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Flow polytopes of signed graphs and the Kostant partition function. (English) Zbl 1307.05097
Summary: 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 W. Baldoni and M. Vergne [Transform. Groups 13, No. 3–4, 447–469 (2008; Zbl 1200.52008)] using techniques of residues. In contrast with their approach, we provide entirely combinatorial proofs inspired by the work of A. Postnikov [Personal communication (2010)] and R. P. Stanley [“Acyclic flow polytopes and Kostant’s partition function”, Conference transparencies, 12 p. (2000), http://math.mit.edu/~rstan/trans.html] on flow polytopes. As an interesting special family of flow polytopes, we study the Chan-Robbins-Yuen (CRY) polytopes. Motivated by the volume formula \(\prod^{n-2}_{k=1}\mathrm{Cat}(k)\) for the type \(A_n\) version, where \(\mathrm{Cat}(k)\) is the \(k\)th Catalan number, we introduce type \(C_{n+1}\) and \(D_{n+1}\) CRY polytopes along with intriguing conjectures about their volumes.

05C22 Signed and weighted graphs
05C21 Flows in graphs
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
05C85 Graph algorithms (graph-theoretic aspects)
52B11 \(n\)-dimensional polytopes
51M25 Length, area and volume in real or complex geometry
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