High orders isotropic exchange in symmetrical clusters

The group-theoretical classification of exchange multiplets is suggested. The exchange levels are characterized by full spin S and irreducible representations of SU(2s+1), R(2s+1) or Sp(2s+1) groups. In the cases s≤$\text{3}\text{2}$ the exchange Hamiltonian can be expressed through Casimir's operators of these groups. It makes possible to find the expression for energy of symmetrical systems in analitical form. It is shown that Schrödinger exchange operator H(s), which is the generalization of the Dirac exchange Hamiltonian, is the Casimir's operator of the corresponding unitary U(2s+1) group.