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

TL;DR

This paper derives explicit formulas for missing label operators in the tensor product of two b1a3a3 representations, revealing a large symmetry group and connections to algebraic structures like E6 and Hahn algebra.

Contribution

It provides explicit formulas for missing label operators in b1a3a3, explores their symmetry group, and links to advanced algebraic frameworks.

Findings

Explicit formulas for missing label operators.

Discovery of a symmetry group of order 144 related to E6.

Connections to Hahn algebra, Heun--Hahn operators, and Bethe ansatz.

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.