dorsal/arxiv
View SchemaThe boundary of Young graph with Jack edge multiplicities
| Authors | Sergei Kerov, Andrei Okounkov, Grigori Olshanski |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9703037 |
| URL | https://arxiv.org/abs/q-alg/9703037 |
| Journal | Intern. Math. Research Notices 1998, no. 4, 173-199. |
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.
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"abstract": "Consider the lattice of all Young diagrams ordered by inclusion, and denote\nby Y its Hasse graph. Using the Pieri formula for Jack symmetric polynomials,\nwe endow the edges of the graph Y with formal multiplicities depending on a\nreal parameter $\\theta$. The multiplicities determine a potential theory on the\ngraph Y. Our main result identifies the corresponding Martin boundary with an\ninfinite-dimensional simplex, the ``geometric boundary\u0027\u0027 of the Young graph Y,\nand provides a canonical integral representation for non-negative harmonic\nfunctions.\n For three particular values of the parameter, the theorem specializes to\nknown results: the Thoma theorem describing characters of the infinite\nsymmetric group, the Kingman\u0027s classification of partition structures, and the\ndescription of spherical functions of the infinite hyperoctahedral Gelfand\npair.",
"arxiv_id": "q-alg/9703037",
"authors": [
"Sergei Kerov",
"Andrei Okounkov",
"Grigori Olshanski"
],
"categories": [
"q-alg",
"math.QA"
],
"journal_ref": "Intern. Math. Research Notices 1998, no. 4, 173-199.",
"title": "The boundary of Young graph with Jack edge multiplicities",
"url": "https://arxiv.org/abs/q-alg/9703037"
},
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