Subspace arrangements of type B n and D n
Anders Björner1
and Bruce E. Sagan2
1Matematiska Institutionen Kungl. Tekniska Högskolan S-100 44 Stockholm Sweden S-100 44 Stockholm Sweden
2Michigan State University Department of Mathematics 48824-1027 East Lansing MI 48824-1027 East Lansing MI
2Michigan State University Department of Mathematics 48824-1027 East Lansing MI 48824-1027 East Lansing MI
DOI: 10.1007/BF00193180
Abstract
Let D \mathcal{D} n,k be the family of linear subspaces of \Bbb R n given by all equations of the form
for 1\leq i l< ... <i k \leq n and (\in 1,...,\in k )\in {+1, - 1} k . Also let B n,k,h be D n, k \mathcal{D}_{n,k} enlarged by the subspaces
for 1 \leq j 1 < ... < jh \leq n.The special cases B n,2,1 and D n \mathcal{D}_n , 2 are well known as the reflection hyperplane arrangements corresponding to the Coxeter groups of type B n and D n, respectively.
| e 1 x l 1 = e 2 x l 2 = \frac{1}{4} = e k x l k , ε_1 x_{l_1 } = ε_2 x_{l_2 } = \cdots = ε_k x_{l_k } , |
| x j 1 = x j 2 = \frac{1}{4} = x j h = 0, x_{j_1 } = x_{j_2 } = \cdots = x_{j_h } = 0, |
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 B n,k,h , 1 \leq h < k, which allows computation of the homology of its intersection lattice and the cohomology groups of the manifold M n,k,h =\Bbb R n \\cup B n,k,h . For instance, it is shown that H d(M n,k,k - 1) is torsion-free and is nonzero if and only if d=t(k - 2) for some t, 0 \leq t \leq [ n/k]. Torsion-free cohomology follows also for the complement in \Bbb C n of the complexification B n,k,h \Bbb C , 1 \leq h < k.
Pages: 291–314
Keywords: cohomology; characteristic polynomial; Coxeter subspace arrangement; homotopy; homology; lexicographic shellability; signed graph
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References
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