Center & Programs
Summer School on Algebraic Combinatorics
Combinatorial Hopf Algebra, quasisymmetric functions and antipode formulas
SPEAKER | Nantel Bergeron
INSTITUTE | York University, Canada
DATE | June 14(Tue), 2016
TIME | 14:00
PLACE | 1114
ABSTRACT | A combinatorial Hopf algebra is a graded-connected Hopf algebra with a given character. Using interesting examples, I will introduce these notions and what we can do with them.
For example take the Hopf algebra of equivalent classes of graphs with character $psi(G)=1$ if the graph is trivial and $0$ otherwise. Given such structure, we can construct a quasisymmetric function $Psi(G)$ that encode some combinatorial invariants of $G$. In particular, the chromatic polynomial $chi_G(x)$ of $G$. All these constructions preserve the Hopf structures, in particular one can see that
$$chi_G(-1) = psi(S(G)) $$
where $S$ is the antipode of the Hopf algebra of graphs. This antipode was explicitly computed by Humpert and Martin and involved acyclic orientation of $G$ and its quotients. From this one can recover a classical theorem of Stanley showing that $chi_G(-1)$ is the number of acyclic orientations of $G$. I will remain as elementary as possible to cover this and will assume only notion of linear algebras and tensor products.