## Abstract

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 ex- tension 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-di- mensional Markov processes with determinantal correlation functions.

Original language | English |
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Pages (from-to) | 193-232 |

Number of pages | 40 |

Journal | Moscow Mathematical Journal |

Volume | 13 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2013 |

Externally published | Yes |

## Keywords

- Characters
- Entrance boundary
- Gelfand-Tsetlin graph
- Gibbs measures
- Infinite symmetric group
- Infinite-dimensional unitary group
- Young graph