Multiplicity of the trivial representation in rank-selected homology of the partition lattice

We study the multiplicity $b_S(n)$ of the trivial representation in the symmetric group representations $\beta_S$ on the (top) homology of the rank-selected partition lattice $\Pi_n^S$. We break the possible rank sets $S$ into three cases: (1) $1\not\in S$, (2) $S=1,..., i$ for $i\ge 1$ and (3) $S=1,..., i,j_1,..., j_l$ for $i,l\ge 1$, $j_1 > i+1$. It was previously shown by Hanlon that $b_S(n)=0$ for $S=1,..., i$. We use a partitioning for $\Delta(\Pi_n)/S_n$ due to Hersh to confirm a conjecture of Sundaram that $b_S(n)>0$ for $1\not\in S$. On the other hand, we use the spectral sequence of a filtered complex to show $b_S(n)=0$ for $S=1,..., i,j_1,..., j_l$ unless a certain type of chain of support $S$ exists. The partitioning for $\Delta(\Pi_n)/S_n$ allows us then to show that a large class of rank sets $S=1,..., i,j_1,..., j_l$ for which such a chain exists do satisfy $b_S(n)>0$. We also generalize the partitioning for $\Delta(\Pi_n)/S_n$ to $\Delta(\Pi_n)/S_{\lambda}$; when $\lambda = (n-1,1)$, this partitioning leads to a proof of a conjecture of Sundaram about $S_1\times S_{n-1}$-representations on the homology of the partition lattice

Chain Decomposition Theorems for Ordered Sets (and Other Musings)

A brief introduction to the theory of ordered sets and lattice theory is given. To illustrate proof techniques in the theory of ordered sets, a generalization of a conjecture of Daykin and Daykin, concerning the structure of posets that can be partitioned into chains in a ``strong'' way, is proved. The result is motivated by a conjecture of Graham's concerning probability correlation inequalities for linear extensions of finite posets

Tableaux on k+1-cores, reduced words for affine permutations, and k-Schur expansions

The $k$-Young lattice $Y^k$ is a partial order on partitions with no part larger than $k$. This weak subposet of the Young lattice originated from the study of the $k$-Schur functions(atoms) $s_\lambda^{(k)}$, symmetric functions that form a natural basis of the space spanned by homogeneous functions indexed by $k$-bounded partitions. The chains in the $k$-Young lattice are induced by a Pieri-type rule experimentally satisfied by the $k$-Schur functions. Here, using a natural bijection between $k$-bounded partitions and $k+1$-cores, we establish an algorithm for identifying chains in the $k$-Young lattice with certain tableaux on $k+1$ cores. This algorithm reveals that the $k$-Young lattice is isomorphic to the weak order on the quotient of the affine symmetric group $\tilde S_{k+1}$ by a maximal parabolic subgroup. From this, the conjectured $k$-Pieri rule implies that the $k$-Kostka matrix connecting the homogeneous basis \{h_\la\}_{\la\in\CY^k} to \{s_\la^{(k)}\}_{\la\in\CY^k} may now be obtained by counting appropriate classes of tableaux on $k+1$-cores. This suggests that the conjecturally positive $k$-Schur expansion coefficients for Macdonald polynomials (reducing to $q,t$-Kostka polynomials for large $k$) could be described by a $q,t$-statistic on these tableaux, or equivalently on reduced words for affine permutations.Comment: 30 pages, 1 figur

Symmetric Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices

We prove that the noncrossing partition lattices associated with the complex reflection groups $G(d,d,n)$ for $d,n\geq 2$ admit symmetric decompositions into Boolean subposets. As a result, these lattices have the strong Sperner property and their rank-generating polynomials are symmetric, unimodal, and $\gamma$-nonnegative. We use computer computations to complete the proof that every noncrossing partition lattice associated with a well-generated complex reflection group is strongly Sperner, thus answering affirmatively a question raised by D. Armstrong.Comment: 30 pages, 5 figures, 1 table. Final version. The results of the initial version were extended to symmetric Boolean decompositions of noncrossing partition lattice