Macdonald polynomials and extended Gelfand–Tsetlin graph

Grigori Olshanski ^[1]
1. [1] Russian Academy of Sciences
  
  Russian Academy of Sciences
  
  Rusia
Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 27, Nº. 3, 2021
Idioma: inglés
DOI: 10.1007/s00029-021-00660-3
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Resumen
- Using Okounkov’s q-integral representation of Macdonald polynomials we construct an infinite sequence Ω1,Ω2,Ω3,… of countable sets linked by transition probabilities from ΩN to ΩN−1 for each N=2,3,…. The elements of the sets ΩN are the vertices of the extended Gelfand–Tsetlin graph, and the transition probabilities depend on the two Macdonald parameters, q and t. These data determine a family of Markov chains, and the main result is the description of their entrance boundaries. This work has its origin in asymptotic representation theory. In the subsequent paper, the main result is applied to large-N limit transition in (q, t)-deformed N-particle beta-ensembles.