Quotient complexes and lexicographic shellability

Hultman, Axel

Journal of Algebraic Combinatorics, Tome 16 (2002) no. 1, pp. 83-96.

Résumé

Summary: Let $D$ mathcalD $_{ n }$ and the $k,$h-equal subspace arrangement of type $B$ mathcalB $_{ n }$ respectively. Denote by $S _{ n} ^{ B}$ S_n^B the group of signed permutations. We show that $S _{ n} ^{ B}$ S_n^B is collapsible. For $S _{ n} ^{ B}$ S_n^B , $h k$, we show the following. If $n \tfrac2 n k$ tfrac2nk . If $n 2\tfrac n - h k - 1 2$tfracn - hk - 1 . Otherwise, it is contractible. Immediate consequences for the multiplicity of the trivial characters in the representations of $S _{ n} ^{ B}$ S_n^B on the homology groups of $S _{ n} ^{ B}$ S_n^B is established using a discrete Morse function. The same method is used to show that $S _{ n} ^{ B}$ S_n^B , $h k$, is homotopy equivalent to a certain subcomplex. The homotopy type of this subcomplex is calculated by showing that it is shellable. To do this, we are led to introduce a lexicographic shelling condition for balanced cell complexes of boolean type. This extends to the non-pure case work of P. Hersh (Preprint, 2001) and specializes to the CL-shellability of A. Björner and M. Wachs ( Trans. Amer. Math. Soc. 4 (1996), 1299-1327) when the cell complex is an order complex of a poset.

Détail

Keywords: quotient complex, cell complex of Boolean type, lexicographic shellability, Coxeter subspace arrangement, homotopy