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 $$\mathfrak {su}_3$$ . The result is seen as a particular representation of the diagonal centraliser of $$\mathfrak {su}_3$$ 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 well-known symmetries of the Clebsch–Gordan coefficients.