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Spectral Measures Associated to Rank two Lie Groups and Finite Subgroups of \({GL(2,\mathbb{Z})}\)

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Abstract

Spectral measures for fundamental representations of the rank two Lie groups SU(3), Sp(2) and G 2 have been studied. Since these groups have rank two, these spectral measures can be defined as measures over their maximal torus \({\mathbb{T}^2}\) and are invariant under an action of the corresponding Weyl group, which is a subgroup of \({GL(2,\mathbb{Z})}\). Here we consider spectral measures invariant under an action of the other finite subgroups of \({GL(2,\mathbb{Z})}\). These spectral measures are all associated with fundamental representations of other rank two Lie groups, namely \({\mathbb{T}^2=U(1) \times U(1)}\), \({U(1) \times SU(2)}\), U(2), \({SU(2) \times SU(2)}\), SO(4) and PSU(3).

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References

  1. Banica T., Bisch D.: Spectral measures of small index principal graphs. Commun. Math. Phys. 269, 259–281 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Banica, T., Bichon, J.: Spectral measure blowup for basic Hadamard subfactors. arXiv:1402.1048v2 [math.OA]

  3. Böckenhauer J., Evans D.E.: Modular invariants from subfactors: type I coupling matrices and intermediate subfactors. Commun. Math. Phys. 213, 267–289 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Böckenhauer, J., Evans, D.E.: Modular invariants and subfactors. In: Mathematical Physics in Mathematics and Physics (Siena, 2000), Fields Inst. Commun., vol. 30, pp. 11–37. Amer. Math. Soc., Providence (2001)

  5. Böckenhauer J., Evans D.E., Kawahigashi Y.: On \({\alpha}\)-induction, chiral generators and modular invariants for subfactors. Commun. Math. Phys. 208, 429–487 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Bousquet-Mélou, M., Mishna, M.: Walks with small steps in the quarter plane. In: Algorithmic Probability and Combinatorics, Contemp. Math., vol. 520, pp. 1–39. Amer. Math. Soc., Providence (2010)

  7. Byrd, P.F., Friedman, M.D.: Handbook of Elliptic Integrals for Engineers and Scientists, Die Grundlehren der Mathematischen Wissenschaften, Band 67, 2nd edn., revised. Springer, New York (1971)

  8. Cleary S., Murray M., Rechnitzer A., Taback J.: Random subgroups of Thompson’s group F. Groups Geom. Dyn. 4, 91–126 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Evans, D.E.: Critical phenomena, modular invariants and operator algebras. In: Operator Algebras and Mathematical Physics (Constanţa, 2001), pp. 89–113. Theta, Bucharest (2003)

  10. Evans D.E., Pugh M.: Spectral measures and generating series for nimrep graphs in subfactor theory. Commun. Math. Phys. 295, 363–413 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Evans D.E., Pugh M.: Spectral measures and generating series for nimrep graphs in subfactor theory II: SU(3). Commun. Math. Phys. 301, 771–809 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Evans D.E., Pugh M.: Spectral measures for G 2. Commun. Math. Phys. 337, 1161–1197 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Evans, D.E., Pugh, M.: Spectral measures for G 2 II: finite subgroups. Preprint, arXiv:1404.1866 [math.OA]

  14. Evans, D.E., Pugh, M.: Spectral measures for Sp(2). Preprint, arXiv:1404.1912 [math.OA]

  15. Evans, D.E., Pugh, M.: Braided subfactors, spectral measures, planar algebras and Calabi–Yau algebras associated to SU(3) modular invariants. In: Progress in Operator Algebras, Noncommutative Geometry, and Their Applications, vol. 15, pp. 17–60. Theta Ser. Adv. Math., Theta, Bucharest (2012)

  16. Gaberdiel M.R., Gannon T.: Boundary states for WZW models. Nuclear Phys. B 639, 471–501 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Hiai, F., Petz, D.: The semicircle law, free random variables and entropy. In: Mathematical Surveys and Monographs, vol. 77. American Mathematical Society, Providence (2000)

  18. Jones V.F.R.: Index for subfactors. Invent. Math. 72, 1–25 (1983)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Klimyk, A.U., Patera, J.: Orbit functions. SIGMA 2, 006, 60 (2006)

  20. Klimyk, A.U., Patera, J.: Antisymmetric orbit functions. SIGMA 3, 023, 83 (2007)

  21. Moody, R.V., Patera, J.: Orthogonality within the families of C-, S-, and E-functions of any compact semisimple Lie group. SIGMA 2, 076, 14 (2006)

  22. Nesterenko, M., Patera, J., Tereszkiewicz, A.: Orthogonal polynomials of compact simple Lie groups. Int. J. Math. Math. Sci. 2011, 23 (Art. ID 969424)

  23. Newman, M.: Integral matrices. In: Pure and Applied Mathematics, vol. 45. Academic Press, New York (1972)

  24. The On-Line Encyclopedia of Integer Sequences, published electronically at http://oeis.org (2010). Sequence A001246

  25. Stanley, R.P.: Enumerative combinatorics. In: Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62. Cambridge University Press, Cambridge (1999)

  26. Takesaki, M.: Theory of operator algebras. I. In: Encyclopaedia of Mathematical Sciences, vol. 124. Springer, Berlin (2002)

  27. Voiculescu, D.V., Dykema, K.J., Nica, A.: Free random variables. In: CRM Monograph Series 1. American Mathematical Society, Providence, (1992)

  28. Wassermann A.: Operator algebras and conformal field theory. III. Fusion of positive energy representations of LSU(N) using bounded operators. Invent. Math. 133, 467–538 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Whittaker, E.T., Watson, G.N.: A course of modern analysis. Reprint of the fourth (1927) edition. In: Cambridge Mathematical Library. Cambridge University Press, Cambridge (1996)

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  1. School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, Wales, CF24 4AG, UK

    David E. Evans & Mathew Pugh

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  1. David E. Evans
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  2. Mathew Pugh
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Correspondence to Mathew Pugh.

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Communicated by Y. Kawahigashi

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Evans, D.E., Pugh, M. Spectral Measures Associated to Rank two Lie Groups and Finite Subgroups of \({GL(2,\mathbb{Z})}\) . Commun. Math. Phys. 343, 811–850 (2016). https://doi.org/10.1007/s00220-015-2434-5

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