2 Boundary of Young Lattice

The description of characters of the innnite symmetric group S1 = lim ?! Sn is considered in the paper as an asymptotical combinatorial problem. It is equivalent to the characterization problem of totally positive sequences. We derive the list of characters, rst obtained by E. Thoma in 1964, using appropriate exact formulae for the number of skew Young tableaux. The character space is treated as a boundary of the Young lattice. The characterproblemfor a locally nite group and the classicalde Finetti theorem on exchangeable random sequences are two particular cases of a more general problem in potential theory on a graph, namely that of the explicite description of Martin boundary of this graph. We show that Selberg type integrals arise naturally in this general setup in connection with Pieri formula for Jack symmetric polynomials. Some remarkable connections between Plancherel measures of symmetric groups and topics in Analysis are indicated.