|
Representations of Lie groups and random matrices
Author(s):
Benoît
Collins;
Piotr
Sniady
Journal:
Trans. Amer. Math. Soc.
361
(2009),
3269-3287.
MSC (2000):
Primary 22E46;
Secondary 46L53, 15A52
Posted:
January 27, 2009
MathSciNet review:
2485426
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
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:
-
- [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)
Similar Articles:
Retrieve articles in Transactions of the American Mathematical
Society
with
MSC (2000):
22E46,
46L53, 15A52
Retrieve articles in all Journals with
MSC (2000):
22E46,
46L53, 15A52
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:
10.1090/S0002-9947-09-04624-8
PII:
S 0002-9947(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
Posted:
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 \text {2 P03A 007 23}, RTN network: QP-Applications contract No. HPRN-CT-2002-00279, and KBN-DAAD project 36/2003/2004.
Copyright of article:
Copyright
2009,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
