A statistical law for multiplicities of SU(3) irreps (lambda, mu) in the plethysm {eta} circle times(3) {m} -> (lambda, mu)

A statistical law for the multiplicities of the SU(3) irreps (lambda, mu) in the reduction of totally symmetric irreducible representations {m} of U(N), N = (eta + 1) (eta + 2)/2 with eta being the three-dimensional oscillator major shell quantum number, is derived in terms of the quadratic and cubic invariants of SU(3), by determining the first three terms of an asymptotic expansion for the multiplicities. To this end, the bivariate Edgeworth expansion known in statistics is used. Simple formulae, in terms of m and eta, for all the parameters in the expansion are derived. Numerical tests with large m and eta = 4, 5 and 6 show good agreement with the statistical formula for the SU(3) multiplicities.