The boundary of Young graph with Jack edge multiplicities

Abstract

Consider the lattice of all Young diagrams ordered by inclusion, and denote by Y its Hasse graph. Using the Pieri formula for Jack symmetric polynomials, we endow the edges of the graph Y with formal multiplicities depending on a real parameter $\theta$. The multiplicities determine a potential theory on the graph Y. Our main result identifies the corresponding Martin boundary with an infinite-dimensional simplex, the ``geometric boundary'' of the Young graph Y, and provides a canonical integral representation for non-negative harmonic functions. For three particular values of the parameter, the theorem specializes to known results: the Thoma theorem describing characters of the infinite symmetric group, the Kingman's classification of partition structures, and the description of spherical functions of the infinite hyperoctahedral Gelfand pair.