Multiplicity of the Trivial Representation in Rank-selected Homology of the Partition Lattice

We study the multiplicity bS(n) of the trivial representation in the symmetric group representations βS on the (top) homology of the rankselected partition lattice ΠSn. We break the possible rank sets S into three cases: (1) 1 6∈ S, (2) S = 1, . . . , i for i ≥ 1 and (3) S = 1, . . . , i, j1, . . . , jl for i, l ≥ 1, j1 > i + 1. It was previously shown by Hanlon that bS(n) = 0 for S = 1, . . . , i. We use a partitioning for ∆(Πn)/Sn due to Hersh to confirm a conjecture of Sundaram [Su] that bS(n) > 0 for 1 6∈ S. On the other hand, we use the spectral sequence of a filtered complex to show bS(n) = 0 for S = 1, . . . , i, j1, . . . , jl unless a certain type of chain of support S exists. The partitioning for ∆(Πn)/Sn allows us then to show that a large class of rank sets S = 1, . . . , i, j1, . . . , jl for which such a chain exists do satisfy bS(n) > 0. We also generalize the partitioning for ∆(Πn)/Sn to ∆(Πn)/Sλ; when λ = (n−1, 1), this partitioning leads to a proof of a conjecture of Sundaram about S1 × Sn−1-representations on the homology of the partition lattice.