The missing label of $\mathfrak{su}_3$ and its symmetry

Abstract

We present explicit formulas for the operators providing missing labels for the tensor product of two irreducible representations of su 3 . The result is seen as a particular representation of the diagonal centraliser of su 3 through a pair of tridiagonal matrices. Using these explicit formulas, we investigate the symmetry of this missing label problem and we find a symmetry group of order 144 larger than what can be expected from the natural symmetries. Several realisations of this symmetry group are given, including an interpretation as a subgroup of the Weyl group of type E 6 , which appeared in an earlier work as the symmetry group of the diagonal centraliser. Using the combinatorics of the root system of type E 6 , we provide a family of representations of the diagonal centraliser by infinite tridiagonal matrices, from which all the finite-dimensional representations affording the missing label can be extracted. Besides, some connections with the Hahn algebra, Heun-Hahn operators and Bethe ansatz are discussed along with some similarities with the wellknown symmetries of the Clebsch-Gordan coefficients.

Chern-Simons theory, link invariants and the Askey-Wilson algebra

“…A specific example would be to consider SU (3), to determine if similar tangle diagrams are also associated to natural elements of the centralizers of U q (su 3 ) in its tensor products, and to look if the approach using tangles and partial traces allows to understand better the algebra formed by these elements. The study of the algebraic structure of the centralizer of U (su 3 ) has been initiated in [10,11]. It could also be interesting to examine how the choice of a different manifold for the CS action affects the results of this paper.…”

The $SU(3)\supset SO(3)$ missing label problem and the analytical Bethe ansatz

“…Remark 2. With an appropriate identification of parameters, it can be shown that the matrix elements of x and y have the same form as those of the missing label operators in the two fold tensor product of su(3) found in [12].…”

“…ii. the determination in [12] with the nested Bethe ansatz of the eigenvalues of an operator providing the missing label in the SU (3) Clebsch-Gordan problem.…”