On Flow Polytopes, nu-Associahedra, and the Subdivision Algebra

Author

Matias von Bell, University of KentuckyFollow

Author ORCID Identifier

https://orcid.org/0000-0003-4288-4329

Date Available

5-22-2022

Year of Publication

2022

Document Type

Doctoral Dissertation

Degree Name

Doctor of Philosophy (PhD)

College

Arts and Sciences

Department/School/Program

Mathematics

Advisor

Dr. Martha Yip

Co-Director of Graduate Studies

Dr. Benjamin Braun

Abstract

This dissertation studies the geometry and combinatorics related to a flow polytope F_car(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 F_car(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 F_car(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 F_car(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 F_car(ν) and a subpolytope U_I,J of a product of two simplices subdivisions of U_I,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)

https://doi.org/10.13023/etd.2022.184

Funding Information

This study was partially supported by the Max Steckler Fellowship in 2020, and the Cliff Swauger Jr. Summer Graduate Fellowship in 2021.

Recommended Citation

von Bell, Matias, "On Flow Polytopes, nu-Associahedra, and the Subdivision Algebra" (2022). Theses and Dissertations--Mathematics. 90.
https://uknowledge.uky.edu/math_etds/90