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

Abstract

We study the multiplicity bS(n) of the trivial representation in the symmetric group representations βS on the (top) homology of the rank-selected partition lattice ΠnS. We break the possible rank sets S into three cases: (1) 1∉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 Δ(Πn)/Sn due to Hersh to confirm a conjecture of Sundaram [S. Sundaram, The homology representations of the symmetric group on Cohen–Macaulay subposets of the partition lattice, Adv. Math. 104 (1994) 225–296] that bS(n)>0 for 1∉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 Δ(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 Δ(Πn)/Sn to Δ(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.