THE SYMPLECTIC AND ALGEBRAIC GEOMETRY OF HORN'S PROBLEM

Description

ar X iv :m at h/ 99 11 08 8v 2 [m ath .R A] 2 5 M ay 20 00 THE SYMPLECTIC AND ALGEBRAIC GEOMETRY OF HORN'S PROBLEM ALLEN KNUTSON ABSTRACT. One version of Horn's problem asks for which ?, µ, ? does H? + Hµ + H? = 0 have solutions, where H?,µ,? are Hermitian matrices with spectra ?, µ, ?. This turns out to be a moment map condition in Hamiltonian geometry. Many of the results around Horn's problem proven with great effort “by hand” are in fact simple consequences of the modern machinery of symplectic geometry, and the subtler ones provable via the connection to geometric invariant theory. We give an overview of this theory (which was not available to Horn), including all definitions, and how it can be used in linear algebra. 1. INTRODUCTION This is an expository paper on the symplectic and algebraic geometry implicit in Horn's problem, which asks the possible spectra of a sum of two Hermitian matrices each with known spectrum. The connection with symplectic geometry is very straightforward: the map (H?, Hµ) 7? H? + Hµ that takes a pair of Hermitian matrices with known spectra to their sum is a moment map for the diagonal conjugation action of U(n) on a certain symplectic manifold (definitions to follow).

• symplectic manifolds


• lie algebra


• borel-weil-bott-kostant theorem


• theorem states


• now let


• let ?


• hamiltonian geometry


• gradient points


• vector


• schur-horn theorem

Sujets

• Denotation (semiotics)
• Lie algebra
• Vector
• Schur?Horn theorem

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 46
• Langue: English

Extrait

THESYMPLECTICANDALGEBRAICGEOMETRYOFHORN’SPROBLEM ALLENKNUTSON A BSTRACT .OneversionofHorn’sproblemasksforwhichλ,,νdoesH λ + H µ + H ν = 0 havesolutions,whereH λ,µ,ν areHermitianmatriceswithspectraλ,,ν.Thisturnsoutto beamomentmapconditioninHamiltoniangeometry.ManyoftheresultsaroundHorn’s problemprovenwithgreateffort“byhand”areinfactsimpleconsequencesofthemodern machineryofsymplecticgeometry,andthesubtleronesprovableviatheconnectionto geometricinvarianttheory.Wegiveanoverviewofthistheory(whichwasnotavailableto Horn),includingalldefinitions,andhowitcanbeusedinlinearalgebra. 1.I NTRODUCTION ThisisanexpositorypaperonthesymplecticandalgebraicgeometryimplicitinHorn’s problem,whichasksthepossiblespectraofasumoftwoHermitianmatriceseachwith knownspectrum. Theconnectionwithsymplecticgeometryisverystraightforward:themap(H λ ,H µ ) 7 → H λ + H µ thattakesapairofHermitianmatriceswithknownspectratotheirsumisa momentmap forthediagonalconjugationactionofU ( n ) onacertain symplecticmanifold (definitionstofollow).Thisisaveryrestrictivepropertyofmaps,andmanythingscanbe provedaboutthem.Theproofs,atheart,arenotreallyanydifferentthanthetechniques Hornhimselfusedtostudythismap.Nonethelesstheframeworkisworthunderstanding inordertorecognizewhatotherlinearalgebraproblemsarelikelytohaveanswersasnice astheonestoHorn’sproblem.InparticulartheSchur-Horntheoremfollowsveryeasily fromthegeneraltheoremsinthisarea(andwasaprimaryinspirationforthem). Someofthemoreesotericconnections–toalgebraicgeometryandrepresentationthe- ory–canalsobeseeninthiscontext,viatheKirwan/Nesstheorem(whichwewillalso state).Again,thebasictechniquesusedarethesame,butintheKirwan/Nesstheorem oneseesthesetechniquespushedtoprovethestatementsinwhatappearstobetheir propergenerality. AlongthewayweexplaintherelationbetweenHermitianmatrices,flagmanifolds, andtheBorel-Weil-Bott-Kostanttheorem. 2.T HE S CHUR -H ORNTHEOREM ,H ORN ’ SPROBLEM , AND H AMILTONIANMANIFOLDS Let λ =( λ 1 ≥ λ 2 ≥ ... ≥ λ n )beaweaklydecreasinglistofrealnumbers,whichwe’ll usetoencodetheeigenvaluespectrumofaHermitianmatrix.TheSchur-Horntheorem statesthefollowing: Date :February1,2008. 1