A complete set of commuting operators for the Bethe ansatz

We have derived an explicit formula for a complete set of commuting operators for the XXX model of the Heisenberg magnetic ring, using an algebraic Bethe ansatz approach of Faddeev and Takhtajan. Each operator turns out to be the sum of increasing l-cycles in the symmetric group acting on the set of nodes of the ring. It is demonstrated that the resulting algebra of operators encloses the total spin S, the Hamiltonian and the quasimomentum. We point out that it is a maximal Abelian subalgebra in the algebra of the symmetric group, associated with the basis of exact Bethe ansatz eigenstates, the latter classified by rigged string configurations of Kerrov, Kirillov and Reshetikhin. This algebra is also conjugated to the Jucys-Murphy algebra, responsible for the Young orthogonal basis of standard tableaux along the Schur-Weyl duality.