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 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 $S_2\wr S_n$ of symmetric groups, and we provide a partitioning for the quotient complex $\Delta ({\Pi }_n)/S_n$. Stanley asked for a description of the symmetric group representation ${\beta }_S$ on the homology of the rank-selected partition lattice ${\Pi }_n^S$ in [St2], 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 $\dps $ is a (fairly complicated) combinatorial interpretation for $b_S(n)$; another is a simple proof of Hanlon's result that $b_{1,...,i}(n)=0$. Using a result of Garsia and Stanton, we deduce from our shelling for $\Delta (B_{2n})/S_2\wr S_n$ that the ring of invariants $k[x_1,...,x_{2n}{]}^{S_2\wr S_n}$ is Cohen-Macaulay over any field $k$.

28 pages, 10 figures

References in corpus (2)

• Group Actions on Posets
• Multiplicity of the trivial representation in rank-selected homology of the partition lattice