Combinatorial Hopf Algebras

Frank Sottile,
University of Massachusetts at Amherst

Many generating functions of enumerative combinatorics are either
symmetric or quasi-symmetric.  Also, there are important eulerian
objects, whose combinatorial invariants satisfy certain linear equations.
The notion of a Combinatorial Hopf algebra attempts to explain this 
ubiquity of quasi-symmetric generating functions, while also giving 
this notion of eulerian an algebraic framework from which it
can be generalized.   

   A Combinatorial Hopf algebras is a graded connected Hopf algebra 
H over a field F equipped with a multiplicative linear functional 
zeta : H --> F.  The terminal object in the category 
of combinatorial Hopf algebras is the algebra of quasi-symmetric 
functions, which explains their ubiquity as generating functions 
in combinatorics.  The M\"obius function of a combinatorial Hopf 
algebra gives rise to natural Dehn-Sommerville relations on the 
Hopf algebra, and a combinatorial Hopf algebra satisfying these 
relations is eulerian.  The relation of this theory to combinatorics 
is through Hopf algebras of combinatorial objects.

   In this talk, we will define combinatorial Hopf algebras, and
show how the algebra of quasi-symmetric functions is a terminal
object in their category.  We then define eulerian combinatorial 
Hopf algebras and illustrate this theory with examples.