Variations on the Theme of Harmonics for the Symmetric Group

Francois Bergeron
(Universite du Quebec a Montreal)

We will present the "classical" theory of harmonics of the symmetric group (as a special case of results of Steinberg and others), and generalizations suggested by the study of representations theoretic models for Macdonald symmetric polynomials (although no knowledge of these polynomials will be used). The only background needed will be basic linear algebra and algebra, and some combinatorics, and even with this restriction we will be able to present exciting new (and not so new) conjectures. If time allows, we will suggest how deeper conjectures can be obtained in the framework of representations of the symmetric group.