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Báo cáo toán học: " Geometrically constructed bases for homology of partition lattices of types A, B and D"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Geometrically constructed bases for homology of partition lattices of types A, B and D. | Geometrically constructed bases for homology of partition lattices of types A B and D Anders Bjorner Royal Institute of Technology Department of Mathematics S-100 44 Stockholm Sweden bjorner@math.kth.se Michelle L. Wachs University of Miami Department of Mathematics Coral Gables FL 33124 USA wachs@math.miami.edu Submitted Jan 1 2004 Accepted Apr 17 2004 Published Jun 3 2004 MR Subject Classifications 05E25 52C35 52C40 Dedicated to Richard Stanley on the occasion of his 60th birthday Abstract We use the theory of hyperplane arrangements to construct natural bases for the homology of partition lattices of types A B and D. This extends and explains the splitting basis for the homology of the partition lattice given in 20 thus answering a question asked by R. Stanley. More explicitly the following general technique is presented and utilized. Let A be a central and essential hyperplane arrangement in Rd. Let R1 . Rk be the bounded regions of a generic hyperplane section of A. We show that there are induced polytopal cycles pRi in the homology of the proper part La of the intersection lattice such that pRi i . k is a basis for Hd-2 LA . This geometric method for constructing combinatorial homology bases is applied to the Coxeter arrangements of types A B and D and to some interpolating arrangements. 1 Introduction In 20 Wachs constructs a basis for the homology of the partition lattice nra via a certain natural splitting procedure for permutations. This basis has very favorable properties Supported in part by Goran Gustafsson Foundation for Research in Natural Sciences and Medicine. Supported in part by National Science Foundation grants DMS-9701407 and DMS-0073760. THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2 2004 R3 1 with respect to the representation of the symmetric group Sn on Hn_3 n n C a representation that had earlier been studied by Stanley 19 Hanlon 14 and many others. It also is the shelling basis for a certain EL-shelling of the partition lattice given in 20