  
  
                                   [1X [5XHAPprime[0m[1X [0m
  
  
                          [1X Datatypes reference manual [0m
  
  
                                  Version 0.3
  
  
                                10 December 2008
  
  
                                   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
  © 2006-2008 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)
  
  
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  [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 HAPZeroRingHomomorphism
      7.2-6 InverseRingHomomorphism
      7.2-7 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
  
  
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