M. B. Porter Lectures, 2006-2007.

Rice University
Department of Mathematics

Alexei Borodin

March 12 - March 21, 2007

Representations of symmetric groups and determinantal point processes

The infinite symmetric group S(\infty) is a big group in the sense that the space of its irreducible characters is infinite-dimensional.  The description of this space is a classical result known as Thoma's theorem (1964). It is (nontrivially) equivalent to another well-known theorem -- classification of totally positive sequences due to Edrei and Schoenberg (1953). The main goal of the first two lectures will be to explain these theorems as well as ideas behind one of the proofs.

The regular representation of S(\infty) is irreducible as opposed to the case of finite groups when the decomposition of the regular representation yields all irreducible ones. In the third lecture we will provide a geometric construction of a deformation of the regular representation of S(\infty) which makes the problem of harmonic analysis (=decomposition on irreducibles) meaningful.

In the last lecture we will solve the problem of harmonic analysis for the generalized regular representations of S(\infty) using tools from probability theory (determinantal point processes). Connections with random matrices will be emphasized.

Monday, March 12
4:00 - 5:00 PM     Keck Lecture Hall 100

Lecture 1. Discrete probabilistic models of random matrix type
In recent years there has been a lot of progress in understanding the objects like increasing subsequences of random permutations, random directed polymers, various models of vicious walkers, random tilings etc. that miraculously exhibit similar behavior as they become ``large''. The goal of the talk is to give a survey of such objects with emphasis on unexpected connections between various domains of mathematics and mathematical physics.

Wednesday, March 14
4:00 - 5:00 PM     Herman Brown 227

Lecture 2. Thoma's theorem I

Thursday, March 15
4:00 - 5:00 PM     Herman Brown 227

Lecture 3. Thoma's theorem II

Monday, March 19
4:00 - 5:00 PM     Herman Brown 227

Lecture 4. Generalized regular representations of the infinite symmetric group

Wednesday, March 21
4:00 - 5:00 PM     Herman Brown 227

Lecture 5. Harmonic analysis on S(\infty) and determinantal point processes

There will be a tea preceding each lecture in the Mathematics Commons Room, Herman Brown 438.

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