Enumerative and algebraic aspects of matroids and hyperplane arrangements

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.

Department dc:contributor.department

Massachusetts Institute of Technology. Dept. of Mathematics.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2003

Author dc:creator

• Ardila, Federico, 1977-

Advisor dc:contributor.advisor

• Richard P. Stanley.

• Mathematics.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/29287

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/29287