Open Problems Concerning Diagonal Harmoniccs

François Bergeron

Université du Québec à Montréal

November 20,
refreshments at 3:45pm


Even after Mark Haiman algebraic-geometric proofs of the $n!$ conjecture and of the more involved fact that the Frobenius characteristic $F_n$ of the bigraded $S_n$-module of Diagonal Harmonics is explicit given by an intricate formula involving Macdonald symmetric polynomials, there are still many open questions concerning both properties of Macdonald polynomials and related $S_n$-modules. Many of these involve an operator, called $\nabla$, that we introduced many years ago to simplify Haiman's expression for $F_n$. After reviewing the basics, we will discuss these open problems, including a program for a combinatorial proof of the $n!$ conjecture and several extensions.

Speaker's Contact Info: bergeron.francois(at-sign)

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