On the (co)homology of the partition lattice and the free Lie algebra

摘要

We explore the well-known relationship between the representations of the symmetric group on the homology of the partition lattice and on the 1~n component of the free Lie algebra. We give two new combinatorial proofs of the sign twisted isomorphism between the two modules. One involves a bijection between bases and the other involves a bijection between generating sets which takes relations to relations.