On Schur problem and Kostka numbers
Abstract
We reconsider the two related problems: distribution of the diagonal elements of a Hermitian n x n matrix of known eigenvalues (Schur) and determination of multiplicities of weights in a given irreducible representation of 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 that complement those given in the literature, 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 B2 algebra, with a few novel conjectures.
Cite
@article{arxiv.2001.08046,
title = {On Schur problem and Kostka numbers},
author = {Robert Coquereaux and Jean-Bernard Zuber},
journal= {arXiv preprint arXiv:2001.08046},
year = {2020}
}
Comments
22 pages, 12 figures. To be published in Integrability, Quantization, and Geometry - B.A. Dubrovin memorial volume, edited by I. Krichever, S. Novikov, O. Ogievetsky and S. Shlosman, AMS, Proceedings of Symposia in Pure Mathematics (PSPUM) book series