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Macdonald polynomials and extended Gelfand–Tsetlin graph

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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.

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Acknowledgements

I am grateful to Cesar Cuenca and an anonymous referee for valuable comments.

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Authors and Affiliations

  1. Institute for Information Transmission Problems of the Russian Academy of Sciences, Moscow, Russia

    Grigori Olshanski

  2. Skolkovo Institute for Science and Technology, Moscow, Russia

    Grigori Olshanski

  3. Faculty of Mathematics, HSE University, Moscow, Russia

    Grigori Olshanski

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  1. Grigori Olshanski
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Correspondence to Grigori Olshanski.

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G. Olshanski: Research supported by the Russian Science Foundation under Project 20-41-09009.

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Olshanski, G. Macdonald polynomials and extended Gelfand–Tsetlin graph. Sel. Math. New Ser. 27, 41 (2021). https://doi.org/10.1007/s00029-021-00660-3

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