Matroidal Combinatorics Intersecting with Algebra and Geometry

Abstract

In this dissertation, we study different perspectives of matroidal combinatorics andanswer the following questions: 1. Canonical resolution of lattice ideals: we write down a canonical minimal free resolu- tion of any lattice ideal. We a give combinatorial description to the resolution using sylvan morphisms constructed in the monomial ideal case. 2. Shellability of posets of lattice path matroids Pn: we show that Pn admits an EL- shelling and compute the Möbius function of some intervals in this poset. We also show that Pn admits Whitney dual labelling. 3. Log-concave property in Schubert calculus: we study the log-concave property that appears in Schubert calculus. We show that multidegree polynomials are covolume and normalized Macaulay inverse polynomials are Lorentzian. In particular, we prove that double Richardson polynomials are covolume and hence the coefficients form a log-concave sequence along type A directions. We also give an alternative proof to the log-concavity of h-vectors of representable matroids using matroid Schubert varieties. Parts of the thesis are based on joint works with Carolina Benedetti-Velásquez, Yairon Cid-Ruiz, Anton Dochtermann, Kolja Knauer, Jacob Matherne, Ezra Miller and Erika Ordog.