Quotient complexes and lexicographic shellability
2002 (English)In: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192, Vol. 16, no 1, 83-96 p.Article in journal (Refereed) Published
Let Pi(n,k,k) and Pi(n,k,h), h < k, denote the intersection lattices of the k-equal subspace arrangement of type D-n and the k, h-equal subspace arrangement of type B-n respectively. Denote by S-n(B) the group of signed permutations. We show that Delta(Pi(n,k,k))/S-n(B) is collapsible. For Delta(Pi(n,k,h))/S-n(B),h < k, we show the following. If n = 0 (mod k), then it is homotopy equivalent to a sphere of dimension 2n/k = 2. If n = h (mod k), then it is homotopy equivalent to a sphere of dimension 2n-h/k-1. Otherwise, it is contractible. Immediate consequences for the multiplicity of the trivial characters in the representations of S-n(B) on the homology groups of Delta(Pi(n,k,k)) and Delta(Pi(n,k,h)) are stated. The collapsibility of Delta (Pi(n,k,k))/S-n(B) is established using a discrete Morse function. The same method is used to show that Delta(Pi(n,k,h))/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. Bjorner and M. Wachs (Trans. Amer. Math. Soc. 4 (1996), 1299-1327) when the cell complex is an order complex of a poset.
Place, publisher, year, edition, pages
2002. Vol. 16, no 1, 83-96 p.
quotient complex, cell complex of boolean type, lexicographic shellability, coxeter subspace arrangement, homotopy, subspace arrangements, simplicial posets
IdentifiersURN: urn:nbn:se:kth:diva-22002ISI: 000178886000006OAI: oai:DiVA.org:kth-22002DiVA: diva2:340700
QC 201005252010-08-102010-08-10Bibliographically approved