On Schur problem and Kostka numbers

Abstract

We reconsider the two related problems: distribution of the diagonal elements of a Hermitian n × n n\times n matrix of known eigenvalues (Schur) and determination of multiplicities of weights in a given irreducible representation of S U ( n ) SU(n) (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 n that complement those given in the literature \cite{GLS,BGR}, 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 B_2 algebra, with a few novel conjectures.