Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 072, 19 pages      arXiv:2210.06024      https://doi.org/10.3842/SIGMA.2023.072

Multiplicative Characters and Gaussian Fluctuation Limits

Ryosuke Sato
Department of Physics, Faculty of Science and Engineering, Chuo University, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan

Received May 09, 2023, in final form September 19, 2023; Published online October 03, 2023

Abstract
It is known that extreme characters of several inductive limits of compact groups exhibit multiplicativity in a certain sense. In the paper, we formulate such multiplicativity for inductive limit quantum groups and provide explicit examples of multiplicative characters in the case of quantum unitary groups. Furthermore, we show a Gaussian fluctuation limit theorem using this concept of multiplicativity.

Key words: asymptotic representation theory; quantum groups; inductive limits; quasi-local algebras.

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References

  1. Accardi L., Bach A., Central limits of squeezing operators, in Quantum Probability and Applications IV, Lecture Notes in Math., Vol. 1396, Springer, Berlin, 1989, 7-19.
  2. Blackadar B., Operator algebras. Theory of $C^*$-algebras and von neumann algebras, Encyclopaedia Math. Sci., Vol. 122, Springer, Berlin, 2006.
  3. Borodin A., Bufetov A., A CLT for Plancherel representations of the infinite-dimensional unitary group, J. Math. Sci. 190 (2013), 419-426, arXiv:1203.3010.
  4. Borodin A., Bufetov A., Plancherel representations of ${\rm U}(\infty)$ and correlated Gaussian free fields, Duke Math. J. 163 (2014), 2109-2158, arXiv:1301.0511.
  5. Borodin A., Olshanski G., Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes, Ann. of Math. 161 (2005), 1319-1422, arXiv:math.RT/0109194.
  6. Borodin A., Olshanski G., The boundary of the Gelfand-Tsetlin graph: A new approach, Adv. Math. 230 (2012), 1738-1779, arXiv:1109.1412.
  7. Borodin A., Olshanski G., Markov processes on the path space of the Gelfand-Tsetlin graph and on its boundary, J. Funct. Anal. 263 (2012), 248-303, arXiv:1009.2029.
  8. Borodin A., Olshanski G., Markov dynamics on the Thoma cone: a model of time-dependent determinantal processes with infinitely many particles, Electron. J. Probab. 18 (2013), 75, 43 pages, arXiv:1303.2794.
  9. Boyer R.P., Infinite traces of AF-algebras and characters of ${\rm U}(\infty )$, J. Operator Theory 9 (1983), 205-236.
  10. Boyer R.P., Characters of the infinite symplectic group—A Riesz ring approach, J. Funct. Anal. 70 (1987), 357-387.
  11. Bratteli O., Robinson D.W., Operator algebras and quantum statistical mechanics. Equilibrium states. Models in quantum statistical mechanics, Theoret. Math. Phys., Springer, Berlin, 1997.
  12. Dereziński J., Boson free fields as a limit of fields of a more general type, Rep. Math. Phys. 21 (1985), 405-417.
  13. Goderis D., Verbeure A., Vets P., Non-commutative central limits, Probab. Theory Related Fields 82 (1989), 527-544.
  14. Goderis D., Verbeure A., Vets P., Quantum central limit and coarse graining, in Quantum Probability and Applications V, Lecture Notes in Math., Vol. 1442, Springer, Berlin, 1990, 178-193.
  15. Goderis D., Verbeure A., Vets P., About the exactness of the linear response theory, Comm. Math. Phys. 136 (1991), 265-283.
  16. Goderis D., Vets P., Central limit theorem for mixing quantum systems and the CCR-algebra of fluctuations, Comm. Math. Phys. 122 (1989), 249-265.
  17. Gorin V., The $q$-Gelfand-Tsetlin graph, Gibbs measures and $q$-Toeplitz matrices, Adv. Math. 229 (2012), 201-266, arXiv:1011.1769.
  18. Kerov S.V., Asymptotic representation theory of the symmetric group and its applications in analysis, Transl. Math. Monogr., Vol. 219, American Mathematical Society, Providence, RI, 2003.
  19. Kerov S.V., Olshanski G., Vershik A., Harmonic analysis on the infinite symmetric group, Invent. Math. 158 (2004), 551-642, arXiv:math.RT/0312270.
