Lexicographic Shellability for Balanced Complexes

Abstract

We introduce a notion of lexicographic shellability for pure, balanced boolean cell complexes, modelled after the CL-shellability criterion of Björner and Wachs (Adv. in Math. 43 (1982), 87-100) for posets and its generalization by Kozlov (Ann. of Comp. 1(1) (1997), 67-90) 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 wreath product sign Sn of symmetric groups, and we provide a partitioning for the quotient complex Δ(Πn)/S n. Stanley asked for a description of the symmetric group representation βS on the homology of the rank-selected partition lattice ΠnS in Stanley (J. Combin. Theory Ser. A 32(2) (1982), 132-161), and in particular he asked when the multiplicity bS(n) of the trivial representation in βS is 0. One consequence of the partitioning for Δ(Πn)/Sn is a (fairly complicated) combinatorial interpretation for bS(n); another is a simple proof of Hanlon's result (European J. Combin. 4(2) (1983), 137-141) that b1,...,i(n) = 0. Using a result of Garsia and Stanton from (Adv. in Math. 51(2) (1984), 107-201), we deduce from our shelling for Δ(B2n)/S2wreath product signSn that the ring of invariants k[x1, ..., X2n]S2wreath product signSn is Cohen-Macaulay over any field k.