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The boundary of Young graph with Jack edge multiplicities

Sergei Kerov·Andrei Okounkov·Grigori Olshanski
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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 $θ$. 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.

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BibTeXRIS

Sergei Kerov, Andrei Okounkov, Grigori Olshanski. 1997-03-21. The boundary of Young graph with Jack edge multiplicities. https://arxiv.org/abs/q-alg/9703037

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