C and D, proposing conjectures on their volume formulas.

Abstract

We establish the relationship between volumes of flow polytopes associated to signed graphs and the Kostant partition function. A special case of this relationship, namely, when the graphs are signless, has been studied in detail by Baldoni and Vergne using techniques of residues. In contrast with their approach, we provide entirely combinatorial proofs inspired by the work of Postnikov and Stanley on flow polytopes. As a fascinating special family of flow polytopes, we study the Chan-Robbins-Yuen polytopes. Motivated by the beautiful volume formula $\prod_{k=1}^{n-2} Cat(k)$ for the type $A_n$ version, where $Cat(k)$ is the $k$th Catalan number, we introduce type $C_{n+1}$ and $D_{n+1}$ Chan-Robbins-Yuen polytopes along with intriguing conjectures pertaining to their properties.