Algebra and Combinatorics Seminar

Date: January 19, 2018

Time: 3:00PM - 4:00PM

Location: BLOC 628

Speaker: Anton Dochtermann, Texas State University

Title: Coparking functions and h-vectors of matroids

Abstract: The h-vector of a simplicial complex X is a well-studied invariant with connections to algebraic aspects of its Stanley-Reisner ring. When X is the independence complex of a matroid Stanley has conjectured that its h-vector is a ‘pure O-sequence’, i.e. the degree sequence of a monomial order ideal where all maximal elements have the same degree. The conjecture has inspired a good deal of research and is proven for some classes of matroids, but is open in general. Merino has established the conjecture for the case that X is a cographical matroid by relating the h-vector to properties of chip-firing and `G-parking functions' on the underlying graph G. We introduce and study the notion of a ‘coparking’ function on a graph (and more general matroids) inspired by a dual version of chip-firing. As an application we establish Stanley’s conjecture for certain classes of binary matroids that admit a well-behaved `circuit covering'. Joint work with Kolja Knauer