Compositions of quadratic forms PDF
Preview Compositions of quadratic forms
Dedicated to Amanda, Becky and Jacob Table of Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xi Chapter0 HistoricalBackground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Part I. Classical Compositions and Quadratic Forms Chapter1 SpacesofSimilarities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Appendix. Compositionalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Chapter2 AmicableSimilarities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Chapter3 CliffordAlgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Chapter4 C-ModulesandtheDecompositionTheorem . . . . . . . . . . . . . . . . . . 73 Appendix. λ-HermitianformsoverC . . . . . . . . . . . . . . . . . . . . . . . . 81 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Chapter5 Small(s,t)-Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .106 viii Chapter6 Involutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .108 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .116 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .118 Chapter7 Unsplittable(σ,τ)-Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . .119 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .131 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .134 Chapter8 TheSpaceofAllCompositions . . . . . . . . . . . . . . . . . . . . . . . . . .135 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .150 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .157 Chapter9 ThePfisterFactorConjecture . . . . . . . . . . . . . . . . . . . . . . . . . . .160 Appendix. Pfisterformsandfunctionfields . . . . . . . . . . . . . . . . . . . .167 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .170 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .175 Chapter10 CentralSimpleAlgebrasandanExpansionTheorem . . . . . . . . . . . . .176 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .196 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .202 Chapter11 HassePrinciples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .204 Appendix. Hasseprinciplefordivisibilityofforms . . . . . . . . . . . . . . . .218 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .220 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .223 Part II. Compositions of Size [r,s,n] Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .227 Chapter12 [r,s,n]-FormulasandTopology . . . . . . . . . . . . . . . . . . . . . . . . . .231 Appendix. Moreapplicationsoftopologytoalgebra . . . . . . . . . . . . . . .252 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .256 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .264 ix Chapter13 IntegerCompositionFormulas . . . . . . . . . . . . . . . . . . . . . . . . . .268 AppendixA.AnewproofofYuzvinsky’stheorem . . . . . . . . . . . . . . . .286 AppendixB.Monomialcompositions . . . . . . . . . . . . . . . . . . . . . . . .288 AppendixC.Knownupperboundsforr ∗s . . . . . . . . . . . . . . . . . . . .291 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .294 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .297 Chapter14 CompositionsoverGeneralFields . . . . . . . . . . . . . . . . . . . . . . . . .299 Appendix. Compositionsofquadraticformsα,β,γ . . . . . . . . . . . . . . .317 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .321 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .327 Chapter15 HopfConstructionsandHiddenFormulas . . . . . . . . . . . . . . . . . . . .329 Appendix. Polynomialmapsbetweenspheres . . . . . . . . . . . . . . . . . . .348 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .353 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .361 Chapter16 RelatedTopics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .363 SectionA.Higherdegreeformspermittingcomposition . . . . . . . . . . . . . .363 SectionB.Vectorproductsandcompositionalgebras . . . . . . . . . . . . . . .368 SectionC.Compositionsoverringsandoverfieldsofcharacteristic2 . . . . . .370 D.Linearspacesofmatricesofconstantrank . . . . . . . . . . . . . . . . . . . .372 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .375 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .380 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .381 ListofSymbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .407 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .413 Introduction This book addresses basic questions about compositions of quadratic forms in the sense of Hurwitz and Radon. The initial question is: For what dimensions can they exist? Subsequentquestionsinvolveclassificationandanalysisofthequadraticforms whichcanoccurinacomposition. This topic originated with the “1, 2, 4, 8 Theorem” concerning formulas for a product of two sums of squares. That theorem, proved by Adolf Hurwitz in 1898, wasgeneralizedinvariouswaysduringthefolowingcentury,leadingtothetheories discussed here. This area is worth studying because it is so centrally located in mathematics: thesecompositionshavecloseconnectionswithmathematicalhistory, algebra,combinatorics,geometry,andtopology. Compositions have deep historical roots: the 1, 2, 4, 8 Theorem settled a long standing question about the existence of “n-square identities” and exhibited some of the power of linear algebra. Compositions are also entwined with the nineteenth centurydevelopmentofquaternions,octonionsandCliffordalgebras. Another attraction of this subject is its fascinating relationship with Clifford al- gebras and the algebraic theory of quadratic forms. A general composition formula involvesarbitraryquadraticformsoverafield,notjusttheclassicalsumsofsquares. SuchcompositionscanbereformulatedintermsofCliffordalgebrasandtheirinvo- lutions. Thereisalsoacloseconnectionbetweentheformsinvolvedincompositions andthemultiplicativequadraticformsintroducedbyPfisterinthe1960s. Alltheknownconstructionsofcompositionformulasforsumsofsquarescanbe achievedusingintegercoefficients. Acompositionformulawithintegercoefficients canberecastasacombinatorialobject: aspecialsortofmatrixofsymbolsandsigns. These“intercalate”matriceshavebeenstudiedintensively,leadingtoaclassification oftheintegercompositionswhichinvolveatmost16squares. Finally this topic is connected with certain deep questions in geometry. For in- stance,compositionformulasprovideexamplesofvectorbundlesonprojectivespaces, of independent vector fields on spheres, of immersions of projective spaces into eu- clideanspaces, andofHopfmapsbetweeneuclideanspheres. Thetopologicaltools developedtoanalyzethesetopicsalsoyieldresultsaboutrealcompositions. Let us now describe the original question with more precision: A composition formulaofsize[r,s,n]isasumofsquaresformulaofthetype (x2+x2+···+x2)·(y2+y2+···+y2)=z2+z2+···+z2 1 2 r 1 2 s 1 2 n
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