Classifying triples of Lagrangians in a Hermitian vector space
type: Item
instance of: scholarly article
title: Classifying triples of Lagrangians in a Hermitian vector space
zbMATH Open document ID: 1086.53095
author: Elisha Falbel
author: Jean-Pierre Marco
author: Florent Schaffhauser
published in: Topology and its Applications
publication date: 2004-11-19T00:00:00Z
review text: 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 \(\pi _{1}( S^{2}\backslash \{s_{1},s_{2},s_{3}\}) \).
Mathematics Subject Classification ID: 53D12
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 2116781
zbMATH Keywords: Lagrangian subspace
zbMATH Keywords: Hermitian vector space
zbMATH Keywords: diagonal action
zbMATH Keywords: inertia index
zbMATH Keywords: unitary representation
reviewed by: Gheorghe Zet
MaRDI profile type: Publication
full work available at URL: j.topol.2003.08.027
OpenAlex ID: W2049703309
cites work: Eigenvalues of products of unitary matrices and quantum Schubert calculus
cites work: Local systems on \({\mathbb{P}}^1-S\) for \(S\) a finite set
cites work: A CRITERION FOR THE EXISTENCE OF A PARABOLIC STABLE BUNDLE OF RANK TWO OVER THE PROJECTIVE LINE
cites work: On the existence of unitary flat connections over the punctured sphere with given local monodromy around the punctures
cites work: Finite Groups Generated by Involutions on Lagrangian Planes of C2
cites work: A Poincaré's polyhedron theorem for complex hyperbolic geometry
cites work: Q4237093
cites work: Q3999005
cites work: On the splitting of invariant manifolds in multidimensional near-integrable Hamiltonian systems
cites work: Classifying pairs of Lagrangians a Hermitian vector space
DOI: 10.1016/J.TOPOL.2003.08.027
description: scientific article; zbMATH DE number 2116781