The goal of this mini-course is to introduce the basic methods of Hopf algebras and their interactions with algebraic combinatorics. The topics presented will include: Combinatorial Hopf algebras, symmetric and quasisymmetric functions, Macdonald polynomials and ties with group representation theory.

The school is intended for graduate students, postdocs and researchers wishing to be introduced to these questions. 

Workshop: Combinatorial Hopf Algebras and Macdonald Polynomials
May 7-11, 2007

Centre de recherches mathématiques

Scientific organizers: M. Aguiar (Texas A&M), F. Bergeron (UQAM), N. Bergeron (York), M. Haiman (Berkeley) and S. van Willigenburg (UBC)

The goal of this workshop is to take stock of ongoing work, and of the many rich problems that still need to be addressed in two area, naturally linked through the combinatorics behind the study of Macdonald polynomials. On one side, the recent past has seen a marked deepened interest in the study of graded Hopf algebras, in part because of their fundamental interactions with algebraic combinatorics, but also because of their importance for Theoretical Physics. In particular, it has recently been made apparent that Hopf Algebras play a crucial role in the study of renormalizations in quantum electrodynamics. On the other hand, in the realm of symmetric and quasisymmetric functions it also appears to play a very significant role, with surprising repercussions in representation theory, algebraic geometry, mathematical physics, and the combinatorics of Macdonald polynomials. From another perspective, there has been a lot of recent developments regarding combinatorial models for Macdonald polynomials and their link to diagonal coinvariant spaces.We expect to link these often complementary point of view.