Skip to main content
SpringerLink
Log in

q-Deformed character theory for infinite-dimensional symplectic and orthogonal groups

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

Abstract

The classification of irreducible, spherical characters of the infinite-dimensional unitary/orthogonal/symplectic groups can be obtained by finding all possible limits of normalized, irreducible characters of the corresponding finite-dimensional groups, as the rank tends to infinity. We solve a q-deformed version of the latter problem for orthogonal and symplectic groups, extending previously known results for the unitary group. The proof is based on novel determinantal and double-contour integral formulas for the q-specialized characters.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Notes

  1. For SO(2N), \(\lambda _N\) is allowed to be negative, but one should have \(\lambda _1\ge \cdots \ge \lambda _{N-1}\ge |\lambda _N|\). In this case we deal instead with the direct sum of two twin representations that differ by a flip of the sign of \(\lambda _N\).

References

  1. Bakalov, B., Kirillov Jr., A.: Lectures on Tensor Categories and Modular Functor, University Lecture Series, vol. 21. American Mathematical Society (2001)

  2. Borodin, A., Olshanski, G.: The boundary of the Gelfand-Tsetlin graph: a new approach. Adv. Math. 230(4–6), 1738–1779 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Boyer, R.P.: Infinite traces of AF-algebras and characters of \(U(\infty )\). J. Oper. Theory 9, 205–236 (1983)

    MathSciNet  MATH  Google Scholar 

  4. Boyer, R.P.: Characters and factor representations of the infinite dimensional classical groups. J. Oper. Theory 28(2), 281–307 (1992)

    MathSciNet  MATH  Google Scholar 

  5. Bufetov, A., Gorin, V.: Representations of classical Lie groups and quantized free convolution. Geom. Funct. Anal. 25(3), 763–814 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bump, D.: Lie Groups. Springer, New York (2004)

    Book  MATH  Google Scholar 

  7. Chary, V., Pressley, A.: A Guide to Quantum Groups. Cambridge University Press, Cambridge (1994)

    Google Scholar 

  8. Cuenca, C.: Asymptotic formulas for macdonald polynomials and the boundary of the \((q, t)\)-Gelfand–Tsetlin graph. SIGMA 14, 001 (2018)

    MathSciNet  MATH  Google Scholar 

  9. Cuenca, C., Gorin, V., Olshanski, G.: The Elliptic Tail Kernel. Preprint (2019). arXiv:1907.11841

  10. Diaconis, P., Freedman. D.: Partial exchangeability and sufficiency. Statistics: Applications and New Directions, pp. 205–236 (1984)

  11. Dynkin, E.B.: The initial and final behaviour of trajectories of Markov processes. Russ. Math. Surv. 26(4), 165–185 (1971)

    Article  MATH  Google Scholar 

  12. Edrei, A.: On the generating function of a doubly-infinite, totally positive sequence. Trans. Am. Math. Soc. 74(3), 367–383 (1953)

    MATH  Google Scholar 

  13. El Samra, N., King, R.C.: Reduced determinantal forms for characters of the classical Lie groups. J. Phys. A Math. General 12(12), 2305 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fulton, W., Harris, J.: Representation Theory: A First Course, vol. 129. Springer, Berlin (2013)

    MATH  Google Scholar 

  15. Gorin, V.: The q-Gelfand–Tsetlin graph, Gibbs measures and q-Toeplitz matrices. Adv. Math. 229(1), 201–266 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gorin, V., Panova, G.: Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory. Ann. Probab. 43(6), 3052–3132 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gorin, V., Olshanski, G.: A quantization of the harmonic analysis on the infinite-dimensional unitary group. J. Funct. Anal. 270(1), 375–418 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gorin, V., Sun, Y.: Gaussian fluctuations for products of random matrices. Preprint (2018). arXiv:1812.06532

  19. Jing, N., Nie, B.: Vertex operators, Weyl determinant formulae and Littlewood duality. Ann. Comb. 19(3), 427–442 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. King, R.C., El-Sharkaway, N.G.I.: Standard Young tableaux and weight multiplicities of the classical Lie groups. J. Phys. A Math. General 16(14), 3153 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  21. Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Oxford University Press, Oxford (1998)

    MATH  Google Scholar 

  22. Mackey, G.W.: Borel structure in groups and their duals. Trans. Am. Math. Soc. 85(1), 134–165 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  23. Olshanski, G.: Unitary representations of infinite-dimensional pairs \((G, K)\) and the formalism. Sov. Math. Dokl. 27(2), 273–285 (1983)

    Google Scholar 

  24. Olshanski, G.: Unitary representations of infinite-dimensional pairs \((G, K)\) and the formalism of R. Howe. In: Vershik, A., Zhelobenko, D. (eds.) Representation of Lie Groups and Related Topics, Advanced Studies in Contemporary Mathematics, vol. 7, pp. 269–463. Gordon and Breach Science Publishers, New York (1990)

