THE SYMPLECTIC AND ALGEBRAIC GEOMETRY OF HORN'S PROBLEM

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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

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Publié par

profil-urra-2012

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THESYMPLECTICANDALGEBRAICGEOMETRYOFHORN’SPROBLEMALLENKNUTSONABSTRACT.OneversionofHorn’sproblemasksforwhichλ,,νdoesHλ+Hµ+Hν=0havesolutions,whereHλ,µ,νareHermitianmatriceswithspectraλ,,ν.ThisturnsouttobeamomentmapconditioninHamiltoniangeometry.ManyoftheresultsaroundHorn’sproblemprovenwithgreateffort“byhand”areinfactsimpleconsequencesofthemodernmachineryofsymplecticgeometry,andthesubtleronesprovableviatheconnectiontogeometricinvarianttheory.Wegiveanoverviewofthistheory(whichwasnotavailabletoHorn),includingalldeﬁnitions,andhowitcanbeusedinlinearalgebra.1.INTRODUCTIONThisisanexpositorypaperonthesymplecticandalgebraicgeometryimplicitinHorn’sproblem,whichasksthepossiblespectraofasumoftwoHermitianmatriceseachwithknownspectrum.Theconnectionwithsymplecticgeometryisverystraightforward:themap(Hλ,Hµ)7→Hλ+HµthattakesapairofHermitianmatriceswithknownspectratotheirsumisamomentmapforthediagonalconjugationactionofU(n)onacertainsymplecticmanifold(deﬁnitionstofollow).Thisisaveryrestrictivepropertyofmaps,andmanythingscanbeprovedaboutthem.Theproofs,atheart,arenotreallyanydifferentthanthetechniquesHornhimselfusedtostudythismap.NonethelesstheframeworkisworthunderstandinginordertorecognizewhatotherlinearalgebraproblemsarelikelytohaveanswersasniceastheonestoHorn’sproblem.InparticulartheSchur-Horntheoremfollowsveryeasilyfromthegeneraltheoremsinthisarea(andwasaprimaryinspirationforthem).Someofthemoreesotericconnections–toalgebraicgeometryandrepresentationthe-ory–canalsobeseeninthiscontext,viatheKirwan/Nesstheorem(whichwewillalsostate).Again,thebasictechniquesusedarethesame,butintheKirwan/Nesstheoremoneseesthesetechniquespushedtoprovethestatementsinwhatappearstobetheirpropergenerality.AlongthewayweexplaintherelationbetweenHermitianmatrices,ﬂagmanifolds,andtheBorel-Weil-Bott-Kostanttheorem.2.THESCHUR-HORNTHEOREM,HORN’SPROBLEM,ANDHAMILTONIANMANIFOLDSLetλ=(λ1≥λ2≥...≥λn)beaweaklydecreasinglistofrealnumbers,whichwe’llusetoencodetheeigenvaluespectrumofaHermitianmatrix.TheSchur-Horntheoremstatesthefollowing:Date:February1,2008.1

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