Thus the isomorphism is a consequence of the linking of (l - 2)-spheres and l-tori in the (2l - 1)-sphere. The canonical isomorphism is shown to coincide with the restriction of the classical Alexander duality mapping. We put this result into a geometric framework by constructing a realization of the order complex of the intersection lattice inside the link of the arrangement. Dividing by the number of ways to label the hyperplanes in A gives N A. Thus the isomorphism is a consequence of the linking of (l - 2)-spheres and l-tori in the (2l - 1)-sphere.ĪB - The top cohomology of the complement of an arrangement of complex hyperplanes is canonically isomorphic to the order homology of the associated intersection lattice. with intersection lattice isomorphic to L (A) that pass through D points in general position in P n. A region is relatively bounded if its intersection with space spanned by the normals of the hyperplanes in. Its self-intersection index is obviously one. N2 - The top cohomology of the complement of an arrangement of complex hyperplanes is canonically isomorphic to the order homology of the associated intersection lattice. bounded region really mean relatively bounded here. The Picard group of Pn is spanned by the divisor H of hyperplane section. Then, A(x) An H(x) (x): Use Whitney’s theorem while considering if H is in B or not. Theorem Let Abe an arrangement in Kn and H be a hyperplane of A. 4, we study in detail the hyperplane arrangements corresponding to hook diagrams. Moreover, denote AH the arrangement of nonempty H \J in the a ne space H for J 2A. the intersection of the coordinate arrangement in a larger space with the subspace corresponding to. and hence in particular the lattice of intersections. The intersection poset of an affine arrangement is a semi-lattice and. A hyperplane arrangement is a finite set of hyperplanes through the origin in a finite-dimensional. This work was partially supported by NSF grant DMS-9004202 and an NAU Organized Research grant. For a hyperplane H 2A, denote AnH the arrangement without the hyperplane H. It follows that the intersection lattice of a central arrangement is indeed a lattice. We will discuss the main open question in this area, namely whether the combinatorics of the intersection lattice determines the monodromy operator of the. Below in the introduction we recall some well known facts about Coxeter groups, weak Bruhat order, and Poincar'e seri.T1 - A geometric duality for order complexes and hyperplane complements The intersection poset of any (real or complex) afflne hyperplane arrangement A is a geometric semilattice. Some of the simplest pictures of the labeled Hasse diagrams in Section 1 have appeared also in connection with Verma modules and Schubert cells and as a graphical device for calculating the homology of the most elementary Artin groups. Basic cycles can be carried over from BC(M). ) - and by (2) deriving simple non-recursive schemes for the computation of standard reduced words for both unsigned and signed permutations. In the present paper we show the usefulness of this partition property by (1) giving a pictorial combinatorial derivation of the Poincar'e polynomials and series for the finite irreducible Coxeter groups and the affine Coxeter groups on three generators - results, which until now have been obtained by invariant theoretic or Lie theoretic methods (cf. But the partitioning property of the weak Bruhat order of Coxeter groups into isomorphic parts as stated in Theorem 0.1 below - though probably known by the experts - has certainly not been fully exploited. The combinatorial properties of weak Bruhat order of Coxeter groups, especially of the finite irreducible and affine ones, have been investigated for quite a time (see for example and the references therein). The algebraic basis for both (1) and (2) is a simple partition property of the weak Bruhat order of Coxeter groups into isomorphic parts. (2) Non-recursive methods for the computation of `standard reduced words' for (signed) permutations are described. showed that Lo uniquely determines the intersection lattice L of the. The Poincar'e polynomials of the finite irreducible Coxeter groups and the Poincar'e series of the affine Coxeter groups on three generators are derived by an elementary combinatorial method avoiding the use of Lie theory and invariant theory. homology of matroids, geometric lattices and linear hyperplane arrangements.
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