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HAPcryst

A HAP extension for crystallographic groups

Version 0.2.1

18 August 2026

Marc Roeder
Email: roeder.marc@gmail.com

Copyright

© 2007 Marc Röder.

This package is distributed under the terms of the GNU General Public License version 2 or later (at your convenience). See the file LICENSE or https://www.gnu.org/copyleft/gpl.html

Acknowledgements

This work was supported by Marie Curie Grant No. MTKD-CT-2006-042685

Contents

1.1 Abstract and Notation

1.1-1 The natural action of crystallographic groups

1.2 Requirements

1.2-1 Recommendation concerning polymake

1.3 Global Variables

1.3-1 InfoHAPcryst

2 Bits and Pieces

2.1 Matrices and Vectors

  2.1-1 SignRat
  2.1-2 VectorModOne
  2.1-3 IsSquareMat
  2.1-4 DimensionSquareMat

2.2 Affine Matrices OnRight

  2.2-1 LinearPartOfAffineMatOnRight
  2.2-2 BasisChangeAffineMatOnRight
  2.2-3 TranslationOnRightFromVector

  2.3-1 GramianOfAverageScalarProductFromFiniteMatrixGroup
  2.3-2 Inequalities
  2.3-3 BisectorInequalityFromPointPair
  2.3-4 WhichSideOfHyperplane
  2.3-5 RelativePositionPointAndPolygon

2.4 Space Groups

2.4-1 PointGroupRepresentatives

3 Algorithms of Orbit-Stabilizer Type

3.1 Orbit Stabilizer for Crystallographic Groups

  3.1-1 OrbitStabilizerInUnitCubeOnRight
  3.1-2 OrbitStabilizerInUnitCubeOnRightOnSets
  3.1-3 OrbitPartInVertexSetsStandardSpaceGroup
  3.1-4 OrbitPartInFacesStandardSpaceGroup
  3.1-5 OrbitPartAndRepresentativesInFacesStandardSpaceGroup
  3.1-6 StabilizerOnSetsStandardSpaceGroup
  3.1-7 RepresentativeActionOnRightOnSets
  3.1-8 Getting other orbit parts
  3.1-9 ShiftedOrbitPart
  3.1-10 TranslationsToOneCubeAroundCenter
  3.1-11 TranslationsToBox

4 Resolutions of Crystallographic Groups

4.1 Fundamental Domains

  4.1-1 FundamentalDomainStandardSpaceGroup
  4.1-2 FundamentalDomainBieberbachGroup
  4.1-3 FundamentalDomainFromGeneralPointAndOrbitPartGeometric
  4.1-4 IsFundamentalDomainStandardSpaceGroup
  4.1-5 IsFundamentalDomainBieberbachGroup

4.2 Face Lattice and Resolution

  4.2-1 ResolutionBieberbachGroup
  4.2-2 FaceLatticeAndBoundaryBieberbachGroup
  4.2-3 ResolutionFromFLandBoundary

A Resolutions in Hap

A.1 The Standard Representation HapResolutionRep

A.2 The HapLargeGroupResolutionRep Representation

B Accessing and Manipulating Resolutions

B.1 Representation-Independent Access Methods

  B.1-1 StrongestValidRepresentationForLetter
  B.1-2 StrongestValidRepresentationForWord
  B.1-3 PositionInGroupOfResolution
  B.1-4 IsValidGroupInt
  B.1-5 GroupElementFromPosition
  B.1-6 MultiplyGroupElts
  B.1-7 MultiplyFreeZGLetterWithGroupElt
  B.1-8 MultiplyFreeZGWordWithGroupElt
  B.1-9 BoundaryOfFreeZGLetter
  B.1-10 BoundaryOfFreeZGWord

B.2 Converting Between Representations

  B.2-1 ConvertStandardLetter
  B.2-2 ConvertStandardWord
  B.2-3 ConvertLetterToStandardRep
  B.2-4 ConvertWordToStandardRep

B.3 Special Methods for HapResolutionRep

  B.3-1 IsFreeZGLetter
  B.3-2 IsFreeZGWord
  B.3-3 MultiplyGroupEltsNC
  B.3-4 MultiplyFreeZGLetterWithGroupEltNC
  B.3-5 MultiplyFreeZGWordWithGroupEltNC
  B.3-6 BoundaryOfFreeZGLetterNC
  B.3-7 BoundaryOfFreeZGWordNC

B.4 The HapLargeGroupResolutionRep Representation

  B.4-1 GroupRingOfResolution
  B.4-2 MultiplyGroupElts_LargeGroupRep
  B.4-3 IsFreeZGLetterNoTermCheck_LargeGroupRep
  B.4-4 IsFreeZGWordNoTermCheck_LargeGroupRep
  B.4-5 IsFreeZGLetter_LargeGroupRep
  B.4-6 IsFreeZGWord_LargeGroupRep
  B.4-7 MultiplyFreeZGLetterWithGroupElt_LargeGroupRep
  B.4-8 MultiplyFreeZGWordWithGroupElt_LargeGroupRep
  B.4-9 GeneratorsOfModuleOfResolution_LargeGroupRep
  B.4-10 BoundaryOfGenerator_LargeGroupRep
  B.4-11 BoundaryOfFreeZGLetterNC_LargeGroupRep
  B.4-12 BoundaryOfFreeZGWord_LargeGroupRep

C Contracting Homotopies

C.1 The PartialContractingHomotopy Data Type

C.1-1 ResolutionOfContractingHomotopy
C.1-2 PartialContractingHomotopyLookup

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