计算不变量理论

Introduction

1ConstructiveIdealTheory

1.1IdealsandGrSbnerBases

1.2EliminationIdeals

1.3SyzygyModules

1.4HilbertSeries

1.5TheRadicalIdeal

1.6Normalization

2InvariantTheory

2.1InvariantRings

2.2ReductiveGroups

2.3CategoricalQuotients

2.4HomogeneousSystemsofParameters

2.5TheCohen-MacaulayPropertyofInvariantRings

2.6HilbertSeriesofInvariantRings

3InvariantTheoryofFiniteGroups

3.1HomogeneousComponents

3.2MoliensFormula..

3.3PrimaryInvariants

3.4Cohen-Macaulayness

3.5SecondaryInvariants

3.6MinimalAlgebraGeneratorsandSyzygies

3.7PropertiesofInvariantRings

3.8NoethersDegreeBound

3.9DegreeBoundsintheModularCase

3.10PermutationGroups

3.11AdHocMethods

4InvariantTheoryofReductiveGroups

4.1ComputingInvariantsofLinearlyReductiveGroups

4.2ImprovementsandGeneralizations

4.3InvariantsofTori

4.4InvariantsofSLnandGLn

4.5TheReynoldsOperator

4.6ComputingHilbertSeries

4.7DegreeBoundsforInvariants

4.8PropertiesofInvariantRings

5ApplicationsofInvariantTheory

5.1CohomologyofFiniteGroups

5.2GaloisGroupComputation

5.3NoethersProblemandGenericPolynomials

5.4SystemsofAlgebraicEquationswithSymmetries

5.5GraphTheory

5.6Combinatorics

5.7CodingTheory

5.8EquivariantDynamicalSystems

5.9MaterialScience

5.10ComputerVision

ALinearAlgebraicGroups

A.1LinearAlgebraicGroups

A.2TheLieAlgebraofaLinearAlgebraicGroup

A.3ReductiveandSemi-simpleGroups

A.4Roots

A.5RepresentationTheory

References

Notation

Index

　　Thisbookisaboutthecomputationalaspectsofinvarianttheory.Thenalgorithmsforcomputinginvariantsoffiniteandreductivegroupsarediscussed.Thetextisenrichedwithnumerousexplicitexampleswhichillustratethetheoiwandshouldbeofmorethanpassinginterest.　　Thisbookisaboutthecomputationalaspectsofinvarianttheory.Ofcentralinterestisthequestionhowtheinvariantringofagivengroupactioncanbecalculated.Algorithmsforthispurposeformthemainpillarsaroundwhichthebookisbuilt.Therearetwointroductorychapters,oneonGrObnerbasismethodsandoneonthebasicconceptsofinvarianttheory,whichpreparethegroundforthealgorithms.Thenalgorithmsforcomputinginvariantsoffiniteandreductivegroupsarediscussed.Particularemphasisliesoninterrelationsbetweenstructuralpropertiesofinvariantringsandcomputationalmethods.Finally,thebookcontainsachapteronapplicationsofinvarianttheory,coveringfieldsasdisparateasgraphtheory,codingtheory,dynamicalsystems,andcomputervision.　　Thebookisintendedforpostgraduatestudentsaswellasresearchersingeometry,computeralgebra,and,ofcourse,invarianttheory.Thetextisenrichedwithnumerousexplicitexampleswhichillustratethetheoiwandshouldbeofmorethanpassinginterest.