Journal of Theoretical Probability

, Volume 31, Issue 3, pp 1411–1428 | Cite as

Two Constructions of Markov Chains on the Dual of U(n)

  • Jeffrey Kuan


We provide two new constructions of Markov chains which had previously arisen from the representation theory of \(U(\infty )\). The first construction uses the combinatorial rule for the Littlewood–Richardson coefficients, which arise from tensor products of irreducible representations of the unitary group. The second arises from a quantum random walk on the von Neumann algebra of U(n), which is then restricted to the center. Additionally, the restriction to a maximal torus can be expressed in terms of weight multiplicities, explaining the presence of tensor products.


Noncommutative random walk Littlewood–Richardson coefficients Group von Neumann algebra Representation theory 

Mathematics Subject Classification (2010)

60B99 60C05 60J10 60J27 


  1. 1.
    Biane, P.: Quantum random walk on the dual of \(SU(n)\). Probab Theory Relat Fields 89, 117–129 (1991)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Biane, P.: Miniscule weights and random walks on lattices. Quantum Probab Relat Top VII, 51–65 (1992)CrossRefMATHGoogle Scholar
  3. 3.
    Borodin, A., Ferrari, P.L.: Anisotropic growth of random surfaces in 2+1 dimensions. J. Stat. Mech. P02009 (2009). arXiv:0804.3035v1
  4. 4.
    Borodin, A., Ferrari, P.L.: Large time asymptotics of growth models on space-like paths I: PushASEP. Elecron. J. Probab. 13(50), 1380–1418 (2008). arXiv:0707.2813 MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Borodin, A., Corwin, I.: Macdonald processes. arXiv:1111.4408
  6. 6.
    Borodin, A., Kuan, J.: Asymptotics of Plancherel measures for the infinite-dimensional unitary group. Adv. Math. 219, 894–931 (2008). arXiv:0712.1848v1 MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Corwn, I.: The Kardar-Parisi-Zhang equation and universality class. arXiv:1106.1596
  8. 8.
    Defosseux, M.: An interacting particle model and a Pieri-type formula for the orthogonal group. J. Theor. Probab. (Feb 2012). arXiv:1012.0117v1
  9. 9.
    Defosseux, M.: Interacting particle models and the Pieri-type formulas : the symplectic case with non equal weights. arXiv:1104.4457
  10. 10.
    König, W., O’Connell, N., Roch, S.: Non-colliding random walks, tandem queues and discrete orthogonal polynomial ensembles. Electron. J. Probab. 7(1), 1–24 (2002)MathSciNetMATHGoogle Scholar
  11. 11.
    Kuan, J.: Discrete-time particle system with a wall and representations of O(infinity). arXiv:1203.1660v1
  12. 12.
    Kuan, J.: A (2+1)—dimensional Gaussian field as fluctuations of quantum random walks on quantum groups. arXiv:1601.04402v1
  13. 13.
    Warren, J., Windridge, P.: Some examples of dynamics for Gelfand–Tsetlin patterns. Electron. J. Probab. 14(59), 1745–1769 (2009). arXiv:0812.0022 MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Columbia UniversityNew YorkUSA

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