On the invariant factors and module structure of the kernel of the Varchenko matrix.

Abstract

Information about the nullspace and Smith normal form of the Varchenko matrix <italic>B</italic>(<italic>q</italic>) of a hyperplane arrangement has been useful in the study of Kac-Moody Lie algebras and quantum groups, hypergeometric functions, and the homology of the Milnor Fibre. Denham and Hanlon completely determined this information for the Boolean arrangement <italic> O_n</italic>, but in the case of the braid arrangement <italic> A_n</italic>_-1 much less is known. For <italic> A_n</italic>_-1, we determine the <italic>S_n</italic>-module structure of the kernel of <italic>T_n</italic>(1), an operator related to <italic>B</italic>(<italic>q</italic>), and provide evidence for a generalization to other roots of unity for which the kernel is nontrivial. As a means of interpolating between <italic>O_n</italic>_-1 and <italic>A_n</italic>_-1, we define and study a family of arrangements that have an interesting correspondence with rook placements on Ferrers boards. Using graph theory, we calculate the determinants of the corresponding Varchenko matrices in order to study the growth of the invariant factors of <italic>B</italic>(<italic>q</italic>) from powers of 1 - <italic>q</italic><super>2</super> to products of other cyclotomic polynomials. We also consider the family of rectangular interpolating arrangements defined by removing an <italic>r</italic> x <italic>s</italic> rectangle from a Ferrers board in an attempt to generalize Hanlon and Stanley's result about <italic>A_n</italic>_-1 (the case <italic> r</italic> = 0). We calculate det(<italic>B</italic>(<italic>q</italic>)) as the product of the invariant factors and use this information to conjecture what the nullity might be as a function of <italic>r</italic>, <italic>s</italic>, and <italic>n</italic>.