Quotient Complexes and Lexicographic Shellability

• Authors: Axel Hultman
• Venue: Journal of Algebraic Combinatorics: An International Journal
• Year: 2002

Let Πin,k,k and Πin,k,h, ih k, denote the intersection lattices of the ik-equal subspace arrangement of type \cal Din and the ik,h-equal subspace arrangement of type \cal Bin respectively. Denote by iSnB the group of signed permutations. We show that Δ(Πin,k,k)/iSnB is collapsible. For Δ(Πin,k,h)/iSnB, ih k, we show the following. If in ≡ 0 (mod ik), then it is homotopy equivalent to a sphere of dimension \frac{2n}{k} - 2. If in ≡ ih (mod ik), then it is homotopy equivalent to a sphere of dimension 2\frac{n-h}{k} - 1. Otherwise, it is contractible. Immediate consequences for the multiplicity of the trivial characters in the representations of iSnB on the homology groups of Δ(Πin,k,k) and Δ(Πin,k,h) are stated.The collapsibility of Δ(Πin,k,k)/iSnB is established using a discrete Morse function. The same method is used to show that Δ(Πin,k,h)/iSnB, ih 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 (iTrans. Amer. Math. Soc. 4 (1996), 1299–1327) when the cell complex is an order complex of a poset.