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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.

MSC:
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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