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...
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| Format: | Article |
| Language: | en_US |
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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 |
| _version_ | 1826201796053303296 |
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| author | Borodin, Alexei Olshanski, Grigori |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Borodin, Alexei Olshanski, Grigori |
| author_sort | Borodin, Alexei |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T11:57:02Z |
| format | Article |
| id | mit-1721.1/81879 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T11:57:02Z |
| publishDate | 2013 |
| publisher | Independent University of Moscow |
| record_format | dspace |
| spelling | mit-1721.1/818792022-09-27T23:01:21Z The Young Bouquet and Its Boundary Borodin, Alexei Olshanski, Grigori Massachusetts Institute of Technology. Department of Mathematics Borodin, Alexei 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. National Science Foundation (U.S.) (NSF-grant DMS-1056390) Simons Foundation (Simons–IUM Fellowship) Russian Foundation for Basic Research (RFBR-CNRS grant 10-01-93114) Universität Bielefeld (project SFB 701) 2013-10-30T15:46:48Z 2013-10-30T15:46:48Z 2013-04 2012-09 Article http://purl.org/eprint/type/JournalArticle http://hdl.handle.net/1721.1/81879 Borodin, Alexei and Grigori Olshanski. "THE YOUNG BOUQUET AND ITS BOUNDARY." Moscow Mathematical Journal 13:2 (2013) Pp.193–232. https://orcid.org/0000-0002-2913-5238 en_US http://www.mathjournals.org/mmj/2013-013-002/2013-013-002-002.pdf Moscow Mathematical Journal Creative Commons Attribution-Noncommercial-Share Alike 3.0 http://creativecommons.org/licenses/by-nc-sa/3.0/ application/pdf Independent University of Moscow arXiv |
| spellingShingle | Borodin, Alexei Olshanski, Grigori The Young Bouquet and Its Boundary |
| title | The Young Bouquet and Its Boundary |
| title_full | The Young Bouquet and Its Boundary |
| title_fullStr | The Young Bouquet and Its Boundary |
| title_full_unstemmed | The Young Bouquet and Its Boundary |
| title_short | The Young Bouquet and Its Boundary |
| title_sort | young bouquet and its boundary |
| url | http://hdl.handle.net/1721.1/81879 https://orcid.org/0000-0002-2913-5238 |
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