## 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.