Quotient complexes and lexicographic shellability

Axel Hultman

To appear at Formal Power Series and Algebraic Combinatorics (FPSAC01), Tempe, Arizona (USA), May 20-26, 2001


Let P(n,k,k) and P(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 SB(n) the group of signed permutations. We show that the quotient complex D(P(n,k,k))/SB(n) is collapsible. For D(P(n,k,h))/SB(n), h < k, we show the following. If n is congruent to 0 modulo k, then it is homotopy equivalent to a sphere of dimension 2n/k - 2. If n is congruent to h modulo k, then it is homotopy equivalent to a sphere of dimension 2(n-h)/k - 1. Otherwise, it is contractible. Immediate consequences for the multiplicity of the trivial characters in the representations of SB(n) on the homology groups of D(P(n,k,k)) and D(P(n,k,h)) are stated. In order to establish these results, we are led to introduce a lexicographic shelling condition for a class of not necessarily pure triangulated spaces. It specializes to the CL-shellability of Bjorner and Wachs when the triangulated space is an order complex of a poset.

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