Lie algebra representations, nilpotent matrices, and the \(C\)-numerical range
This is a survey paper on some applications of the representation theory of Lie algebras to linear algebra. The authors illustrate the power of representation theoretic methods by discussing their role in three different areas: Jordan canonical forms for tensor products of matrices; Geometry and parametrization of unitary orbits of nilpotent matrices; The calculation of the \(C\)-numerical range of nilpotent matrices. For example, using the Clebsch-Gordan decomposition for Lie group representation of \(SL_2({\mathbb C})\), the authors obtain an explicit formula for the Jordan form of the Kronecker product of two invertible Jordan blocks. They discuss the relations of classification of Lie algebra representations of \( \mathfrak{gl}_2({\mathbb C})\) to the task of parametrizing nilpotent similarity orbits. Further, applications to quantum mechanics are discussed, i.e.\ the maximization of the transfer function of \(N+1\) weakly coupled spin-\(\frac 12\)-systems arising in NMR-spectroscopy and quantum computing. Their approach consists in reformulating the task as a least squares matching problem for two unitary representations of \(\mathfrak{su}_2({\mathbb C})\). This then leads us to explicit formulas for the conjectured maxima inn terms of the Clebsch--Gordan decomposition. It is shown that unitary transformations exist that achieve these values.
scientific article; zbMATH DE number 5014299
scientific article