**
Symmetric Functions 2001:
Surveys of Developments and Perspectives
**

**
Lectures**

**Persi Diaconis**,
Stanford University, U.S.A.

**Lecture 1:** Kerov's work on the Markov moment problem

**Abstract:**

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

**Abstract:**

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

**Abstract:**

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.

1. Eigenvalues, Invariant factors, highest weights, and Schubert calculus, Bull. Amer. Math. Soc. 37, 2000, 209--249.

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.

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.

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.

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, math.CO/0003031.

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.

*compiled by
Sergey Fomin
on June 19, 2001*