Braid group and 𝑞-Racah polynomials

Crampé, Nicolas; Vinet, Luc; Zaimi, Meri

doi:10.1090/proc/15811

Braid group and $q$-Racah polynomials
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Proc. Amer. Math. Soc. 150 (2022), 951-966 Request permission

Abstract:

The irreducible representations of two intermediate Casimir elements associated to the recoupling of three identical irreducible representations of $U_q(\mathfrak {sl}_2)$ are considered. It is shown that these intermediate Casimirs are related by a conjugation involving braid group representations. Consequently, the entries of the braid group matrices are explicitly given in terms of the $q$-Racah polynomials which appear as $6j$-symbols in the Racah problem for $U_q(\mathfrak {sl}_2)$. Formulas for these polynomials are derived from the algebraic relations satisfied by the braid group representations.

References

E. Artin, Theory of braids, Ann. of Math. (2) 48 (1947), 101–126. MR 19087, DOI 10.2307/1969218
L. C. Biedenharn and J. D. Louck, Angular momentum in quantum physics, Encyclopedia of Mathematics and its Applications, vol. 8, Addison-Wesley Publishing Co., Reading, Mass., 1981. Theory and application; With a foreword by Peter A. Carruthers. MR 635121
Nicolas Crampé, Julien Gaboriaud, Luc Vinet, and Meri Zaimi, Revisiting the Askey-Wilson algebra with the universal $R$-matrix of $U_q(\mathfrak {sl}_2)$, J. Phys. A 53 (2020), no. 5, 05LT01, 10. MR 4054740, DOI 10.1088/1751-8121/ab604e
Nicolas Crampé, Luc Vinet, and Meri Zaimi, Temperley-Lieb, Birman-Murakami-Wenzl and Askey-Wilson algebras and other centralizers of $U_q(\mathfrak {sl}_2)$, Ann. Henri Poincaré 22 (2021), no. 10, 3499–3528. MR 4314133, DOI 10.1007/s00023-021-01064-x
Nicolas Crampé, Luc Vinet, and Meri Zaimi, Bannai-Ito algebras and the universal $R$-matrix of $\mathfrak {osp}(1|2)$, Lett. Math. Phys. 110 (2019), no. 5, 1043–1055. MR 4082204, DOI 10.1007/s11005-019-01249-w
Brian Curtin, Spin Leonard pairs, Ramanujan J. 13 (2007), no. 1-3, 319–332. MR 2281169, DOI 10.1007/s11139-006-0255-z
V. G. Drinfel′d, Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 798–820. MR 934283
George Gasper and Mizan Rahman, Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 96, Cambridge University Press, Cambridge, 2004. With a foreword by Richard Askey. MR 2128719, DOI 10.1017/CBO9780511526251
Ya. A. Granovskii and A. S. Zhedanov, Hidden symmetry of the Racah and Clebsch-Gordan problems for the quantum algebra $sl_q(2)$, J. Group Theory Phys. 1 (1993), 161–171, arXiv:hep-th/9304138.
Hau-Wen Huang, An embedding of the universal Askey-Wilson algebra into $U_q(\mathfrak {sl}_2)\otimes U_q(\mathfrak {sl}_2)\otimes U_q(\mathfrak {sl}_2)$, Nuclear Phys. B 922 (2017), 401–434. MR 3689722, DOI 10.1016/j.nuclphysb.2017.07.007
V. F. R. Jones, Baxterization, Proceedings of the Conference on Yang-Baxter Equations, Conformal Invariance and Integrability in Statistical Mechanics and Field Theory, 1990, pp. 701–713. MR 1064744, DOI 10.1142/S021797929000036X
A. N. Kirillov and N. Yu. Reshetikhin, Representations of the algebra ${U}_q(\textrm {sl}(2)),\;q$-orthogonal polynomials and invariants of links, Infinite-dimensional Lie algebras and groups (Luminy-Marseille, 1988) Adv. Ser. Math. Phys., vol. 7, World Sci. Publ., Teaneck, NJ, 1989, pp. 285–339. MR 1026957
Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw, Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. With a foreword by Tom H. Koornwinder. MR 2656096, DOI 10.1007/978-3-642-05014-5
G. I. Lehrer and R. B. Zhang, Strongly multiplicity free modules for Lie algebras and quantum groups, J. Algebra 306 (2006), no. 1, 138–174. MR 2271576, DOI 10.1016/j.jalgebra.2006.03.043
Kazumasa Nomura and Paul Terwilliger, Leonard pairs, spin models, and distance-regular graphs, J. Combin. Theory Ser. A 177 (2021), Paper No. 105312, 59. MR 4139111, DOI 10.1016/j.jcta.2020.105312
G. Racah, Theory of complex spectra. II, Phys. Rev. 62 (1942), 438–462.
Hjalmar Rosengren, An elementary approach to $6j$-symbols (classical, quantum, rational, trigonometric, and elliptic), Ramanujan J. 13 (2007), no. 1-3, 131–166. MR 2281159, DOI 10.1007/s11139-006-0245-1
A. S. Zhedanov, “Hidden symmetry” of Askey-Wilson polynomials, Teoret. Mat. Fiz. 89 (1991), no. 2, 190–204 (Russian, with English summary); English transl., Theoret. and Math. Phys. 89 (1991), no. 2, 1146–1157 (1992). MR 1151381, DOI 10.1007/BF01015906