Applications of Macdonald Polynomials

s for Talks Speaker: Nick Loehr (Virginia Tech, USA) (talk describes joint work with Jim Haglund and Mark Haiman) Title: Symmetric and Non-symmetric Macdonald Polynomials Abstract: Macdonald polynomials have played a central role in symmetric function theory ever since their introduction by Ian Macdonald in 1988. The original algebraic definitions of these polynomials are very nonexplicit and difficult to work with. Haglund conjectured an explicit combinatorial formula for the Macdonald polynomials. This was later extended to a combinatorial formula for non-symmetric Macdonald polynomials in type A. This talk will discuss the algebraic and combinatorial definitions of both symmetric and nonsymmetric Macdonald polynomials. We also sketch the main ideas in the proofs that the algebraic and combinatorial constructions are equal. Speaker: Jim Haglund (Univ. of Pennsylvania, USA) will deliver a talk prepared by Greg Warrington (Wake Forest, USA) who had to cancel his trip Title: Combinatorical structures associated to the nabla operator Abstract: Over the past ten years, there has been a rich interplay among the modified Macdonald polynomials, the diagonal harmonics modules, the nabla operator, and the combinatorics of q,t-weighted lattice paths. In this talk, we review these connections, paying particular attention to the q,t-Catalan numbers. We finish with recent joint work of N. Loehr and G. Warrington regarding a nested-lattice-path interpretation for nabla applied to arbitrary Schur functions. Speaker: Sami Assaf (Univ. of Pennsylvania, USA) Title: A combinatorial proof of Macdonald positivity Abstract:Taking Haglund’s formula for the transformed Macdonald polynomials expressed in terms of monomials as the definition, we present a self-contained, combinatorial proof of symmetry and Schur positivity of Macdonald polynomials, and give a combinatorial interpretation of the Schur coefficients. The method of the proof uses the theory of dual equivalence graphs and a new generalization of them called D graphs. Speaker: Jennifer Morse (Drexel Univ., USA) Title: An update on the k-Schur approach to statistics problems Abstract: We will review the k-Schur role in the theory of Macdonald polynomials and talk about some related open problems and new conjectures. Speaker:Thomas Lam (Harvard Univ., USA) Title:k-Schur functions and the homology of the affine Grassmannian Abstract: I will explain the relationship between Lapointe, Lascoux and Morse’s k-Schur functions and the Schubert basis of the homologyH∗(Gr) of the affine Grassmannian of SL(n). I will state some general facts about H∗(Gr) then describe Peterson’s work on affine Schubert calculus. Peterson’s work can be connected to k-Schur functions via the Fomin-Stanley subalgebra and the theory of Stanley symmetric functions. Speaker: John Stembridge (Univ. of Michigan, USA) Title: Kostka-Foulkes polynomials of general type and their variations Abstract: In this talk we plan to discuss the general features of Kostka-Foulkes polynomials for finite root systems. We will pose several problems or conjectures aimed at developing a general framework for explaining the nonnegativity of their coefficients in a combinatorial way. If there is time, we will also discuss some additional families of univariate polynomials that also occur in representation theoretic contexts and have the same combinatorial flavor– one related to the Blattner multiplicity formula, and another related to Demazure modules. Speaker: Iain Gordon (University of Edinburgh, United Kingdom) Title: Rational Cherednik algebras, diagonal coinvariants, and other animals Abstract: I will explain how the representation theory of rational Cherednik algebras is used to get a handle on diagonal coinvariants for Weyl groups. This is quite well understood, but may only be part of a broad scheme. Beyond diagonal invariants there is a dream that the representation theory could shed new light on the n! theorem and its conjectural generalisations to wreath products.