Subspace arrangements of type $B\sb n$ and $D\sb n$

Björner, Anders ; Sagan, Bruce E.

Journal of Algebraic Combinatorics, Tome 5 (1996) no. 4, pp. 291-314.

Résumé

Summary: Let $D$ mathcalD $_{n,k}$ be the family of linear subspaces of $\Bbb R ^{n}$ given by all equations of the form e $_{1} x _{ l $_1 $ = e _{2} x _{ l $_2 = frac14 = e $_{ k} x _{ l $_ k $ , \varepsilon $_1 x_l_1 = $\varepsilon $_2 x_l_2 = $\cdots = \varepsilon $_k x_l_k , for $1\leq i _{l} \dots$ $_{ k }\leq n$ and $(\in _{1},\dots ,\in _{ k })\in {+1, - 1} _{ k }$. Also let $B _{ n,k,h }$ be $D _{ n, k}$ mathcalD_n,k enlarged by the subspaces $x _{ j $_1 $ = x _{ j $_2 = frac14 = $x _{ j $_ h $ = 0$, x_j_1 = x_j_2 = cdots= x_j_h = 0, for $1 \leq j _{1} \dots jh \leq n$.The special cases $B _{ n,2,1}$ and $D _{ n}$ mathcalD_n , $_{2}$ are well known as the reflection hyperplane arrangements corresponding to the Coxeter groups of type $B _{n}$ and $D _{n}$, 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 $B _{ n,k,h }, 1 \leq h k$, 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 $B _{ n,k,h } ^{\Bbb C} , 1 \leq h k$.

Détail

Classification : 52B30, 05A15, 05E25, 06A07, 20F55
Keywords: cohomology, characteristic polynomial, Coxeter subspace arrangement, homotopy, homology, lexicographic shellability, signed graph