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HAPcryst – An extension of the GAP package HAP for crystallographic groups

( Version 0.1.11 )

Marc Röder
Email: marc_roeder(at)web.de

Address:
Marc Röder, Department of Mathematics, NUI Galway, Irleland

Abstract

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.txt or http://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

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