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Báo cáo toán học: "Geometrically constructed bases for homology of non-crossing partition lattices"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Geometrically constructed bases for homology of non-crossing partition lattices. | Geometrically constructed bases for homology of non-crossing partition lattices Aisling Kenny School of Mathematical Sciences Dublin City University Glasnevin Dublin 9 Ireland aisling.kenny9@mail.dcu.ie Submitted Jun 25 2008 Accepted Apr 10 2009 Published Apr 22 2009 Mathematics Subject Classifications 20F55 Abstract For any finite real reflection group W we construct a geometric basis for the homology of the corresponding non-crossing partition lattice. We relate this to the basis for the homology of the corresponding intersection lattice introduced by Bjorner and Wachs in 4 using a general construction of a generic affine hyperplane for the central hyperplane arrangement defined by W . 1 Introduction Let W be a finite real reflection group acting effectively on R . In 4 Bjorner and Wachs construct a geometric basis for the homology of the intersection lattice associated to W. There is another lattice associated to W called the non-crossing partition lattice. In 2 Athanasiadis Brady and Watt prove that the non-crossing partition lattice is shellable for any finite Coxeter group W. Zoque constructs a basis for the top homology of the non-crossing partition lattice for the An case in 11 where the basis elements are in bijection with binary trees. A geometric model X c of the non-crossing partition lattice is constructed in 7 . In this paper we use X c to construct a geometric basis for the homology of the noncrossing partition lattice that corresponds to W . We construct the basis by defining a homotopy equivalence between the proper part of the non-crossing partition lattice and the n 2 -skeleton of X c . We exhibit an explicit embedding of the homology of the non-crossing partition lattice in the homology of the intersection lattice using the general construction of a generic affine hyperplane Hv. THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2009 R48 1 2 Preliminaries We refer the reader to 5 and 8 for standard facts and notation about finite reflection groups. As .