    Google Scholar 

  25. Olshanskii, G.I.: Twisted Yangians and infinite-dimensional classical Lie algebras. In: Kulish, P.P. (ed.) Quantum Groups Lecture Notes in Mathematics, vol. 1510, pp. 104–119. Springer, Berlin (1992)

    Google Scholar 

  26. Olshanski, G.: The problem of harmonic analysis on the infinite-dimensional unitary group. J. Funct. Anal. 205, 464–524 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Olshanski, G.: Extended Gelfand–Tsetlin graph, its q-boundary, and q-B-splines. Funct. Anal. Appl. 50(2), 107–130 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  28. Olshanski, G.: Asymptotics of characters in type C-B-D (in preparation)

  29. Olshanski, G.: Private communications (Unpublished manuscript)

  30. Okounkov, A., Olshanski, G.: Shifted Schur functions II. The binomial formula for characters of classical groups and its applications, pp. 245–271. American Mathematical Society Translations (1998)

  31. Okounkov, A., Olshanski, G.: Asymptotics of jack polynomials as the number of variables goes to infinity. Math. Res. Notices 13, 641–682 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  32. Okounkov, A., Olshansky, G.: Limits of BC-type orthogonal polynomials as the number of variables goes to infinity. In: Jack, Hall-Littlewood and Macdonald Polynomials, American Mathematical Society Contemporary Mathematics Series 417, 281–318 (2006)

  33. Olshanski, G.I., Osinenko, A.A.: Multivariate Jacobi polynomials and the Selberg integral. Funct. Anal. Appl. 46(4), 262–278 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  34. Petrov, L.: The boundary of the Gelfand–Tsetlin graph: new proof of Borodin–Olshanski’s formula, and its q-analogue. Moscow Math. J. 14(1), 121–160 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. Proctor, R.A.: Young tableaux, Gelfand patterns, and branching rules for classical groups. J. Algebra 164(2), 299–360 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  36. Sato, R.: Quantized Vershik–Kerov Theory and Quantized Central Measures on Branching Graphs. Preprint (2018). arXiv:1804.02644

  37. Sato, R.: Type classification of extremal quantized characters. Preprint (2019). arXiv:1903.07454

  38. Sergeev, A.N., Veselov, A.P.: Jacobi–Trudy formula for generalized Schur polynomials. Moscow Math. J. 14(1), 161–168 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  39. Vershik, A.M.: Description of invariant measures for the actions of some infinite-dimensional groups. Sov. Math. Dokl. 15, 1396–1400 (1974)

    MATH  Google Scholar 

  40. Vershik, A.M., Kerov, S.V.: Asymptotic theory of characters of the symmetric group. Funct. Anal. Appl. 15(4), 246–255 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  41. Vershik, A.M., Kerov, S.V.: Characters and factor representations of the inifinite unitary group. Sov. Math. Dokl. 26, 570–574 (1982)

    MATH  Google Scholar 

  42. Voiculescu, D.: Représentations factorielles de type II\({}_1\) de \(U(\infty )\). J. Math. Pures et Appl. 55, 1–20 (1976)

    MathSciNet  MATH  Google Scholar 

  43. Winkler, G.: Choquet Order and Simplices: With Applications in Probabilistic Models, vol. 1145. Springer, Berlin (2006)

    MATH  Google Scholar 

  44. Zhelobenko, D.P.: The classical groups. Spectral analysis of their finite-dimensional representations. Russ. Math. Surv. 17(1), 1–94 (1962)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank G. Olshanski for encouraging us to study whether the extension of [15] to orthogonal and symplectic groups is possible and for a number of fruitful discussions. V.G. was partially supported by the NSF Grant DMS-1664619, NSF Grant DMS-1949820, by the NEC Corporation Fund for Research in Computers and Communications, and by the Sloan Research Fellowship. The authors also thank the organizers of the Park City Mathematics Institute research program on Random Matrix Theory, where part of this work was carried out.

Author information

Authors and Affiliations

  1. Department of Mathematics, MIT, Cambridge, MA, USA

    Cesar Cuenca & Vadim Gorin

  2. Institute for Information Transmission Problems, Moscow, Russia

    Vadim Gorin

  3. Department of Mathematics, University of Wisconsin - Madison, WI, USA

    Vadim Gorin

  4. Department of Mathematics, Caltech, Pasadena, CA, USA

    Cesar Cuenca

Authors
  1. Cesar Cuenca
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vadim Gorin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cesar Cuenca.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cuenca, C., Gorin, V. q-Deformed character theory for infinite-dimensional symplectic and orthogonal groups. Sel. Math. New Ser. 26, 40 (2020). https://doi.org/10.1007/s00029-020-00572-8

Download citation

  • Published:

  • DOI: https://doi.org/10.1007/s00029-020-00572-8

Mathematics Subject Classification

Search

Navigation