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Home > Vol 16, No 1 (2002)

Axel Hultman

Kungl. Tekniska Hogskolan SE-100 44 Stockholm Sweden

Let D \mathcal{D} _n and the k,h-equal subspace arrangement of type B \mathcal{B} _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 \tfrac{{2n}}{k} . If n 2\tfrac n - h k - 1 2\tfrac{{n - h}}{k} - 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.

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