On the Smith normal form of the Varchenko bilinear form of a hyperplane arrangement

Abstract

Let A = {H1, . . . , H`} be an arrangement of hyperplanes in R and let r(A) = {R1, . . . , Rm} denote the set of regions in the complement of the union of A. Let L(A) denote the collection of intersections of hyperplanes in A including the empty intersection which we take to be R. We order the elements of L(A) by reverse inclusion thus making it into a poset. It is well known that this poset is a semilattice and is a geometric lattice if the arrangement is central. We will abbreviate L(A) to L when the arrangement is clear. For regions S, T ∈ R(A), define n(S, T ) to be the number of hyperplanes in A which separate S from T . In [6], Varchenko defines a matrix B = B(A) with rows and columns indexed by the regions in R(A) by saying that the S, T entry in B is q . Example 1.1. An important example is the arrangement A consisting of the (n 2 ) hyperplanes Hi,j in R given by Hi,j = {(x1, . . . , xn) : xi = xj}. The reader will note that A consists of the reflecting hyperplanes for the root system An−1 and so we denote this arrangement by An−1. Two points (x1, . . . , xn) and (y1, . . . , yn) are in the same region of the complement if and only if the relative orders of their coordinates are the same. So, the permutations in Sn index the regions of the complement via the correspondence σ ↔ {(x1, . . . , xn) : xσ1 < xσ2 < . . . < xσn}. For σ, τ in Sn, the exponent of q in the σ, τ entry of B is i(στ−1), the number of inversions of στ−1. Equivalently, B is the matrix for left multiplication by ∑ α∈Sn q α in CSn (the matrix with respect to the standard basis). It is interesting to note that this matrix is studied by Zagier [10] for quite different reasons.