Representations of classical Lie groups and quantized free convolution
We study the decompositions into irreducible components of tensor products and restrictions of irreducible representations for all series of classical Lie groups as the rank of the group goes to infinity. We prove the Law of Large Numbers for the random counting measures describing the decomposition...
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| Format: | Article |
| Language: | English |
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Springer Basel
2017
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| Online Access: | http://hdl.handle.net/1721.1/106922 https://orcid.org/0000-0002-9828-5862 |
| _version_ | 1826193309971775488 |
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| author | Bufetov, Alexey Gorin, Vadim |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Bufetov, Alexey Gorin, Vadim |
| author_sort | Bufetov, Alexey |
| collection | MIT |
| description | We study the decompositions into irreducible components of tensor products and restrictions of irreducible representations for all series of classical Lie groups as the rank of the group goes to infinity. We prove the Law of Large Numbers for the random counting measures describing the decomposition. This leads to two operations on measures which are deformations of the notions of the free convolution and the free projection. We further prove that if one replaces counting measures with others coming from the work of Perelomov and Popov on the higher order Casimir operators for classical groups, then the operations on the measures turn into the free convolution and projection themselves. We also explain the relation between our results and limit shape theorems for uniformly random lozenge tilings with and without axial symmetry. |
| first_indexed | 2024-09-23T09:36:57Z |
| format | Article |
| id | mit-1721.1/106922 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T09:36:57Z |
| publishDate | 2017 |
| publisher | Springer Basel |
| record_format | dspace |
| spelling | mit-1721.1/1069222022-09-26T12:41:15Z Representations of classical Lie groups and quantized free convolution Bufetov, Alexey Gorin, Vadim Massachusetts Institute of Technology. Department of Mathematics Gorin, Vadim We study the decompositions into irreducible components of tensor products and restrictions of irreducible representations for all series of classical Lie groups as the rank of the group goes to infinity. We prove the Law of Large Numbers for the random counting measures describing the decomposition. This leads to two operations on measures which are deformations of the notions of the free convolution and the free projection. We further prove that if one replaces counting measures with others coming from the work of Perelomov and Popov on the higher order Casimir operators for classical groups, then the operations on the measures turn into the free convolution and projection themselves. We also explain the relation between our results and limit shape theorems for uniformly random lozenge tilings with and without axial symmetry. Russian Foundation for Basic Research (Centre National de la Recherche Scientifique (France) RFBR-CNRS grant 11-01-93105) National Science Foundation (U.S.) (grant DMS-1407562) 2017-02-10T23:49:33Z 2017-02-10T23:49:33Z 2015-03 2016-08-18T15:40:22Z Article http://purl.org/eprint/type/JournalArticle 1016-443X 1420-8970 http://hdl.handle.net/1721.1/106922 Bufetov, Alexey, and Vadim Gorin. “Representations of Classical Lie Groups and Quantized Free Convolution.” Geometric and Functional Analysis 25, no. 3 (March 6, 2015): 763–814. https://orcid.org/0000-0002-9828-5862 en http://dx.doi.org/10.1007/s00039-015-0323-x Geometric and Functional Analysis Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Springer Basel application/pdf Springer Basel Springer Basel |
| spellingShingle | Bufetov, Alexey Gorin, Vadim Representations of classical Lie groups and quantized free convolution |
| title | Representations of classical Lie groups and quantized free convolution |
| title_full | Representations of classical Lie groups and quantized free convolution |
| title_fullStr | Representations of classical Lie groups and quantized free convolution |
| title_full_unstemmed | Representations of classical Lie groups and quantized free convolution |
| title_short | Representations of classical Lie groups and quantized free convolution |
| title_sort | representations of classical lie groups and quantized free convolution |
| url | http://hdl.handle.net/1721.1/106922 https://orcid.org/0000-0002-9828-5862 |
| work_keys_str_mv | AT bufetovalexey representationsofclassicalliegroupsandquantizedfreeconvolution AT gorinvadim representationsofclassicalliegroupsandquantizedfreeconvolution |