Selecta Mathematica

, Volume 24, Issue 3, pp 2571–2623 | Cite as

Semiclassical asymptotics of \(\mathbf {GL}_N({\mathbb {C}})\) tensor products and quantum random matrices

  • Benoît Collins
  • Jonathan NovakEmail author
  • Piotr Śniady


The Littlewood–Richardson process is a discrete random point process arising from the isotypic decomposition of tensor products of irreducible representations of the linear group \(\mathrm {GL}_N({\mathbb {C}})\). Biane–Perelomov–Popov matrices are quantum random matrices obtained as the geometric quantization of random Hermitian matrices with deterministic eigenvalues and uniformly random eigenvectors. As first observed by Biane, the correlation functions of certain global observables of the LR process coincide with the correlation functions of linear statistics of sums of classically independent BPP matrices, thereby enabling a random matrix approach to the statistical study of \(\mathrm {GL}_N({\mathbb {C}})\) tensor products. In this paper, we prove an optimal result: classically independent BPP matrices become freely independent in any semiclassical/large-dimension limit. This proves and generalizes a conjecture of Bufetov and Gorin, and leads to a Law of Large Numbers for the BPP observables of the LR process which holds in any and all semiclassical scalings.


Asymptotic representation theory Random matrix theory Free probability Representations of general linear groups Quantization 

