UC Berkeley

This thesis is a compendium of three studies on which matroids and convex geometry play a central role and show their connections to Catalan combinatorics, tiling theory, and factorization theory. First, we study positroids in connection with rational Dyck paths. Then, we study certain matroids on the lattice points of a regular triangle in connection with lozenge tilings. Finally, we explore the connection between the atomic structure of submonoids of (N^d,+) and the geometric properties of the cones they generate.

Positroids, first studied by Postnikov in 2006, are matroids that parameterize the cells of the totally nonnegative part of a Grassmannian variety. The first part of this thesis concerns with the study of a family of positroids that can be parameterized by (rational) Dyck paths. We call such positroids (rational) Dyck positroids. Using work of Reed and Skandera, we show that Dyck positroids on the ground set [2n] are in natural bijection with unit interval orders of size n. We also offer recipes to read the decorated permutation of a Dyck positroid directly from either the antiadjacency matrix representation or the interval representation of the corresponding unit interval order. Finally, for the family of rational Dyck positroids, we provide combinatorial descriptions for some of the most relevant combinatorial objects that are in bijection with positroids.

The second part of this thesis pertains to the study of certain class of matroids which naturally appear in the set of one-dimensional intersections of complete complex flag arrangements. More specifically, these matroids encode the dependency relations among the lines of such flag arrangements. The bases of such matroids can be thought of as certain n-subsets of lattice points of a regular n-simplex. For dimension 2, we provide various cryptomorphic characterizations of these matroids in connection with lozenge tilings of a regular triangle. We also study the connectivity of members of this family of matroids in any dimension.

Finitely generated submonoids of (N^d,+), also known as affine monoids, are crucial in the study of combinatorial commutative algebra and, in particular, toric geometry. Let C denote the class consisting of all submonoids of (N^d,+) (not necessarily finitely generated). The last part of this thesis is devoted to explore how atomic properties of a monoid M in C (and the monoid algebras M induces) are connected with the geometry of its conic hull cone(M) and with the combinatorial structure of the face lattice of cone(M). For monoids in C, we investigate two of the most important arithmetic invariants in factorization theory: the system of sets of lengths and the elasticity. We conclude this thesis studying the atomicity of monoid algebras, including the algebras induced by monoids in C. We shall provide a partial answer to a fundamental question about the atomicity of monoid algebras that Robert Gilmer asked back in the 1980's.