Lexicographic shellability for balanced complexes

Hersh, Patricia

Journal of Algebraic Combinatorics, Tome 17 (2003) no. 3, pp. 225-254.

Résumé

Summary: We introduce a notion of lexicographic shellability for pure, balanced boolean cell complexes, modelled after the $C$L-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 $C$C-shellability. We give a lexicographic shelling for the quotient of the order complex of a Boolean algebra of rank $2 n$ by the action of the wreath product $S_{2} wreathS _{n}$ of symmetric groups, and we provide a partitioning for the quotient complex $Delta( Pgr_{ n })/ S _{n}$. Stanley asked for a description of the symmetric group representation $beta_{ S }$ on the homology of the rank-selected partition lattice $Pgr _{n} ^{S}$ in Stanley ( J. Combin. Theory Ser. A $32(2) (1982)$, 132-161), and in particular he asked when the multiplicity $b _{S}( n)$ of the trivial representation in $beta_{ S }$ is 0. One consequence of the partitioning for $Delta( Pgr_{ n })/ S _{n}$ is a (fairly complicated) combinatorial interpretation for $b _{S}( n)$; another is a simple proof of Hanlon"s result ( European J. Combin. $4(2) (1983)$, 137-141) that $b _{1}, ctdot, i( n) = 0$. Using a result of Garsia and Stanton from ( Adv. in Math. $51(2) (1984)$, 107-201), we deduce from our shelling for $Delta( B _{2 n })/ S_{2} wreathS _{n}$ that the ring of invariants $k[ x _{1}, ctdot, x _{2} _{ n }] ^{ S2}wreath^{ Sn}$ is Cohen-Macaulay over any field $k$.

Détail

Keywords: shellability, Boolean cell complex, simplicial poset, partition lattice, wreath product