[1X [5XHAPprime[0m[1X [0m [1X Datatypes reference manual [0m Version 0.3.2 13 February 2009 Paul Smith Paul Smith Email: [7Xmailto:paul.smith@nuigalway.ie[0m Homepage: [7Xhttp://www.maths.nuigalway.ie/~pas[0m Address: Department of Mathematics, National University of Ireland, Galway Galway, Ireland. ------------------------------------------------------- [1XCopyright[0m © 2006-2009 Paul Smith [5XHAPprime[0m is released under the GNU General Public License (GPL). This file is part of [5XHAPprime[0m, though as documentation it is released under the GNU Free Documentation License (see [7Xhttp://www.gnu.org/licenses/licenses.html#FDL[0m). [5XHAPprime[0m is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. [5XHAPprime[0m is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with [5XHAPprime[0m; if not, write to the Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA For more details, see [7Xhttp://www.fsf.org/licenses/gpl.html[0m. ------------------------------------------------------- [1XAcknowledgements[0m [5XHAPprime[0m is supported by a Marie Curie Transfer of Knowledge grant based at the Department of Mathematics, NUI Galway (MTKD-CT-2006-042685) ------------------------------------------------------- [1XContents (HAPprime Datatypes)[0X 1 Introduction 2 Resolutions 2.1 The [9XHAPResolution[0m datatype in [5XHAPprime[0m 2.2 Implementation: Constructing resolutions 2.3 Resolution construction functions 2.3-1 LengthOneResolutionPrimePowerGroup 2.3-2 LengthZeroResolutionPrimePowerGroup 2.4 Resolution data access functions 2.4-1 ResolutionLength 2.4-2 ResolutionGroup 2.4-3 ResolutionFpGModuleGF 2.4-4 ResolutionModuleRank 2.4-5 ResolutionModuleRanks 2.4-6 BoundaryFpGModuleHomomorphismGF 2.4-7 ResolutionsAreEqual 2.5 Example: Computing and working with resolutions 2.6 Miscellaneous resolution functions 2.6-1 BestCentralSubgroupForResolutionFiniteExtension 3 Graded algebras 3.1 Graded algebras in [5XHAP[0m (and [5XHAPprime[0m) 3.2 Data access functions 3.2-1 ModPRingGeneratorDegrees 3.2-2 ModPRingNiceBasis 3.2-3 ModPRingNiceBasisAsPolynomials 3.2-4 ModPRingBasisAsPolynomials 3.3 Other functions 3.3-1 PresentationOfGradedStructureConstantAlgebra 3.4 Example: Graded algebras and mod-p cohomology rings 4 Presentations of graded algebras 4.1 The [9XGradedAlgebraPresentation[0m datatype 4.2 Construction function 4.2-1 GradedAlgebraPresentation construction functions 4.3 Data access functions 4.3-1 BaseRing 4.3-2 CoefficientsRing 4.3-3 IndeterminatesOfGradedAlgebraPresentation 4.3-4 GeneratorsOfPresentationIdeal 4.3-5 PresentationIdeal 4.3-6 IndeterminateDegrees 4.3-7 Example: Constructing and accessing data of a [9XGradedAlgebraPresentation[0m 4.4 Other functions 4.4-1 TensorProduct 4.4-2 IsIsomorphicGradedAlgebra 4.4-3 IsAssociatedGradedRing 4.4-4 DegreeOfRepresentative 4.4-5 MaximumDegreeForPresentation 4.4-6 SubspaceDimensionDegree 4.4-7 SubspaceBasisRepsByDegree 4.4-8 CoefficientsOfPoincareSeries 4.4-9 HilbertPoincareSeries 4.4-10 LHSSpectralSequence 4.5 Example: Computing the Lyndon-Hoschild-Serre spectral sequence and mod-p cohomology ring for a small p-group 5 FG-modules 5.1 The [9XFpGModuleGF[0m datatype 5.2 Implementation details: Block echelon form 5.2-1 Generating vectors and their block structure 5.2-2 Matrix echelon reduction and head elements 5.2-3 Echelon block structure and minimal generators 5.2-4 Intersection of two modules 5.3 Construction functions 5.3-1 FpGModuleGF construction functions 5.3-2 FpGModuleFromFpGModuleGF 5.3-3 MutableCopyModule 5.3-4 CanonicalAction 5.3-5 Example: Constructing a [9XFpGModuleGF[0m 5.4 Data access functions 5.4-1 ModuleGroup 5.4-2 ModuleGroupOrder 5.4-3 ModuleAction 5.4-4 ModuleActionBlockSize 5.4-5 ModuleGroupAndAction 5.4-6 ModuleCharacteristic 5.4-7 ModuleField 5.4-8 ModuleAmbientDimension 5.4-9 AmbientModuleDimension 5.4-10 DisplayBlocks 5.4-11 Example: Accessing data about a [9XFpGModuleGF[0m 5.5 Generator and vector space functions 5.5-1 