Incidence Hopf algebras: Antipodes, forest formulas, and noncrossing partitions

This dissertation focuses on incidence Hopf algebras of hereditary families of posets. In the first half we study antipodes of incidence Hopf algebras of arbitrary families of posets and lattices, and in the second half we investigate various aspects of the incidence Hopf algebra of noncrossing partition lattices. We present a new forest formula for the antipode of incidence Hopf algebras of hereditary families of posets. The antipode can be expressed as a sum over all chains of a poset, and we prove our new antipode formula by exhibiting a map from these chains to certain forests. We characterize those families for which our formula is cancellation-free.We then consider the incidence Hopf algebra of noncrossing partition lattices. We find several formulas for the antipode of this Hopf algebra in terms of different sets of generators. We define a closure operator on noncrossing partition lattices and show that the incidence Hopf algebra corresponding to the subposets of closed elements is isomorphic to the linear Faa di Bruno Hopf algebra. Using an edge labeling of the noncrossing partition lattices, which was first defined by Stanley, we construct a map from the noncrossing partition Hopf algebra to the incidence Hopf algebra of linear orders. We use these results to find explicit formulas for the coproduct and a projection onto the subspace of primitive elements of the noncrossing partition Hopf algebra.