Open Problems Concerning Diagonal
Université du Québec
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)uqam.ca
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