Math and Stats Colloquium: The valuation polytope of the zig-zag poset

Friday, May 10, 2024, 4–5 p.m.

Anastasia Chavez - Saint Mary's College of California

Title: The valuation polytope of the zig-zag poset

Abstract: The summer 2021 Latinx Mathematical Research Community (LMRC) served as a catalyst for several research projects in various areas of mathematics. This talk will introduce the research of one such project. Geissinger defined the valuation polytope as the set of all [0,1]-valuations on a finite distributive lattice. We study the valuation polytope, V(Z_n), arising from the height 2 up-down poset on n elements, referred to as the zig-zag poset Z_n. Dobbertin showed that the valuation polytope of any poset can be described as the convex hull of vertices characterized by all the chains of that poset. It follows that the dimension of the valuation polytope is the number of elements of the corresponding poset. We discuss our combinatorial results of V(Z_n) such as its normalized volume, the existence of a unimodular triangulation, and facet enumeration. Additionally, we discuss current work towards describing the f-vector and underlying matroidal description. This is joint work with Federico Ardila, Anastasia Chavez, Jessica De Silva, Pamela E. Harris, Jose Luis Herrera Bravo, and Andrées R. Vindas-Meléndez.

Mathematics and Statistics Colloquium Series