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On the Clebsch–Gordan coefficients for the quantum group \({\varvec{U}}_{\varvec{q}}\varvec{(2)}\)

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Abstract

We consider the compact quantum group \(U_q(2)\) for \(q\in \mathbb {C}\setminus \{0\}\) with \(|q|\ne 1\), and decompose the tensor product of two irreducible representations into irreducible components. The decomposition is realized in terms of a basis of homogeneous polynomials in four variables involving the matrix elements of the irreducible representations of \(U_q(2)\). Then, we compute the Clebsch–Gordan coefficients in terms of the q-hypergeometric series \({}_3\phi _2\). When q is real, the Clebsch–Gordan coefficients are real and its expression can be written in terms of the q-Hahn polynomial \({\mathscr {Q}}_n(x;a,b,N|q^2)\).

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Acknowledgements

The first author, Satyajit Guin acknowledges the support of SERB Grant MTR/2021/000818 and the second author, Bipul Saurabh acknowledges the support of SERB Grant SRG/2020/000252.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, 208 016, India

    Satyajit Guin

  2. Indian Institute of Technology Gandhinagar, Palaj, Gandhinagar, 382 355, India

    Bipul Saurabh

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  1. Satyajit Guin
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  2. Bipul Saurabh
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Correspondence to Satyajit Guin.

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Communicated by S Viswanath.

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Guin, S., Saurabh, B. On the Clebsch–Gordan coefficients for the quantum group \({\varvec{U}}_{\varvec{q}}\varvec{(2)}\). Proc Math Sci 133, 44 (2023). https://doi.org/10.1007/s12044-023-00762-2

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