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Algebras and Representation Theory

, Volume 20, Issue 4, pp 821–842 | Cite as

Branching Rules for Finite-Dimensional \(\mathcal {U}_{q}(\mathfrak {su}(3))\)-Representations with Respect to a Right Coideal Subalgebra

  • Noud Aldenhoven
  • Erik Koelink
  • Pablo RománEmail author
Open Access
Article

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.

Keywords

Quantum groups Coideal subalgebras Quantum symmetric pairs Branching rules 

Notes

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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© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.IMAPPRadboud UniversiteitNijmegenThe Netherlands
  2. 2.CIEM, FaMAFUniversidad Nacional de Córdoba, Medina Allende s/n Ciudad UniversitariaCórdobaArgentina

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