Symmetries of Clebsch–Gordan coefficients of the quantum group U q(n)

Abstract

Permutation symmetries for Clebsch–Gordan coefficients of the quantum group Uq(n) and for Uq(n−1)-reduced Clebsch–Gordan coefficients of Uq(n) are derived. In particular, the formula relating Clebsch–Gordan coefficients for the tensor product Tm1⊗Tm2 with Clebsch–Gordan coefficients for the tensor product with permuted representations Tm1 and Tm is obtained, where Tm is one of the irreducible representations in the decomposition of Tm1⊗Tm2. Contrary to the classical case, coefficients connecting these Clebsch–Gordan coefficients depend on Gel’fand–Tsetlin patterns of one of irreducible representations. This dependence is explicitly determined.