Macdonald polynomials and extended Gelfand-Tsetlin graph

TL;DR

This paper constructs a family of Markov chains on extended Gelfand-Tsetlin graph vertices using Macdonald polynomials, describing their entrance boundaries and connecting to asymptotic representation theory.

Contribution

It introduces a novel construction of transition probabilities on the extended Gelfand-Tsetlin graph via Macdonald polynomials, advancing the understanding of asymptotic representation theory.

Findings

Describes the entrance boundaries of the constructed Markov chains.

Links Macdonald polynomials to the structure of the extended Gelfand-Tsetlin graph.

Provides a foundation for analyzing large-N limits in deformed particle ensembles.

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.