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2019 서울대학교 수리과학부 세미나 



Macdonald Polynomials: Representation Theory and Combinatorics I 

Cristian Lenart (State University of New York at Albany ) 
Macdonald polynomials are associated with an irreducible affine root system, and are of two types: symmetric and nonsymmetric. The former are orthogonal polynomials with rational function coefficients in q,t, which are invariant under the corresponding finite Weyl group; upon setting q=t=0, they specialize to the irreducible characters of semisimple Lie algebras, in particular to Schur polynomials in type A. Macdonald polynomials have deep connections with: double affine Hecke algebras (DAHA), padic groups, integrable systems, conformal field theory, statistical mechanics, Hilbert schemes etc. This series of lectures will explore two closely related sides of the story of Macdonald polynomials: their central role in the representation theory of affine Lie algebras, and combinatorial constructions. I will start with the definition of Macdonald polynomials, and their construction in terms of the DAHA. I will continue with two classes of combinatorial formulas for Macdonald polynomials and the connection between them: the typeindependent RamYip formula, based on the socalled alcove model, and tableau formulas in classical types. Then I will discuss the way in which various specializations of Macdonald polynomials occur in representation theory, particularly as graded characters of certain modules for affine Lie algebras (Demazure modules, KirillovReshetikhin modules, and several variations of them). The mentioned alcove model leads to a combinatorial model for the corresponding Kashiwara crystals; these are colored directed graphs encoding representations of quantum algebras in the limit of the quantum parameter going to 0. I will conclude with several recent developments in the area. The lectures contain joint work with my collaborators: Satoshi Naito, Daisuke Sagaki, Anne Schilling, Travis Scrimshaw, and Mark Shimozono, as well as my students Arthur Lubovsky and Adam Schultze. They will be largely selfcontained, and only basic knowledge of the representation theory of Lie algebras is assumed. 







