Abstract:
The classification results for the extreme characters of two basic “big” groups, the infinite symmetric group S(∞)S(∞) and the infinite-dimensional unitary group U(∞)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(∞)S(∞) and U(∞)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.
Key words and phrases:
Young graph, Gelfand–Tsetlin graph, entrance boundary, infinite symmetric group, infinite-dimensional unitary group, characters, Gibbs measures.
Received:October 19, 2011; in revised form September 18, 2012
\Bibitem{BorOls13}
\by Alexei~Borodin, Grigori~Olshanski
\paper The Young bouquet and its boundary
\jour Mosc. Math.~J.
\yr 2013
\vol 13
\issue 2
\pages 193--232
\mathnet{http://mi.mathnet.ru/mmj495}
\crossref{https://doi.org/10.17323/1609-4514-2013-13-2-193-232}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3134905}
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Linking options:
https://www.mathnet.ru/eng/mmj495
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