### The Varchenko Matrix for Cones

Consider an arrangement of hyperplanes and assign to each hyperplane a weight. By using this weights Varchenko defines a bilinear form on the vector space freely generated by the regions of the arrangement. We define this bilinear form for cones of the arrangement. Then we show that the determinant...

Ausführliche Beschreibung

Gespeichert in:
1. Verfasser: Welker, Volkmar (Prof. Dr.) (BetreuerIn (Doktorarbeit)) Dissertation Englisch Philipps-Universität Marburg 2013 Mathematik und Informatik http://dx.doi.org/10.17192/z2013.0480 PDF-Volltext Keine Tags, Fügen Sie den ersten Tag hinzu!
Zusammenfassung: Consider an arrangement of hyperplanes and assign to each hyperplane a weight. By using this weights Varchenko defines a bilinear form on the vector space freely generated by the regions of the arrangement. We define this bilinear form for cones of the arrangement. Then we show that the determinant of the matrix of the bilinear form restricted to the cone is determined by the combinatorics of the arrangement inside the cone and factors nicely. The resulting theorem induces Varchenko's thereom about the determinant of the matrix of the Varchenko bilinear form. We consider cones of the braid arrangement which are defined by partially ordered sets. We give a formular for the determinant of a matrices of the bilinear form restricted to one this cones by using properties of the partially ordered set. http://dx.doi.org/10.17192/z2013.0480