  20. Kerov S.V., Vershik A.M., Characters, factor representations and $K$-functor of the infinite symmetric group, in Operator Algebras and Group Representations II, Monogr. Stud. Math., Vol. 18, Pitman, Boston, MA, 1984, 23-32.
  21. Klimyk A., Schmüdgen K., Quantum groups and their representations, Texts Monogr. Phys., Springer, Berlin, 1997.
  22. Matsui T., Bosonic central limit theorem for the one-dimensional $XY$ model, Rev. Math. Phys. 14 (2002), 675-700.
  23. Matsui T., On the algebra of fluctuation in quantum spin chains, Ann. Henri Poincaré 4 (2003), 63-83, arXiv:math-ph/0202011.
  24. Neshveyev S., Tuset L., Compact quantum groups and their representation categories, Cours Spéc., Vol. 20, Société Mathématique de France, Paris, 2013.
  25. Noumi M., Yamada H., Mimachi K., Finite-dimensional representations of the quantum group ${\rm GL}_q(n;{\bf C})$ and the zonal spherical functions on ${\rm U}_q(n-1)\backslash{\rm U}_q(n)$, Japan. J. Math. (N.S.) 19 (1993), 31-80.
  26. Okounkov A., Olshanski G., Asymptotics of Jack polynomials as the number of variables goes to infinity, Int. Math. Res. Not. 1998 (1998), 641-682, arXiv:q-alg/9709011.
  27. Olshanski G., The representation ring of the unitary groups and Markov processes of algebraic origin, Adv. Math. 300 (2016), 544-615, arXiv:1504.01646.
  28. Petrov L., The boundary of the Gelfand-Tsetlin graph: new proof of Borodin-Olshanski's formula, and its $q$-analogue, Mosc. Math. J. 14 (2014), 121-160, arXiv:1208.3443.
  29. Petz D., An invitation to the algebra of canonical commutation relations, Leuven Notes Math. Theor. Phys. Ser. A., Vol. 2, Leuven University Press, Leuven, 1990.
  30. Sato R., Markov semigroups on unitary duals generated by quantized characters, arXiv:2102.09082.
  31. Sato R., Quantized Vershik-Kerov theory and quantized central measures on branching graphs, J. Funct. Anal. 277 (2019), 2522-2557, arXiv:1804.02644.
  32. Sato R., Inductive limits of compact quantum groups and their unitary representations, Lett. Math. Phys. 111 (2021), 122, 20 pages, arXiv:1908.03988.
  33. Stokman J.V., Dijkhuizen M.S., Some limit transitions between $BC$ type orthogonal polynomials interpreted on quantum complex Grassmannians, Publ. Res. Inst. Math. Sci. 35 (1999), 451-500, arXiv:math.QA/9806123.
  34. Tomatsu R., A characterization of right coideals of quotient type and its application to classification of Poisson boundaries, Comm. Math. Phys. 275 (2007), 271-296, arXiv:math.OA/0611327.
  35. Ueda Y., Spherical representations of $C^*$-flows II: Representation system and quantum group setup, SIGMA 18 (2022), 050, 43 pages, arXiv:2201.10931.
  36. Ueda Y., Spherical representations of $C^*$-flows I, Münster J. Math. 16 (2023), 201-263, arXiv:2010.15324.
  37. Vershik A.M., Kerov S.V., Asymptotic theory of the characters of a symmetric group, Funct. Anal. Appl. 15 (1981), 246-255.
  38. Vershik A.M., Kerov S.V., Characters and factor representations of the infinite symmetric group, Sov. Math. Dokl. 23 (1981), 389-392.
  39. Vershik A.M., Kerov S.V., Characters and factor-representations of the infinite unitary group, Sov. Math. Dokl. 26 (1982), 570-574.
  40. Vershik A.M., Kerov S.V., The $K$-functor (Grothendieck group) of the infinite symmetric group, J. Sov. Math. 28 (1985), 549-568.
  41. Vershik A.M., Kerov S.V., Locally semisimple algebras. Combinatorial theory and the $K_0$-functor, J. Sov. Math. 38 (1987), 1701-1733.
  42. Vershik A.M., Kerov S.V., The Grothendieck group of infinite symmetric group and symmetric functions (with the elements of the theory of $K_0$-functor of AF-algebras), in Representation of Lie Groups and Related Topics, Adv. Stud. Contemp. Math., Vol. 7, Gordon and Breach, New York, 1990, 39-117.
  43. Voiculescu D., Sur les représentations factorielles finies de ${\rm U}(\infty )$ et autres groupes semblables, C. R. Acad. Sci. Paris Sér. A 279 (1974), 945-946.
  44. Voiculescu D., Représentations factorielles de type ${\rm II}_1$ de ${\rm U}(\infty )$, J. Math. Pures Appl. 55 (1976), 1-20.

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