Partition complexes, duality and integral tree representations

Abstract

We show that the poset of non-trivial partitions of $\left\{1,2,\dots ,n\right\}$ has a fundamental homology class with coefficients in a Lie superalgebra. Homological duality then rapidly yields a range of known results concerning the integral representations of the symmetric groups ${\Sigma }_n$ and ${\Sigma }_{n+1}$ on the homology and cohomology of this partially-ordered set.