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) $1\notin S$, (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 $1\notin S$. 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.

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