On Schur problem and Kostka numbers

Abstract

We reconsider the two related problems: distribution of the diagonal elements of a Hermitian n ˆn matrix of known eigenvalues (Schur) and determination of multiplicities of weights in a given irreducible representation of SUpnq (Kostka). It is well known that the former yields a semi-classical picture of the latter. We present explicit expressions for low values of n that complement those given in the literature [11,1], recall some exact (non asymptotic) relation between the two problems, comment on the limiting procedure whereby Kostka numbers are obtained from Littlewood-Richardson coefficients, and finally extend these considerations to the case of the B 2 algebra, with a few novel conjectures.