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

We will discuss how two very different methods for getting at the multiplicity of the trivial representation in the rank-selected homology of the partition lattice provide complementary results, indicating along the way how the two very different approaches turn out both to be sensitive to the same poset structure. On the one hand, a partitioning for the quotient complex Delta (Pi_n)/S_n allows us to give lower bounds on the multiplicity for various rank sets S by exhibiting minimal faces of support S in the partitioning. On the other hand, spectral sequences of filtered complexes provide upper bounds on these same multiplicities. In particular, we use the partitioning to verify a conjecture of Sundaram that certain multiplicities are positive and to give related positivity results, and we use spectral sequences to show that many of the remaining multiplicities are zero.

This is joint work with Phil Hanlon.