ModuleGenerators 5.5-2 ModuleGeneratorsAreMinimal 5.5-3 ModuleGeneratorsAreEchelonForm 5.5-4 ModuleIsFullCanonical 5.5-5 ModuleGeneratorsForm 5.5-6 ModuleRank 5.5-7 ModuleVectorSpaceBasis 5.5-8 ModuleVectorSpaceDimension 5.5-9 MinimalGeneratorsModule 5.5-10 RadicalOfModule 5.5-11 Example: Generators and basis vectors of a [9XFpGModuleGF[0m 5.6 Block echelon functions 5.6-1 EchelonModuleGenerators 5.6-2 ReverseEchelonModuleGenerators 5.6-3 Example: Converting a [9XFpGModuleGF[0m to block echelon form 5.7 Sum and intersection functions 5.7-1 DirectSumOfModules 5.7-2 DirectDecompositionOfModule 5.7-3 IntersectionModules 5.7-4 SumModules 5.7-5 Example: Sum and intersection of [9XFpGModuleGF[0ms 5.8 Miscellaneous functions 5.8-1 = 5.8-2 IsModuleElement 5.8-3 IsSubModule 5.8-4 RandomElement 5.8-5 Random Submodule 6 FG-module homomorphisms 6.1 The [9XFpGModuleHomomorphismGF[0m datatype 6.2 Calculating the kernel of a FG-module homorphism by splitting into two homomorphisms 6.3 Calculating the kernel of a FG-module homorphism by column reduction and partitioning 6.4 Construction functions 6.4-1 FpGModuleHomomorphismGF construction functions 6.4-2 Example: Constructing a [9XFpGModuleHomomorphismGF[0m 6.5 Data access functions 6.5-1 SourceModule 6.5-2 TargetModule 6.5-3 ModuleHomomorphismGeneratorMatrix 6.5-4 DisplayBlocks 6.5-5 DisplayModuleHomomorphismGeneratorMatrix 6.5-6 DisplayModuleHomomorphismGeneratorMatrixBlocks 6.5-7 Example: Accessing data about a [9XFpGModuleHomomorphismGF[0m 6.6 Image and kernel functions 6.6-1 ImageOfModuleHomomorphism 6.6-2 PreImageRepresentativeOfModuleHomomorphism 6.6-3 KernelOfModuleHomomorphism 6.6-4 Example: Kernel and Image of a [9XFpGModuleHomomorphismGF[0m 7 Ring homomorphisms 7.1 The [9XHAPRingHomomorphism[0m datatype 7.1-1 Implementation details 7.1-2 Elimination orderings 7.2 Construction functions 7.2-1 HAPRingToSubringHomomorphism 7.2-2 HAPSubringToRingHomomorphism 7.2-3 HAPRingHomomorphismByIndeterminateMap 7.2-4 HAPRingReductionHomomorphism 7.2-5 PartialCompositionRingHomomorphism 7.2-6 HAPZeroRingHomomorphism 7.2-7 InverseRingHomomorphism 7.2-8 CompositionRingHomomorphism 7.3 Data access functions 7.3-1 SourceGenerators 7.3-2 SourceRelations 7.3-3 SourcePolynomialRing 7.3-4 ImageGenerators 7.3-5 ImageRelations 7.3-6 ImagePolynomialRing 7.4 General functions 7.4-1 ImageOfRingHomomorphism 7.4-2 PreimageOfRingHomomorphism 7.5 Example: Constructing and using a [9XHAPRingHomomorphism[0m 8 Derivations 8.1 The [9XHAPDerivation[0m datatype 8.2 Computing the kernel and homology of a derivation 8.3 Construction function 8.3-1 HAPDerivation construction functions 8.4 Data access function 8.4-1 DerivationRing 8.4-2 DerivationImages 8.4-3 DerivationRelations 8.4-4 Example: Constructing and accessing data of a [9XHAPDerivation[0m 8.5 Image, kernel and homology functions 8.5-1 ImageOfDerivation 8.5-2 KernelOfDerivation 8.5-3 HomologyOfDerivation 8.5-4 Example: Homology of a [9XHAPDerivation[0m 9 Poincaré series 9.1 Computing the Poincaré series using spectral sequences 9.2 Computing the Poincaré series using a minimal resolution 9.2-1 PoincareSeriesAutoMem 9.3 Example Poincaré series computations 9.4 The Poincaré series of groups of order 64 and 128 10 General Functions 10.1 Matrices 10.1-1 SumIntersectionMatDestructive 10.1-2 SolutionMat 10.1-3 IsSameSubspace 10.1-4 PrintDimensionsMat 10.1-5 Example: matrices and vector spaces 10.2 Polynomials 10.2-1 TermsOfPolynomial 10.2-2 IsMonomial 10.2-3 UnivariateMonomialsOfMonomial 10.2-4 IndeterminateAndExponentOfUnivariateMonomial 10.2-5 IndeterminatesOfPolynomial 10.2-6 ReduceIdeal 10.2-7 ReducedPolynomialRingPresentation 10.2-8 Example: monomials, polynomials and ring presentations 10.3 Singular 10.3-1 SingularSetNormalFormIdeal 10.3-2 SingularPolynomialNormalForm 10.3-3 SingularGroebnerBasis 10.3-4 SingularReducedGroebnerBasis 10.4 Groups 10.4-1 HallSeniorNumber -------------------------------------------------------