Subspace Arrangements of Type B N and D N Send Proofs To
نویسندگان
چکیده
Let Dn,k be the family of linear subspaces of R given by all equations of the form 1xi1 = 2xi2 = . . . = kxik , for 1 ≤ i1 < . . . < ik ≤ n and ( 1, . . . , k) ∈ {+1,−1}. Also let Bn,k,h be Dn,k enlarged by the subspaces xj1 = xj2 = . . . = xjh = 0, for 1 ≤ j1 < . . . < jh ≤ n. The special cases Bn,2,1 and Dn,2 are well known as the reflection hyperplane arrangements corresponding to the Coxeter groups of type Bn and Dn, respectively. In this paper we study combinatorial and topological properties of the intersection lattices of these subspace arrangements. Expressions for their Möbius functions and characteristic polynomials are derived. Lexicographic shellability is established in the case of Bn,k,h, 1 ≤ h < k, which allows computation of the homology of its intersection lattice and the cohomology groups of the manifold Mn,k,h = R \ ∪Bn,k,h. For instance, it is shown that H(Mn,k,k−1) is torsion-free and is nonzero if and only if d = t(k − 2) for some t, 0 ≤ t ≤ bn/kc. Torsionfree cohomology follows also for the complement in C of the complexification BC n,k,h, 1 ≤ h < k.
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