TY - JOUR

T1 - Macdonald polynomials and extended Gelfand–Tsetlin graph

AU - Olshanski, Grigori

N1 - Funding Information:
G. Olshanski: Research supported by the Russian Science Foundation under Project 20-41-09009.
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

PY - 2021/7

Y1 - 2021/7

N2 - Using Okounkov’s q-integral representation of Macdonald polynomials we construct an infinite sequence Ω 1, Ω 2, Ω 3, ⋯ of countable sets linked by transition probabilities from Ω N to Ω N-1 for each N= 2 , 3 , ⋯. The elements of the sets Ω N 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.

AB - Using Okounkov’s q-integral representation of Macdonald polynomials we construct an infinite sequence Ω 1, Ω 2, Ω 3, ⋯ of countable sets linked by transition probabilities from Ω N to Ω N-1 for each N= 2 , 3 , ⋯. The elements of the sets Ω N 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.

KW - Entrance boundary

KW - Gelfand–Tsetlin graph

KW - Macdonald polynomials

KW - Markov chains

KW - Okounkov’s q-integral formula

UR - http://www.scopus.com/inward/record.url?scp=85107803448&partnerID=8YFLogxK

U2 - 10.1007/s00029-021-00660-3

DO - 10.1007/s00029-021-00660-3

M3 - Article

AN - SCOPUS:85107803448

VL - 27

JO - Selecta Mathematica, New Series

JF - Selecta Mathematica, New Series

SN - 1022-1824

IS - 3

M1 - 41

ER -