Representations of Lie groups and random matrices
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- by Benoît Collins and Piotr Śniady PDF
- Trans. Amer. Math. Soc. 361 (2009), 3269-3287 Request permission
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
- Philippe Biane, Representations of unitary groups and free convolution, Publ. Res. Inst. Math. Sci. 31 (1995), no. 1, 63–79. MR 1317523, DOI 10.2977/prims/1195164791
- Philippe Biane, Representations of symmetric groups and free probability, Adv. Math. 138 (1998), no. 1, 126–181. MR 1644993, DOI 10.1006/aima.1998.1745
- Theodor Bröcker and Tammo tom Dieck, Representations of compact Lie groups, Graduate Texts in Mathematics, vol. 98, Springer-Verlag, New York, 1995. Translated from the German manuscript; Corrected reprint of the 1985 translation. MR 1410059
- Benoît Collins and Piotr Śniady. Representations of Lie groups, random matrices and free probability. In preparation, 2008.
- 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 (2002), no. 1, 55–68. MR 1939941, DOI 10.1016/S0034-4877(02)80044-1
- William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249, DOI 10.1007/978-1-4612-0979-9
- N. Giri and W. von Waldenfels, An algebraic version of the central limit theorem, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 42 (1978), no. 2, 129–134. MR 467880, DOI 10.1007/BF00536048
- Roe Goodman and Nolan R. Wallach, Representations and invariants of the classical groups, Encyclopedia of Mathematics and its Applications, vol. 68, Cambridge University Press, Cambridge, 1998. MR 1606831
- G. J. Heckman, Projections of orbits and asymptotic behavior of multiplicities for compact connected Lie groups, Invent. Math. 67 (1982), no. 2, 333–356. MR 665160, DOI 10.1007/BF01393821
- Kurt Johansson, Discrete orthogonal polynomial ensembles and the Plancherel measure, Ann. of Math. (2) 153 (2001), no. 1, 259–296. MR 1826414, DOI 10.2307/2661375
- Anthony W. Knapp, Lie groups beyond an introduction, 2nd ed., Progress in Mathematics, vol. 140, Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1920389
- Vladimir A. Kazakov, Matthias Staudacher, and Thomas Wynter, Almost flat planar diagrams, Comm. Math. Phys. 179 (1996), no. 1, 235–256. MR 1395223, DOI 10.1007/BF02103721
- Vladimir A. Kazakov, Matthias Staudacher, and Thomas Wynter, Character expansion methods for matrix models of dually weighted graphs, Comm. Math. Phys. 177 (1996), no. 2, 451–468. MR 1384144, DOI 10.1007/BF02101902
- Greg Kuperberg, Random words, quantum statistics, central limits, random matrices, Methods Appl. Anal. 9 (2002), no. 1, 99–118. MR 1948465, DOI 10.4310/MAA.2002.v9.n1.a3
- Greg Kuperberg, A tracial quantum central limit theorem, Trans. Amer. Math. Soc. 357 (2005), no. 2, 459–471. MR 2095618, DOI 10.1090/S0002-9947-03-03449-4
- Peter Littelmann, Paths and root operators in representation theory, Ann. of Math. (2) 142 (1995), no. 3, 499–525. MR 1356780, DOI 10.2307/2118553
- Madan Lal Mehta, Random matrices, 2nd ed., Academic Press, Inc., Boston, MA, 1991. MR 1083764
- Paul-André Meyer, Quantum probability for probabilists, Lecture Notes in Mathematics, vol. 1538, Springer-Verlag, Berlin, 1993. MR 1222649, DOI 10.1007/978-3-662-21558-6
- Marc A. Rieffel, Gromov-Hausdorff distance for quantum metric spaces, Mem. Amer. Math. Soc. 168 (2004), no. 796, 1–65. Appendix 1 by Hanfeng Li; Gromov-Hausdorff distance for quantum metric spaces. Matrix algebras converge to the sphere for quantum Gromov-Hausdorff distance. MR 2055927, DOI 10.1090/memo/0796
- Piotr Śniady, Gaussian fluctuations of characters of symmetric groups and of Young diagrams, Probab. Theory Related Fields 136 (2006), no. 2, 263–297. MR 2240789, DOI 10.1007/s00440-005-0483-y
- Piotr Śniady and Roland Speicher. Permutationally invariant random matrices. In preparation, 2008.
- D. V. Voiculescu, K. J. Dykema, and A. Nica, Free random variables, CRM Monograph Series, vol. 1, American Mathematical Society, Providence, RI, 1992. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups. MR 1217253, DOI 10.1090/crmm/001
- S. L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), no. 4, 613–665. MR 901157, DOI 10.1007/BF01219077
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 Śniady
- Affiliation: Institute of Mathematics, University of Wroclaw, pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland
- Email: Piotr.Sniady@math.uni.wroc.pl
- 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 \text{2 P03A 007 23}, RTN network: QP-Applications contract No. HPRN-CT-2002-00279, and KBN-DAAD project 36/2003/2004. - © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2485426