Braid group and $q$-Racah polynomials
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- by Nicolas Crampé, Luc Vinet and Meri Zaimi PDF
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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
Additional Information
- Nicolas Crampé
- Affiliation: Institut Denis-Poisson CNRS/UMR 7013 - Université de Tours - Université d’Orléans, Parc de Grandmont, 37200 Tours, France
- ORCID: 0000-0002-3754-4074
- Email: crampe1977@gmail.com
- Luc Vinet
- Affiliation: Centre de recherches mathématiques, Université de Montréal, P.O. Box 6128, Centre-ville Station, Montréal, Québec H3C 3J7, Canada
- MR Author ID: 178665
- ORCID: 0000-0001-6211-7907
- Email: vinet@crm.umontreal.ca
- Meri Zaimi
- Affiliation: Centre de recherches mathématiques, Université de Montréal, P.O. Box 6128, Centre-ville Station, Montréal, Québec H3C 3J7, Canada
- MR Author ID: 1359410
- ORCID: 0000-0001-7275-2007
- Email: meri.zaimi@umontreal.ca
- Received by editor(s): June 4, 2021
- Published electronically: December 7, 2021
- Additional Notes: The first author was partially supported by Agence Nationale de la Recherche Projet AHA ANR-18-CE40-0001. The work of the second author was funded in part by a discovery grant of the Natural Sciences and Engineering Research Council (NSERC) of Canada. The third author held a graduate scholarship from the Fonds de recherche du Québec – Nature et technologies (FRQNT) and was also partly funded by an Alexander-Graham-Bell scholarship from the NSERC
- Communicated by: Mourad Ismail
- © Copyright 2021 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 951-966
- MSC (2020): Primary 17B37, 20F36, 33C45
- DOI: https://doi.org/10.1090/proc/15811
- MathSciNet review: 4375695