Abstract
We consider the quantum symmetric pair \((\mathcal {U}_{q}(\mathfrak {su}(3)), \mathcal {B})\) where \(\mathcal {B}\) is a right coideal subalgebra. We prove that all finite-dimensional irreducible representations of \(\mathcal {B}\) are weight representations and are characterised by their highest weight and dimension. We show that the restriction of a finite-dimensional irreducible representation of \(\mathcal {U}_{q}(\mathfrak {su}(3))\) to \(\mathcal {B}\) decomposes multiplicity free into irreducible representations of \(\mathcal {B}\). Furthermore we give explicit expressions for the highest weight vectors in this decomposition in terms of dual q-Krawtchouk polynomials.
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Aldenhoven, N., Koelink, E., Román, P.: Matrix valued orthogonal polynomials for the quantum analogue of (SU(2) ×SU(2),diag). In: Ramanujan, J. (ed.) . doi:10.1007/s11139-016-9788-y (2016)
Bao, H.: Kazhdan-Lusztig theory of super type D and quantum symmetric pairs. arXiv:1603.05105
Delius, G.W., MacKay, N.: Affine quantum groups, Encyclopedia of Mathematical Physics. In: Françoise, J.-P., Naber, G.L., Tsun, T.S. (eds.) , pp 183–190 (2006)
Dijkhuizen, M.S., Noumi, M.: A family of quantum projective spaces and related q-hypergeometric orthogonal polynomials. Trans. Amer. Math. Soc. 350, 3269–3296 (1998)
Dijkhuizen, M.S., Stokman, J.V.: Some limit transitions between BC type orthogonal polynomials interpreted on quantum complex Grassmannians. Publ. Res. Inst. Math. Sci. 35, 451–500 (1999)
Ehrig, M., Stroppel, C.: Nazarov-Wenzl algebras, coideal subalgebras and categorified skew Howe duality. arXiv:1310.1972v2
Gasper, G., Rahman, M.: Basic Hypergeometric Series, 2nd edition, vol. 96. Cambridge University Press (2004)
Heckman, G., van Pruijssen, M.: Matrix valued orthogonal polynomials for Gelfand pairs of rank one. Tohoku Math. J. 68(2), 407–437 (2006). to appear, arXiv:1310.5134
Klimyk, A., Schmüdgen, K.: Quantum Groups and Their Representations Texts and Monographs in Physics. Springer-Verlag, Berlin (1997)
Koekoek, R., Swarttouw, R.F.: The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, online at http://aw.twi.tudelft.nl/~koekoek/askey.html, Report 98-17. Technical University Delft (1998)
Koelink, E., van Pruijssen, M., Román, P.: Matrix-valued orthogonal polynomials related to (SU(2) ×SU(2),diag). Int. Math. Res. Not. IMRN 24, 5673–5730 (2012)
Koelink, E., van Pruijssen, M., Román, R.: Matrix-valued orthogonal polynomials related to (SU(2) ×SU(2),diag), II. Publ. Res. Inst. Math. Sci. 49, 271–312 (2013)
Kolb, S.: Quantum symmetric Kac-Moody pairs. Adv. Math. 267, 395–469 (2014)
Kolb, S., Letzter, G.: The center of quantum symmetric pair coideal subalgebras. Rep. Theory 12, 294–326 (2008)
T: Koornwinder, Askey-Wilson polynomials as zonal spherical functions on the SU(2) quantum group. SIAM J. Math. Anal. 24, 795–813 (1993)
Letzter, G.: Symmetric pairs for quantized enveloping algebras. J. Alg. 220, 729–767 (1999)
Letzter, G.: Harish-Chandra modules for quantum symmetric pairs. Rep. Theory 4, 64–96 (2000)
Letzter, G.: Coideal subalgebrasc and quantum symmetric pairs, vol. 43, pp 117–166. New Directions in Hopf Algebras, Cambridge University Press, Cambridge (2002)
Letzter, G.: Quantum symmetric pairs and their zonal spherical functions. Transform. Groups 8, 261–292 (2003)
Letzter, G.: Quantum zonal spherical functions and Macdonald polynomials. Adv. Math. 189, 88–147 (2004)
Letzter, G.: Invariant differential operators for quantum symmetric spaces. Mem. Amer. Math. Soc., 193 (2008)
Mudrov, A.: Orthogonal basis for the Shapovalov form on \(U_{q}(\mathfrak {sl}(n + 1))\). Rev. Math. Phys. 27(2), 1550004 (2015). 23
Noumi, M.: Macdonald’s symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces. Adv. Math. 123, 16–77 (1996)
Noumi, M., Dijkhuizen, M.S., Sugitani, T.: Multivariable Askey-Wilson polynomials and quantum complex Grassmannians. Fields Inst. Commun. 14, 167–177 (1997)
Noumi, M., Sugitani, T.: Quantum symmetric spaces and related q-orthogonal polynomials, Group Theoretical Methods in Physics, pp 28–40. World Science Publishing, River Edge (1994)
Oblomkov, A.A., Stokman, J.V.: Vector valued spherical functions and Macdonald-Koornwinder polynomials. Compos. Math. 141(5), 1310–1350 (2005)
van Pruijssen, M.: Matrix valued orthogonal polynomials related to compact Gel’fand paris of rank one. PhD Thesis, Radboud Universiteit (2012)
Sugitani, T.: Zonal spherical functions on quantum Grassmann manifolds. J. Math. Sci Univ. Tokyo 6, 335–369 (1999)
Acknowledgments
We thank Stefan Kolb for helpful discussions on this paper. Noud Aldenhoven also thanks him for his hospitality during his visit to Newcastle. We thank J. Stokman for pointing out reference [3].
The research of Noud Aldenhoven is supported by the Netherlands Organization for Scientific Research (NWO) under project number 613.001.005 and by the Belgian Interuniversity Attraction Pole Dygest P07/18.
The research of Pablo Román is supported by the Radboud Excellence Fellowship. P. Román was partially supported by CONICET grant PIP 112-200801-01533, FONCyT grant PICT 2014-3452 and by SeCyT-UNC.
We would like to thank the anonymous referees for their comments and remarks, that have helped us to improve the paper.
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Presented by Catharina Stroppel.
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Aldenhoven, N., Koelink, E. & Román, P. Branching Rules for Finite-Dimensional \(\mathcal {U}_{q}(\mathfrak {su}(3))\)-Representations with Respect to a Right Coideal Subalgebra. Algebr Represent Theor 20, 821–842 (2017). https://doi.org/10.1007/s10468-017-9678-z
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DOI: https://doi.org/10.1007/s10468-017-9678-z