Isaac Newton Institute for Mathematical Sciences

Symmetric Functions 2001: Surveys of Developments and Perspectives


Persi Diaconis, Stanford University, U.S.A.
Lecture 1: Kerov's work on the Markov moment problem
The Markov moment problem characterizes measures with a density by their moments. Kerov expanded and deeply understood this little known result and its applications. I will explain Markov's and Kerov's results and then some simplifications and new variations.
Lecture 2: Applications of symmetric function theory to random matrix theory
Random matrix theory studies the eigenvalue distribution of typical large matrices. These are used to model things like particle scattering and the zeros of the zeta function. In this talk I will show how basic facts of symmetric function theory yield very precise results in random matrix theory. This is joint work with Mehrdad Shahshahani and Steve Evans.
Lecture 3: Applications of symmetric function theory to Szego's theorem
Szego's theorem studies the eigenvalues of large "striped" matrices. It is used by statisticians for covariance matrices, by physicists for analysis of Ising models and by operator algebra types. I will show how deep theorems in this subject are simple consequences of elementary properties of Schur functions. More refined properties lead to new theorems in the Toeplitz world. This is joint work with Dan Bump.

William Fulton, University of Michigan, U.S.A.
Title: Eigenvalues and Schubert calculus
1. Eigenvalues, Invariant factors, highest weights, and Schubert calculus, Bull. Amer. Math. Soc. 37, 2000, 209--249.
Mark Haiman, University of California at San Diego, U.S.A.
Title: Macdonald polynomials and the geometry of Hilbert schemes
References (these papers are available from the lecturer's homepage):
1. Hilbert schemes, polygraphs, and the Macdonald positivity conjecture. Journal of the A.M.S., to appear.
2. Vanishing theorems and character formulas for the Hilbert scheme of points in the plane. Abbreviated version to appear in Proceedings of the International Workshop on Physics and Combinatorics, Nagoya, August, 2000.
Phil Hanlon, University of Michigan, U.S.A.
Title: The Laplacian Method
Alexander Klyachko, Bilkent University, Turkey
Title: Principal bundles, representation theory, and spectral problems for semisimple groups

Bernard Leclerc, Université de Caen, France
Title: Symmetric functions and Fock space representations of quantum affine algebras
References (these papers are available from the lecturer's homepage):
1. C. Carre, B. Leclerc, Splitting the square of a Schur function into its symmetric and antisymmetric parts, Journal of Algebraic Combinatorics, 4 (1995), 201-231.
2. A.Lascoux, B. Leclerc, J-Y Thibon, Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Commun. Math. Phys. 181 (1996), 205-263.
3. A. Lascoux, B. Leclerc, J-Y Thibon, Ribbon tableaux, Hall-Littlewood functions, quantum affine algebras and unipotent varieties, J. of Math. Phys., 38 (1997), 1041-1068.
4. B. Leclerc, J-Y Thibon, Littlewood-Richardson coefficients and Kazhdan-Lusztig polynomials, in "Combinatorial Methods in Representation Theory", Adv. Stud. Pure Math. Vol. 28, (2000), 155-220.
Ian G. Macdonald, QMW, University of London, U.K.
Title: Symmetric functions and orthogonal polynomials: problems and prospects
Masatoshi Noumi, Kobe University, Japan
Title: Weyl group approach to nonlinear integrable systems
I will give an overview on recent developments in Weyl group approach to nonlinear (discrete) integrable systems, with emphasis on special polynomials related to Painleve type equations.
Andrei Okounkov, University at California at Berkeley, U.S.A.
Title: Symmetric functions and random partitions
Abstract: I will explain how to use symmetric functions to derive probabilistically interesting results about large random partitions. General constructions will be illustrated by two main examples: the Plancherel measure on usual partitions and the uniform measure on plane partitions. Special focus will be on the analysis of local shape fluctuations.
Grigori Olshanski, Dobrushin Lab, IPPI, Russia
Title: Polynomial functions on the set of Young diagrams and Kerov's central limit theorem
1. A. M. Vershik and S. V. Kerov, Asymptotic theory of characters of the symmetric group, Funct. Anal. Appl. 15 (1981), No. 4, 246--255.
2. S. Kerov, Gaussian limit for the Plancherel measure of the symmetric group, Comptes Rendus Acad. Sci. Paris Ser. I, vol. 316 (1993), 303-308.
3. S. Kerov and G. Olshanski Polynomial functions on the set of Young diagrams Comptes Rendus Acad. Sci. Paris Ser. I, vol. 319 (1994), 121-126.
4. A. Okounkov and G. Olshanski, Shifted Schur functions, Algebra i Analiz 9 (1997), No. 2, 73-146 (Russian); English transl.: St. Petersburg Math. J., 9 (1998), 239--300.
5. A. Hora, Central limit theorem for the adjacency operators on the infinite symmetric group, Comm. Math. Phys. 195 (1998), 405-416.
6. G. Olshanski, A. Regev and A. Vershik, Frobenius-Schur functions: summary of results,
Eric Opdam, Korteweg-de Vries Institute, The Netherlands
Lecture 1: On the rational Dunkl-Cherednik algebra
Lectures 2 and 3: Spectral theory of affine Hecke algebras
1. E.M.Opdam, A generating function for the trace of the Iwahori-Hecke algebra, math.RT/0101006.
2. E.M.Opdam, On the spectral decomposition of affine Hecke algebras, math.RT/0101007.
Anatoly Vershik, Steklov Institute, St.Petersburg Branch, Russia
Lecture 1: Character theory of the group of infinite matrices over a finite field
Lecture 2: Our mathematical work with Sergei Kerov

Andrei Zelevinsky, Northeastern University, U.S.A.
Lecture 1: Generalizing the Littlewood-Richardson rule
Lecture 2: Canonical bases and total positivity
Lecture 3: Introduction to cluster algebras

compiled by Sergey Fomin on June 19, 2001