Mathematics Colloquia and Seminars

Return to Colloquia & Seminar listing

Flow polytopes and the Kostant partition function are inherently related. This surprising connection has been studied with combinatorial methods by Postnikov and Stanley, and subsequently by Baldoni and Vergne using residue techniques. The combinatorial nature of flow polytopes and the partition function is showcased by the well-known Chan-Robbins- Yuen polytope, which is the flow polytope of the complete graph. Namely, an evaluation of the Kostant partition function is equal to the volume of this polytope, which in turn is the product of consecutive Catalan numbers by a theorem of Zeilberger. We use combinatorial techniques similar to those of Postnikov and Stanley to establish the relationship between volumes of flow polytopes associated to signed graphs and a variant of the Kostant partition function. We then use this relationship to study a generalization of the Chan-Robbins-Yuen polytope. This is joint work with Karola M\'esz\'aros.