Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Colored partitions and a generalization of the braid arrangement. | Colored partitions and a generalization of the braid arrangement Vqlkmar Welker1 Fachbereich 6 Mathematik Universitat GH-Essen D-45117 Essen Germany welker@ Submitted November 11 1996 Accepted November 22 1996. Abstract We study the topology and combinatorics of an arrangement of hyperplanes in Cn that generalizes the classical braid arrangement. The arrangement plays in important role in the work of Schechtman Varchenko 12 Part II on Lie algebra homology where it appears in a generic hber of a projection of the braid arrangement. The study of the intersection lattice of the arrangement leads to the dehnition of lattices of colored partitions. A detailed combinatorial analysis then provides algebro-geometric and topological properties of the complement of the arrangement. Using results on the character of Sn on the cohomology of these arrangements we are able to deduce the rational cohomology of certain spaces of polynomials in the complement of the standard discriminant that have no root in the hrst s integers. 1 Introduction In this paper we study the arrangement An l s of all affine hyperplanes Hj z Zj 1 i j n and Hị zi r 1 i n and 1 r s. This arrangement appears in the work of Schechtman Varchenko 12 Part II as a generic hber of projections of the braid space in the context of Lie-algebra homology. We investigate the combinatorics of the intersection lattice LAc s of An l s . the set of all subspaces that are intersections of hyperplanes in the arrangement ordered by reversed inclusion . This leads to the dehnition of colored partitions. Via the analysis of the homology of the order complex of the intersection lattice and using a formula by Orlik Solomon 10 1Supported by the DFG through Habiliationsstipendium We 1479 3 Keywords Partition hyperplane arrangement intersection lattice configuration space Mathematics Subject Classification. Primary 05C40 52B30. Secondary 05C40 05E25 1 THE ELECTRONIC .JOURNAL OF COmBINATORICS 4 1997 R4 2 we .
