Author ORCID Identifier
Year of Publication
Doctor of Philosophy (PhD)
Arts and Sciences
Dr. Martha Yip
Dr. Benjamin Braun
This dissertation studies the geometry and combinatorics related to a flow polytope Fcar(ν) constructed from a lattice path ν, whose volume is given by the ν-Catalan numbers. It begins with a study of the ν-associahedron introduced by Ceballos, Padrol, and Sarmiento in 2019, but from the perspective of Schröder combinatorics. Some classical results for Schröder paths are extended to the ν-setting, and insights into the geometry of the ν-associahedron are obtained by describing its face poset with two ν-Schröder objects. The ν-associahedron is then shown to be dual to a framed triangulation of Fcar(ν), which is a geometric realization of the ν-Tamari complex. The dual graph of this triangulation is the Hasse diagram of the ν-Tamari lattice due to Préville-Ratelle and Viennot. The dual graph of a second framed triangulation of Fcar(ν) is shown to be the Hasse diagram of a principal order ideal of Young’s lattice generated by ν, and is used to show that the h∗-vector of Fcar(ν) is given by ν-Narayana numbers. This perspective serves to unify these two important lattices associated with ν-Dyck paths through framed triangulations of a flow polytope. Via an integral equivalence between Fcar(ν) and a subpolytope UI,J of a product of two simplices subdivisions of UI,J are shown to be obtainable with Mészáros’ subdivision algebra, which answers a question of Ceballos, Padrol, and Sarmiento. Building on this result, the subdivision algebra is extended to encode subdivisions of a product of two simplices, giving a new tool for their future study.
Digital Object Identifier (DOI)
This study was partially supported by the Max Steckler Fellowship in 2020, and the Cliff Swauger Jr. Summer Graduate Fellowship in 2021.
von Bell, Matias, "On Flow Polytopes, nu-Associahedra, and the Subdivision Algebra" (2022). Theses and Dissertations--Mathematics. 90.