Resolution of the GL(3) - O(3) state labelling problem via O(3)-invariant Bethe subalgebra of the twisted Yangian

Abstract

The labelling of states of irreducible representations of GL(3) in an O(3) basis is well known to require the addition of a single O(3)-invariant operator, to the standard diagonalisable set of Casimir operators in the subgroup chain GL(3) - O(3) - O(2). Moreover, this `missing label' operator must be a function of the two independent cubic and quartic invariants which can be constructed in terms of the angular momentum vector and the quadrupole tensor. It is pointed out that there is a unique (in a well-defined sense) combination of these which belongs to the O(3) invariant Bethe subalgebra of the twisted Yangian Y(GL(3);O(3)) in the enveloping algebra of GL(3).