Deformations of Coxeter hyperplane arrangements and their characteristic polynomials

Deformations of Coxeter Hyperplane Arrangements

Deformations of Coxeter Hyperplane Arrangements

We investigate several hyperplane arrangements that can be viewed as deformations of Coxeter arrangements. In particular, we prove a conjecture of Linial and Stanley that the number of regions of the arrangement xi − xj = 1, 1 ≤ i < j ≤ n, is equal to the number of alternating trees on n + 1 vertices. Remarkably, these numbers have several additional combinatorial interpretations in terms of bi...

Tutte polynomials of hyperplane arrangements and the finite field method

The Tutte polynomial is a fundamental invariant associated to a graph, matroid, vector arrangement, or hyperplane arrangement, which answers a wide variety of questions about its underlying object. This short survey focuses on some of the most important results on Tutte polynomials of hyperplane arrangements. We show that many enumerative, algebraic, geometric, and topological invariants of a h...

Relation Spaces of Hyperplane Arrangements and Modules Defined by Graphs of Fiber Zonotopes

We study the exactness of certain combinatorially defined complexes which generalize the Orlik-Solomon algebra of a geometric lattice. The main results pertain to complex reflection arrangements and their restrictions. In particular, we consider the corresponding relation complexes and give a simple proof of the n-formality of these hyperplane arrangements. As an application, we are able to bou...

Enumeration in Algebra and Geometry

This thesis is devoted to solution of two classes of enumerative problems. The first class is related to enumeration of regions of hyperplane arrangements. We investigate deformations of Coxeter arrangements. In particular, we prove a conjecture of Stanley on the numbers of regions of Linial arrangements. These numbers have several additional combinatorial interpretations in terms of trees, par...