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.

References in corpus (5)

• The Askey-Wilson algebra and its avatars
• Diagonalization of the Heun-Askey-Wilson operator, Leonard pairs and the algebraic Bethe ansatz
• Heun operator of Lie type and the modified algebraic Bethe ansatz
• Racah algebras, the centralizer $Z_n({\mathfrak{sl}}_2)$ and its Hilbert-Poincaré series
• The Heun-Racah and Heun-Bannai-Ito algebras

Cited by in corpus (1)

• Loop-string-hadron approach to SU(3) lattice Yang-Mills theory: I. Hilbert space of a trivalent vertex