Lie algebra representations, nilpotent matrices, and the C–numerical range

In this paper we survey some applications of the representation theory of Lie algebras to Linear Algebra. This includes the derivation of the Jordan form for tensor products of invertible matrices, the study of normal form problems for nilpotent matrices, as well as the derivation of explicit formulas for the C-numerical radius of certain nilpotent block-shift matrices, arising in quantum mechanics and nuclear magnetic resonance (NMR-) spectroscopy. In this latter case, a conjecture concerning an explicit formula for the C-numerical radius is stated. We show the existence of unitary transformations that realize the prospective maxima. Our approach depends on the Clebsch-Gordan decomposition for unitary representations of su2(C).