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Moscow Mathematical Journal

Volume 13, Issue 2, April–June 2013  pp. 193–232.

The Young Bouquet and its Boundary

Authors Alexei Borodin (1) and Grigori Olshanski (2)
Author institution: (1) California Institute of Technology, USA; Massachusetts Institute of Technology, USA; Institute for Information Transmission Problems, Moscow, Russia
(2) Institute for Information Transmission Problems, Moscow, Russia; Independent University of Moscow, Russia; National Research University Higher School of Economics, 20 Myasnitskaya Ulitsa, Moscow 101000, Russia

Summary:  The classification results for the extreme characters of two basic “big” groups, the infinite symmetric group S(∞) and the infinite-dimensional unitary group U(∞), 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(∞) and U(∞), 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.

2010 Mathematics Subject Classification. 05E05, 20C32, 60C05, 60J50.

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