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Representations of Lie groups and random matrices


Authors: Benoît Collins and Piotr Sniady
Journal: Trans. Amer. Math. Soc. 361 (2009), 3269-3287
MSC (2000): Primary 22E46; Secondary 46L53, 15A52
DOI: https://doi.org/10.1090/S0002-9947-09-04624-8
Published electronically: January 27, 2009
MathSciNet review: 2485426
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Abstract: We study the asymptotics of representations of a fixed compact Lie group. We prove that the limit behavior of a sequence of such representations can be described in terms of certain random matrices; in particular operations on representations (for example: tensor product, restriction to a subgroup) correspond to some natural operations on random matrices (respectively: sum of independent random matrices, taking the corners of a random matrix). Our method of proof is to treat the canonical block matrix associated to a representation as a random matrix with non-commutative entries.


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Additional Information

Benoît Collins
Affiliation: Department of Mathematics and Statistics, University of Ottawa, 585 King Edward, Ottawa, Ontario, Canada K1N 6N5 – and – CNRS, UMR 5208, Institut Camille Jordan, Université Lyon 1, 21 av Claude Bernard, 69622 Villeurbanne, France
Email: collins@math.univ-lyon1.fr

Piotr Sniady
Affiliation: Institute of Mathematics, University of Wroclaw, pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland
Email: Piotr.Sniady@math.uni.wroc.pl

DOI: https://doi.org/10.1090/S0002-9947-09-04624-8
Received by editor(s): October 10, 2006
Received by editor(s) in revised form: June 5, 2007, June 26, 2007, and August 22, 2007
Published electronically: January 27, 2009
Additional Notes: The research of the first author was partly supported by a RIMS fellowship and by CNRS
The research of the second author was supported by State Committee for Scientific Research (KBN) grant 2 P03A 007 23, RTN network: QP-Applications contract No. HPRN-CT-2002-00279, and KBN-DAAD project 36/2003/2004.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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