On Schur problem and Kostka numbers

Robert Coquereaux (1, 2) , Jean-Bernard Zuber (3)

1 CPT - E2 Géométrie, Physique et Symétries
2 CPT - Centre de Physique Théorique - UMR 7332
3 LPTHE - Laboratoire de Physique Théorique et Hautes Energies

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.

Domaines: Physique mathématique [math-ph]; Théorie des groupes [math.GR]; Probabilités [math.PR]; Théorie des représentations [math.RT]

Robert Coquereaux, Jean-Bernard Zuber. On Schur problem and Kostka numbers. I. Krichever, S. Novikov, O. Ogievetsky and S. Shlosman. Integrability, Quantization, and Geometry : II. Quantum Theories and Algebraic Geometry. B.A. Dubrovin memorial volume., Volume 103.2, AMS, 2021, AMS Books series PSPUM/103.2, 978-1-4704-5592-7 978-1-4704-6435-6. ⟨10.1090/pspum/103.2/01855⟩. ⟨hal-02465963⟩