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Triangulations of flow polytopes

Algebra & Discrete Mathematics

Speaker: Martha Yip, University of Kentucky
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Location: zoom
Start time: Mon, Feb 28 2022, 2:10PM

Flow polytopes are polytopes defined on flows for directed acyclic graphs. One method to obtain unimodular triangulations of these polytopes is through a framing of the graph. We present results on framed triangulations of two families of flow polytopes. 1) Given a lattice path $\nu$, we study a family of flow polytopes whose normalized volumes are $\nu$-Catalan numbers, and show that these polytopes possess interesting triangulations whose dual graphs are the $\nu$-Tamari lattice, and the principal order ideal in Young's lattice generated by $\nu$. 2) The class of graphs whose inner vertices have in- and out-degrees exactly 2 have ample framings. We show that top-dimensional simplices of an amply framed triangulation correspond to support $\tau$-tilting modules of an associated path algebra over a quiver. As a consequence, we conclude that amply-framed triangulations of these flow polytopes have dual graphs which inherit a poset structure coming from the support $\tau$-tilting modules.