The Gelfand-Tsetlin graph and Markov processes

Abstract

The goal of the paper is to describe new connections between representation theory and algebraic combinatorics on one side, and probability theory on the other side. The central result is a construction, by essentially algebraic tools, of a family of Markov processes. The common state space of these processes is an infinite dimensional (but locally compact) space Omega. It arises in representation theory as the space of indecomposable characters of the infinite-dimensional unitary group U(infinity). Alternatively, Omega can be defined in combinatorial terms as the boundary of the Gelfand-Tsetlin graph --- an infinite graded graph that encodes the classical branching rule for characters of the compact unitary groups U(N). We also discuss two other topics concerning the Gelfand-Tsetlin graph: (1) Computation of the number of trapezoidal Gelfand-Tsetlin schemes (one could also say, the number of integral points in a truncated Gelfand-Tsetlin polytope). The formula we obtain is well suited for asymptotic analysis. (2) A degeneration procedure relating the Gelfand-Tsetlin graph to the Young graph by means of a new combinatorial object, the Young bouquet. At the end we discuss a few related works and further developments.Comment: Extended version of invited sectional lecture at ICM 2014, Seou