Homotopy of Non-Modular Partitions and the Whitehouse Module

• Authors: Sheila Sundaram
• Venue: Journal of Algebraic Combinatorics: An International Journal
• Year: 1999

We present a class of subposets of the partition lattice Πn with the following property:The order complex is homotopy equivalent to the order complex of Πn − 1, and the Sn-module structure of the homology coincides with a recently discovered lifting of the Sn − 1-action on the homology of Πn − 1. This is the Whitehouse representation on Robinson‘s space of fully-grown trees, and has also appeared in work of Getzler and Kapranov, Mathieu, Hanlon and Stanley, and Babson et al.One example is the subposet Pnn − 1 of the lattice of set partitionsΠn, obtained by removing all elements with a unique nontrivial block. More generally, for 2 ≤ k ≤ n − 1, let Qnk denote the subposet of the partition lattice Πn obtained by removing all elements with a unique nontrivial block of size equal to k, and let Pnk = \bigcapi = 2k Qni. We show that Pnk is Cohen-Macaulay, and that Pnk and Qnk are both homotopy equivalent to a wedge of spheres of dimension (n − 4), with Betti number (n − 1) \!{n - k\over k}. The posets Qnk are neither shellable nor Cohen-Macaulay. We show that the Sn-module structure of the homology generalises the Whitehouse module in a simple way.We also present a short proof of the well-known result that rank-selection in a poset preserves the Cohen-Macaulay property.