Geodesic
Racah coefficients for the group $\mathrm{SL}(2,\mathbb{R})$
Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 28, Tome 509 (2021), pp. 99-112 Cet article a éte moissonné depuis la source Math-Net.Ru

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The paper is devoted to the derivation of a universal integral representation for $6j$-symbols, or Racah coefficients, for the tensor product of three unitary representations of the main series of the group $\mathrm{SL}(2,\mathbb{R})$. The problem of calculating $6j$-symbols admits a natural reformulation in the language of Feynman diagrams. The original diagram can be significantly simplified and reduced to a basic diagram using the Gorishnii–Isaev method. In the case of representations of the main series, a closed expression in the form of the Mellin–Barnes integral is obtained for the basic diagram. 
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S. E. Derkachev; A. V. Ivanov. Racah coefficients for the group $\mathrm{SL}(2,\mathbb{R})$. Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 28, Tome 509 (2021), pp. 99-112. http://geodesic.mathdoc.fr/item/ZNSL_2021_509_a6/

[1] V. A. Bargmann, “Irreducible unitary representations of the Lorentz group”, Ann. Math., 48:3 (1947), 568–640 | DOI | MR | Zbl

[2] S. Lang, $\mathrm{SL}_2(\mathbf{R})$, Addison-Wesley, Reading, Mass., 1975

[3] L. Pukánszky, “On the Kronecker products of irreducible representations of the $2\times2$ real unimodular group. I”, Trans. Amer. Math. Soc., 100 (1961), 116–152 | MR | Zbl

[4] R. P. Martin, “On the decomposition of tensor products of principal series representations for real-rank one semisimple groups”, Trans. Amer. Math. Soc., 201 (1975), 177–211 | DOI | MR | Zbl

[5] J. Repka, “Tensor products of unitary representations of $\mathrm{SL}_2(\mathbf{R})$”, Bull. Amer. Math. Soc., 82:6 (1976), 930–932 | DOI | MR | Zbl

[6] W. Groenevelt, “Wilson function transforms related to Racah coefficients”, Acta Appl. Math., 91:2 (2006), 133–191 | DOI | MR | Zbl

[7] W. Groenevelt, “The Wilson function transform”, Int. Math. Research Notices, 2003:52 (2003), 2779–2817 | DOI | Zbl

[8] Funct. Anal. Appl., 40:3 (2006), 222–224 | DOI | DOI | MR | Zbl

[9] Sb. Math., 198:3 (2007), 369–381 | DOI | MR | Zbl

[10] S. E. Derkachov, V. P. Spiridonov, “On the $6j$-symbols for $\mathrm{SL}(2,\mathbb{C})$ group”, Theor. Math. Phys., 198:1 (2019), 29–47 | DOI | Zbl

[11] M. Kirch, A. N. Manashov, “Noncompact SL(2,R) spin chain”, JHEP, 0406 (2004), 035 | DOI | MR

[12] A. W. Knapp, Representation theory of semisimple groups: an overview based on examples, Princeton Univ. Press, Princeton, N.J., 1986 | Zbl

[13] I. M. Gelfand, M. I. Graev, N. Ya. Vilenkin, Generalized Functions, v. 5, Integral Geometry and Representation Theory, Academic Press, 1966 | Zbl

[14] A. V. Ivanov, “On the completeness of projectors for tensor product decomposition of continuous series representations groups $\mathrm{SL}(2,\mathbb{R})$”, J. Math. Sci., 242:5 (2019), 692–700 | DOI | MR | Zbl

[15] S. G. Gorishnii, A. P. Isaev, “An approach to the calculation of many-loop massless Feynman integrals”, Theor. Math. Phys., 62:3 (1985), 232–240 | DOI | MR