Hopf Algebras of Labeled Posets

Frank Sottile

University of Toronto

[This is joint work with N. Bergeron]

A labeled poset is a partially ordered set where each cover is labeled with an integer. This structure arises naturally in the theory of shellability of posets, as well as the weak order on the symmetric group and in the study of Schubert polynomials.

In this talk, we will introduce a generating function for maximal chains whose sequence of edge labels have fixed descents. The association of this generating function to such a poset gives a Hopf algebra map from a Hopf algebra of such posets to the algebra of quasi-symmetric functions. We relate this construction to the theory of symmetric functions.

No prior knowledge is assumed; definitions, constructions, and proofs (which are elementary) will be given.