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JOURNALS // Moscow Mathematical Journal // Archive

Mosc. Math. J., 2013 Volume 13, Number 2, Pages 193–232 (Mi mmj495)

This article is cited in 23 papers

The Young bouquet and its boundary

Alexei Borodinabc, Grigori Olshanskidce

a Massachusetts Institute of Technology, USA
b California Institute of Technology, USA
c Institute for Information Transmission Problems, Moscow, Russia
d Independent University of Moscow, Russia
e National Research University Higher School of Economics, 20 Myasnitskaya Ulitsa, Moscow 101000, Russia

Abstract: The classification results for the extreme characters of two basic “big” groups, the infinite symmetric group $S(\infty)$ and the infinite-dimensional unitary group $U(\infty)$, are remarkably similar. It does not seem to be possible to explain this phenomenon using a suitable extension of the Schur–Weyl duality to infinite dimension. We suggest an explanation of a different nature that does not have analogs in the classical representation theory.
We start from the combinatorial/probabilistic approach to characters of “big” groups initiated by Vershik and Kerov. In this approach, the space of extreme characters is viewed as a boundary of a certain infinite graph. In the cases of $S(\infty)$ and $U(\infty)$, those are the Young graph and the Gelfand–Tsetlin graph, respectively. We introduce a new related object that we call the Young bouquet. It is a poset with continuous grading whose boundary we define and compute. We show that this boundary is a cone over the boundary of the Young graph, and at the same time it is also a degeneration of the boundary of the Gelfand–Tsetlin graph.
The Young bouquet has an application to constructing infinite-dimensional Markov processes with determinantal correlation functions.

Key words and phrases: Young graph, Gelfand–Tsetlin graph, entrance boundary, infinite symmetric group, infinite-dimensional unitary group, characters, Gibbs measures.

MSC: 05E05, 20C32, 60C05, 60J50

Received: October 19, 2011; in revised form September 18, 2012

Language: English

DOI: 10.17323/1609-4514-2013-13-2-193-232



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