Enumerative and algebraic aspects of matroids and hyperplane arrangements

Abstract

This thesis consists of three projects on the enumerative and algebraic properties of matroids and hyperplane arrangements. In particular, a central object of study is the Tutte polynomial, which stores much of the enumerative information of these objects. The first project is the study of the Tutte polynomial of an arrangement and, more generally, of a semimatroid. It has two components: an enumerative one and a matroid-theoretic one. We start by considering purely enumerative questions about the Tutte polynomial of a hyperplane arrangement. We introduce a new method for computing it, which generalizes several known results. We apply our method to several specific arrangements, thus relating the computation of Tutte polynomials to problems in enumerative combinatorics. As a consequence, we obtain several new results about classical combinatorial objects such as labeled trees, Dyck paths, semiorders and alternating trees. We then address matroid-theoretic aspects of arrangements and their Tutte polynomials. We start by defining semimatroids, a class of objects which abstracts the dependence properties of an affine hyperplane arrangement. After discussing these objects in detail, we define and investigate their Tutte polynomial. In particular, we prove that it is the universal Tutte-Grothendieck invariant for semimatroids, and we give a combinatorial interpretation for its non-negative coefficients. The second project is the beginning of an attempt to study the Tutte polynomial from an algebraic point of view.

(cont.) Given a matroid representable over a field of characteristic zero, we construct a graded algebra whose Hilbert-Poincar6 series is a simple evaluation of the Tutte polynomial of the matroid. This construction is joint work with Alex Postnikov. The third project involves a class of matroids with very rich enumerative properties. We show how the set of Dyck paths of length 2n naturally gives rise to a matroid, which we call the Catalan matroid Cn. We describe this matroid in detail; among several other results, we show that Cn is self-dual, it is representable over the rationals but not over finite fields Fq with q < n - 2, and it has a nice Tutte polynomial. We then introduce a more general family of matroids, which we call shifted matroids. They are precisely the matroids whose independence complex is a shifted simplicial complex.