The Young Bouquet and Its Boundary
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...
| Main Authors: | , |
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| Other Authors: | |
| Format: | Article |
| Language: | en_US |
| Published: |
Independent University of Moscow
2013
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| Online Access: | http://hdl.handle.net/1721.1/81879 https://orcid.org/0000-0002-2913-5238 |
| 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. |
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