Flow polytopes of signed graphs and the Kostant partition function

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.

29 pages, 13 figures

References in corpus (1)

• Product formulas for volumes of flow polytopes

Cited by in corpus (12)

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• A combinatorial model for computing volumes of flow polytopes
• Flow-oriented perturbation theory
• h-Polynomials of Reduction Trees
• h-Polynomials via Reduced Forms
• Kostant's partition function and magic multiplex juggling sequences
• Flow polytopes with Catalan volumes
• The $q$-analog of Kostant's partition function and the highest root of the classical Lie algebras
• Pipe dream complexes and triangulations of root polytopes belong together
• Calculating Greene's function via root polytopes and subdivision algebras
• Refinements and Symmetries of the Morris identity for volumes of flow polytopes
• Volumes of generalized Chan-Robbins-Yuen polytopes