Two-body homogeneous rational Gaudin models and the missing label problem (Q2847993)

The authors consider the rational \(N=2\) Gaudin model (without an external magnetic field) based on the Lie algebra \(\mathfrak{sl}(3)\) and show that it can be interpreted in natural form as the labelling operators for the reduction chain \(\mathfrak{sl}(3)\subset \mathfrak{sl}(3)\oplus \mathfrak{sl}(3)\). Moreover, it is proved that the traces of the Lax matrix can be expressed as linear combinations of missing label operators arising from decomposed Casimir operators of \(\mathfrak{sl}(3)\oplus \mathfrak{sl}(3)\), and this offers to these spectral invariants an algebraic meaning and a justification for their pairwise commutativity. The corresponding labelling operators are explicitly computed and shown to separate degeneracies in the Clebsch-Gordan series of \(\mathfrak{sl}(3)\). Also, numerical results with all the irreducible representations of \(\mathfrak{sl}(3)\oplus \mathfrak{sl}(3)\) up to dimension \(400\) are given in a tabular form.