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

Abstract

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.