q,t-Kostka polynomials and the affine Weyl group

Symmetric functions called k-Schur functions arose in our study of an open problem on Macdonald polynomials. We will discuss properties of these functions that led us to the refinement of classical ideas in symmetric function theory such as Pieri rules, Kostka numbers, the Young lattice, and Young tableaux. Further, we will show how these functions enabled us to find a bijection between refined Young tableaux and reduced words for affine permutations. We will finish by indicating a possible connection between the coefficients in the k-Schur function expansion of Macdonald polynomials and the affine symmetric group. All necessary background for this talk will be provided in the pre-talk.

The topic of the pre-talk will be the combinatorics of Schur functions. We will discuss combinatorial properties of Schur functions revealing their fundamental role in symmetric function theory and leading to an open problem in the field.