Mathematics Subject Classification

Primary 22E46 Secondary 60B20 46L54 34L20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anderson, G., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics, vol. 118. Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
  2. 2.
    Brouwer, P.W., Beenaker, C.W.J.: Diagrammatic method of integration over the unitary group, with applications to quantum transport in mesoscopic systems. J. Math. Phys. 37, 4904–4934 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bufetov, A., Gorin, V.: Representations of classical Lie groups and quantized free convolution. Geom. Funct. Anal. 25(25), 763–814 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bürgisser, P., Ikenmeyer, C.: Deciding positivity of Littlewood–Richardson coefficients. SIAM J. Discrete Math. 4(27), 1639–1681 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Biane, P.: Representations of unitary groups and free convolution. Publ. Res. Inst. Math. Sci. 31(1), 63–79 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Biane, P.: Representations of symmetric groups and free probability. Adv. Math. 138(1), 126–181 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Biane, P.: Parking functions of types A and B. Electron. J. Combin. 9(1), 7 (2002)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Collins, B., Matsumoto, S., Novak, J.: An invitation to the Weingarten calculus (2017) (in preparation) Google Scholar
  9. 9.
    Collins, B.: Moments and cumulants of polynomial random variables on unitary groups, the Itzykson–Zuber integral, and free probability. Int. Math. Res. Not. 17, 953–982 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Collins, B., Śniady, P.: Integration with respect to the Haar measure on unitary, orthogonal and symplectic group. Commun. Math. Phys. 264(3), 773–795 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Collins, B., Śniady, P.: Asymptotic fluctuations of representations of the unitary groups. Preprint arXiv:0911.5546 (2009)
  12. 12.
    Collins, B., Śniady, P.: Representations of Lie groups and random matrices. Trans. Am. Math. Soc. 361(6), 3269–3287 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Goulden, I.P., Guay-Paquet, M., Novak, J.: On the convergence of monotone Hurwitz generating functions. Ann. Comb. 21, 73–81 (2017)Google Scholar
  14. 14.
    Goulden, I.P., Guay-Paquet, M., Novak, J.: Toda equations and piecewise polynomiality for mixed double Hurwitz numbers. SIGMA Symmetry Integrability Geom. Methods Appl. 12, 1–10 (2016)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Gross, D.J., Taylor, W.: Two-dimensional QCD is a string theory. Nuclear Phys. B 400(1–3), 181–208 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kirillov, A.A.: Lectures on the Orbit Method. Graduate Studies in Mathematics, vol. 64. American Mathematical Society, Providence (2004)zbMATHGoogle Scholar
  17. 17.
    Knutson, A., Tao, T.: Honeycombs and sums of hermitian matrices. Not. AMS 48, 175–186 (2001)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Kuperberg, G.: Random words, quantum statistics, central limits, random matrices. Methods Appl. Anal. 9(1), 99–118 (2002)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Matsumoto, S., Novak, J.: Jucys–Murphy elements and unitary matrix integrals. Int. Math. Res. Not. IMRN 2, 362–397 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Morozov, A.: Unitary Integrals and Related Matrix Models. The Oxford Handbook of Random Matrix Theory, pp. 353–375. Oxford University Press, Oxford (2011)zbMATHGoogle Scholar
  21. 21.
    Mingo, J.A., Speicher, R.: Free Probability and Random Matrices. Fields Institute Publications, Toronto (2016)zbMATHGoogle Scholar
  22. 22.
    Mingo, J.A., Śniady, P., Speicher, R.: Second order freeness and fluctuations of random matrices. II. Unitary random matrices. Adv. Math. 209(1), 212–240 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Narayanan, H.: On the complexity of computing Kostka numbers and Littlewood–Richardson coefficients. J. Algebraic Comb. 24, 347–354 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Novak, J.I.: Jucys–Murphy elements and the unitary Weingarten function. In: Noncommutative Harmonic Analysis with Applications to Probability II. Banach Center Publication, vol. 89, pp. 231–235. Polish Academy of Scientific Institute and Mathematics, Warsaw (2010)Google Scholar
  25. 25.
    Novak, J.: Three lectures on free probability. In: Random Matrix Theory, Interacting Particle Systems, and Integrable Systems. Mathematical Sciences Research Institute Publications, vol. 65, pp. 309–383. Cambridge University Press, New York (2014) (with illustrations by Michael LaCroix) Google Scholar
  26. 26.
    Nica, A., Speicher, R.: Lectures on the Combinatorics of Free Probability. London Mathematical Society Lecture Note Series, vol. 335. Cambridge University Press, Cambridge (2006)CrossRefzbMATHGoogle Scholar
  27. 27.
    Novak, J., Śniady, P.: What is...a free cumulant? Not. Am. Math. Soc. 58, 300–301 (2011)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Perelomov, A.M., Popov, V.S.: Casimir operators for the classical groups. Dokl. Akad. Nauk SSSR 174, 287–290 (1967)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Samuel, S.: \({\rm U}(N)\) integrals, \(1/N\), and the De Wit–’t Hooft anomalies. J. Math. Phys. 21(12), 2695–2703 (1980)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Shlyakhtenko, D.: Notes on free probability. ArXiv preprint arXiv:math/0504063 (2005)
  31. 31.
    Stanley, R.P.: Parking functions and noncrossing partitions. Electron J. Comb. 45, R20 (1997)MathSciNetzbMATHGoogle Scholar
  32. 32.
    ’t Hooft, G., De Wit, B.: Nonconvergence of the \(1/n\) expansion for SU\((n)\) gauge fields on a lattice. Phys. Lett. 69B, 61–64 (1977)MathSciNetGoogle Scholar
  33. 33.
    Tao, T.: Topics in Random Matrix Theory. Graduate Studies in Mathematics, vol. 132. American Mathematical Society, Providence (2012)zbMATHGoogle Scholar
  34. 34.
    Voiculescu, D.V., Dykema, K.J., Nica, A.: A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups. In: Free Random Variables. CRM Monograph Series, vol. 1. American Mathematical Society, Providence (1992)Google Scholar
  35. 35.
    Voiculescu, D.: Limit laws for random matrices and free products. Invent. Math. 104(1), 201–220 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Weyl, H.: The Classical Groups. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ (1997). (their invariants and representations, Fifteenth printing, Princeton Paperbacks) zbMATHGoogle Scholar
  37. 37.
    Xu, F.: A random matrix model from two-dimensional Yang–Mills theory. Commun. Math. Phys. 190(2), 287–307 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Želobenko, D.P.: Compact Lie Groups and Their Representations. American Mathematical Society, Providence (1973). (Translated from the Russian by Israel Program for Scientific Translations, Translations of Mathematical Monographs, vol. 40) CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Graduate School of ScienceKyoto UniversityKyotoJapan
  2. 2.CNRSParisFrance
  3. 3.Department of MathematicsUC San DiegoLa JollaUSA
  4. 4.Institute of MathematicsPolish Academy of SciencesWarszawaPoland

Personalised recommendations