Classifying triples of Lagrangians in a Hermitian vector space
The diagonal action of the unitary group \(U\left( n\right) \) on triples of Lagrangian subspaces of \(\mathbb C^{n}\) is studied using the associated involutions. First, it is recalled how pairs \(( L_{1},L_{2}) \) of Lagrangian subspaces are classified by the eigenvalues of the unitary map \(\sigma _{L_{2}}\circ \sigma _{L_{1}}\) obtained by composing the Lagrangian involutions relative to \(L_{1}\) and \(L_{2}\). The results are then extended to triples \(( L_{1},L_{2},L_{3}) \) of Lagrangian subspaces of \(\mathbb C^{2}\). A complete classification and a description of the orbit space is given in this case. The notions of angle and measure associated to a pair of Lagrangian subspaces are introduced and a necessary and sufficient condition of existence of a triple \(( L_{1},L_{2},L_{3}) \) with a given property relative to measure is established. As an application one gives a method of computing the inertia index of a triple \(( L_{1},L_{2},L_{3}) \) of Lagrangian subspaces of \(\mathbb C^{n}\) from the measure of the angles \(( L_{j},L_{k}) \). The classification of Lagrangian triples of \(\mathbb C^{2}\) is related to the classification of two-dimensional unitary representations of the fundamental group \(\pi _{1}( S^{2}\backslash \{s_{1},s_{2},s_{3}\}) \).
scientific article; zbMATH DE number 2116781