Luc Vinet, McGill University, Montréal, Canada
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Advances in multivariable special functions and mathematical physics

Abstract

My lectures will be based on recent work of L.~Lapointe and J.~Morse, done in collaboration with A.~Lascoux.  This work introduces new families of symmetric polynomials, called atoms, indexed by partitions and depending on a parameter $t$ and a positive integer level $k$. The atoms are defined as sums of tableaux using the charge statistics encountered in the combinatorial interpretation of the $t$-Kostka polynomials, and are such that they become smaller and smaller as the level grows, reducing to a single Schur functions when $k$ is large enough. For a given level, the atoms form a basis of a subspace of the symmetric function space.  It is conjectured that the Macdonald polynomials expand positively in terms of this new basis, which can be seen as a generalization of Macdonald's conjecture on the positivity of the $q,t$-Kostka polynomials.

The first lecture will give a general overview of the subject, with a substantial part devoted to the presentation of basic combinatorial tools and concepts, such as the Knuth equivalence, the Robinson-Schensted correspondance, katabolism, etc.  The second lecture will give appropriate background on symmetric polynomials and creation operators and will introduce more thoroughly the atoms and discuss some of their properties.