Flow Polytopes of Signed Graphs and the Kostant Partition Function

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 an interesting special family of flow polytopes, we study the Chan–Robbins–Yuen (CRY) polytopes. Motivated by the volume formula for the type An version, where Cat(k) is the kth Catalan number, we introduce type Cn+1 and Dn+1 CRY polytopes along with intriguing conjectures about their volumes.