The homology and Varchenko matrix of a generalized braid arrangement.

Abstract

Define the hyperplane arrangement $A\sb{n,k}$ to be the set of hyperplanes of the form $x\sb{i}-x\sb{j}=m$ for all $1\le i<j\le n$ and integers ${-}k\le m\le k.$ This arrangement is a generalization of the well known braid arrangement $A\sb{n}.$ In this dissertation we show that both the regions of $A\sb{n,k}$ and the top homology of its intersection lattice can be indexed by certain classes of rooted trees. The Varchenko matrix of a hyperplane arrangement with weighted hyperplanes is a square matrix whose rows and columns are indexed by the regions of the arrangement and whose (i,j) entry is equal to the product of the weights of the hyperplanes separating the $i\sp{th}$ and $j\sp{th}$ regions of the arrangement. We use the indexing of the regions of $A\sb{n,k}$ to calculate the determinant of its Varchenko matrix. We also show that the Smith Normal Form of the additive Varchenko matrix (obtained by adding rather than multiplying weights) depends only on the number of hyperplanes in the arrangement and their weights.