2003/11/16 by Patricia Hersh
Mathematics · #math.CO #msc:05E25 #msc:05A18
published as J. Algebraic Combinatorics, 17 (2003), no. 1, 27-52 · 28 pages, 10 figures
arxiv created 2003/11/16 · arxiv updated 2009/12/01
We introduce a notion of lexicographic shellability for pure, balanced boolean cell complexes, modelled after the CL-shellability criterion of Björner and Wachs for posets and its generalization by Kozlov called CC-shellability. We give a lexicographic shelling for the quotient of the order complex of a Boolean algebra of rank 2n by the action of the wreath product S2\wr Sn of symmetric groups, and we provide a partitioning for the quotient complex Δ(Πn)/Sn . Stanley asked for a description of the symmetric group representation βS on the homology of the rank-selected partition lattice ΠnS in [St2], and in particular he asked when the multiplicity bS(n) of the trivial representation in βS is 0. One consequence of the partitioning for \dps is a (fairly complicated) combinatorial interpretation for bS(n) ; another is a simple proof of Hanlon's result that b1,..., i(n)=0. Using a result of Garsia and Stanton, we deduce from our shelling for Δ(B2n)/S2 \wr Sn that the ring of invariants k[x1,..., x2n]S2\wr Sn is Cohen-Macaulay over any field k.