The (co)homology of lattices of partitions with restricted block size
Anne E. Browdy · 1996
For $n, k, d\in {\bf Z}\sp+$ such that n is a multiple of $d, \Pi\sbsp{n}{d,k}$ is the poset of partitions of $\lbrack n\rbrack = \{ 1, 2,\..., n\}$ whose block sizes are divisible by d and are at least dk, ordered by refinement, with bottom element $ 0$ adjoined when $dk > 1.$ If k = 1 then $\Pi\sbsp{n}{d,1}$ is the d-divisible partition lattice $\Pi\sbsp{n}{d}$ (each block size is divisible by d) which has been studied by Stanley (St1) who computed its Mobius number; Calderbank, Hanlon, Robinson (CHR) who derived plethystic formulas yielding the character of the representation of the symmetric group on the top homology of $\Pi\sbsp{n}{d};$ and Wachs (W1) who determined its homotopy type. In (W2), Wachs gives explicit bases for the (co)homology of the d-divisible partition lattice. The action of $S\sb{n-1}$ on the top homology of the lattice is studied and the representation matrices for the action of $S\sb{n}$ are constructed. If d = 1 then $\Pi\sbsp{n}{1,k}$ is the at least k partition lattice $\Pi\sb{n,k}$ (each block size is at least k). This lattice was studied previously by Linusson (L) who computed its Mobius number; Sundaram (Su) who computed the virtual representation of $S\sb{n}$ on the alternating sum of its homology; and Bjorner and Wachs (BW3) who determined its homotopy type. Wachs (W4) has shown that $\Pi\sbsp{n}{d,k}$ has the homotopy type of a wedge of spheres of varying dimensions. We construct explicit bases for the cohomology and homology of $\Pi\sbsp{n}{d,k}.$ Each basis element for (co)homology is naturally indexed by a permutation in $S\sb{n-1}$ with specific descents and ascents. This yields a combinatorial description of the Betti numbers of $\Pi\sbsp{n}{d,k}.$ (A generating function for the alternating sum of the Betti numbers was previously computed by Linusson (L) and Sundaram (Su).) The dimensions for which the (co)homology of $\Pi\sbsp{n}{d,k}$ is nonvanishing are calculated. The action of $S\sb{n-1}$ on the cohomology of $\Pi\sbsp{n}{d,k}$ is then studied. We show that in each dimension for which it is nonvanishing, the (co)homology of $\Pi\sbsp{n}{d,k}$ as an $S\sb{n-1}$-module is isomorphic to the direct sum of Specht modules corresponding to certain skew shapes. Representation matrices for the action of $S\sb{n}$ are then constructed.