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Transactions of the American Mathematical Society

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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
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.

References [Enhancements On Off] (What's this?)

  • [Bia95] Philippe Biane.
    Representations of unitary groups and free convolution.
    Publ. Res. Inst. Math. Sci., 31(1):63-79, 1995. MR 1317523 (96c:22021)
  • [Bia98] Philippe Biane.
    Representations of symmetric groups and free probability.
    Adv. Math., 138(1):126-181, 1998. MR 1644993 (2001b:05225)
  • [BtD95] Theodor Bröcker and Tammo tom Dieck.
    Representations of compact Lie groups, volume 98 of Graduate Texts in Mathematics.
    Springer-Verlag, New York, 1995. MR 1410059 (97i:22005)
  • [CŚ08] Benoıt Collins and Piotr Śniady.
    Representations of Lie groups, random matrices and free probability.
    In preparation, 2008.
  • [DLY02] Anatolij Dvurečenskij, Pekka Lahti, and Kari Ylinen.
    The uniqueness question in the multidimensional moment problem with applications to phase space observables.
    Rep. Math. Phys., 50(1):55-68, 2002. MR 1939941 (2003j:44010)
  • [FH91] William Fulton and Joe Harris.
    Representation theory, volume 129 of Graduate Texts in Mathematics.
    Springer-Verlag, New York, 1991. MR 1153249 (93a:20069)
  • [GvW78] N. Giri and W. von Waldenfels.
    An algebraic version of the central limit theorem.
    Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 42(2):129-134, 1978. MR 0467880 (57:7731a)
  • [GW98] Roe Goodman and Nolan R. Wallach.
    Representations and invariants of the classical groups, volume 68 of Encyclopedia of Mathematics and its Applications.
    Cambridge University Press, Cambridge, 1998. MR 1606831 (99b:20073)
  • [Hec82] G. J. Heckman.
    Projections of orbits and asymptotic behavior of multiplicities for compact connected Lie groups.
    Invent. Math., 67(2):333-356, 1982. MR 665160 (84d:22019)
  • [Joh01] Kurt Johansson.
    Discrete orthogonal polynomial ensembles and the Plancherel measure.
    Ann. of Math. (2), 153(1):259-296, 2001. MR 1826414 (2002g:05188)
  • [Kna02] Anthony W. Knapp.
    Lie groups beyond an introduction, volume 140 of Progress in Mathematics.
    Birkhäuser Boston Inc., Boston, MA, second edition, 2002. MR 1920389 (2003c:22001)
  • [KSW96a] Vladimir A. Kazakov, Matthias Staudacher, and Thomas Wynter.
    Almost flat planar diagrams.
    Comm. Math. Phys., 179(1):235-256, 1996. MR 1395223 (98e:82032)
  • [KSW96b] Vladimir A. Kazakov, Matthias Staudacher, and Thomas Wynter.
    Character expansion methods for matrix models of dually weighted graphs.
    Comm. Math. Phys., 177(2):451-468, 1996. MR 1384144 (97b:81083)
  • [Kup02] Greg Kuperberg.
    Random words, quantum statistics, central limits, random matrices.
    Methods Appl. Anal., 9(1):99-118, 2002. MR 1948465 (2003k:60020)
  • [Kup05] Greg Kuperberg.
    A tracial quantum central limit theorem.
    Trans. Amer. Math. Soc., 357(2):459-471 (electronic), 2005. MR 2095618 (2005k:46171)
  • [Lit95] Peter Littelmann.
    Paths and root operators in representation theory.
    Ann. of Math. (2), 142(3):499-525, 1995. MR 1356780 (96m:17011)
  • [Meh91] Madan Lal Mehta.
    Random matrices.
    Academic Press Inc., Boston, MA, second edition, 1991. MR 1083764 (92f:82002)
  • [Mey93] Paul-André Meyer.
    Quantum probability for probabilists, volume 1538 of Lecture Notes in Mathematics.
    Springer-Verlag, Berlin, 1993. MR 1222649 (94k:81152)
  • [Rie04] Marc A. Rieffel.
    Gromov-Hausdorff distance for quantum metric spaces. Matrix algebras converge to the sphere for quantum Gromov-Hausdorff distance.
    American Mathematical Society, Providence, RI, 2004.
    Mem. Amer. Math. Soc. 168 (2004), no. 796. MR 2055927
  • [Śni06] Piotr Śniady.
    Gaussian fluctuations of characters of symmetric groups and of Young diagrams.
    Probab. Theory Related Fields, 136(2):263-297, 2006. MR 2240789 (2007d:20020)
  • [ŚS08] Piotr Śniady and Roland Speicher.
    Permutationally invariant random matrices.
    In preparation, 2008.
  • [VDN92] D. V. Voiculescu, K. J. Dykema, and A. Nica.
    Free random variables.
    American Mathematical Society, Providence, RI, 1992. MR 1217253 (94c:46133)
  • [Wor87] S. L. Woronowicz.
    Compact matrix pseudogroups.
    Comm. Math. Phys., 111(4):613-665, 1987. MR 901157 (88m:46079)

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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

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

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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