The missing label of su(3) and its symmetry

Nicolas Crampé (1) , Loïc Poulain D’andecy (2) , Luc Vinet (3)

1 IDP - Institut Denis Poisson
2 LMR - Laboratoire de Mathématiques de Reims
3 CRM - Centre de Recherches Mathématiques [Montréal]

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.

Domains: Physics [physics]; Mathematical Physics [math-ph]; Mathematics [math]; Representation Theory [math.RT]

Nicolas Crampé, Loïc Poulain D’andecy, Luc Vinet. The missing label of su(3) and its symmetry. Communications in Mathematical Physics, 2023, ⟨10.1007/s00220-022-04596-3⟩. ⟨hal-03873119⟩