Macdonald polynomials and extended Gelfand-Tsetlin graph

Abstract

Using Okounkov's $q$-integral representation of Macdonald polynomials we construct an infinite sequence $\Omega_1,\Omega_2,\Omega_3,\dots$ of countable sets linked by transition probabilities from $\Omega_N$ to $\Omega_{N-1}$ for each $N=2,3,\dots$. The elements of the sets $\Omega_N$ 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.