The young bouquet and its boundary

Alexei Borodin, Grigori Olshanski

Research output: Contribution to journalArticlepeer-review

23 Citations (Scopus)

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 languageEnglish
Pages (from-to)193-232
Number of pages40
JournalMoscow Mathematical Journal
Volume13
Issue number2
DOIs
Publication statusPublished - 2013
Externally publishedYes

Keywords

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

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