FiniteGroups package

FiniteGroups is a package for manipulating discrete groups such as space groups.

Download source or library or ini-file. Tested for Maple 2025.

Highlighted features

• Support of custom group algebras. Predefined are matrices and affine transformations.
• Support of symbolic lattice coordinates.
• Large list of predefined groups: space, layer, rod, point (also 5- and 8-fold axes), in different settings.
• Representations of point groups.
• Sites and dimers: black-box construction of a tight-binding Hamiltonian.

Documentation

• Notations and notes
• Setup and auxiliary routines (including FG_simplify and FG_floor)
Setup,setfloordef,setfloorfast,setfloorback,FG_g2q,FG_GFE
• Group algebra
FG_GroupAlgebra,FGS,FG_unit,FG_ident,FG_inv,FG_mul,FG_app,FG_appt,FG_apl,
unitcellon,unitcelloff,unitcellback
• Manipulations with sets
FG_member,FG_in,FG_equal,FG_set,FG_prod,FG_transform
• Manipulations with groups
FG_test,FG_testnumeric,FG_group,FG_mulTable,FG_invTable,FG_mulinvTable,
FG_cyclic,FG_orbitC,FG_factor,FG_testInvSG,FG_testSDP,FG_testDP
• Group representations
FG_orbit,FG_RegRep
• Representations of predefined groups
FGR_ChTb,FGR_RepU,FGR_RepU2,FGR_RepDec,FGR_Ch,FGR_RepDecCh,FGR_indexC,FGR_dimR
• Sites and dimers
FGS_init,FGS_get,FGS_correctit,FGS_correctost,FGS_it2g,FGS_g2it,FGS_inv,FGS_mul,
FGS_s2it,FGS_app,FGS_s2p,FGS_p2s,FGS_s2m,FGS_m2s,FGS_m2p,FGS_spick,
FGS_d2oost,FGS_d2oost1,FGS_d2oost2,FGS_dorb,FGS_dorb2,FGS_dpick,
FGS_dimersR,FGS_dimersR2,Encodesoost,Decodesoost
• Group tables
SymmetryGroup,PrintSymmetryElement,xyz2MV,SymmetryElement,
TransformationTable,FG_gens,FG_generate,kPoints,
HM2Scho,Scho2HM,PointGroupName,SpaceGroupName,LayerGroupName,RodGroupName,OtherGroupName,
SpaceGroupNumber,LayerGroupNumber,RodGroupNumber,ArithmeticClass,GeometricClass

Notebooks of examples and gray-box solutions

• exam_PG.mw – point groups.
• exam_SG.mw – space groups.

References

• Handbooks on Point Groups and Space Groups
• Bilbao Crystallographic Server
• See also Symmetry in Library

Acknowledgements

• Artem Naumov – rod groups

Requires implementation

• Custom remember table for SymmetryGroup because currently it regenerates groups instead of saving it to tbG table.
• Symmetry groups in all ITA settings.
• Support of nonscalar objects such as multiple degenerate orbitals per site.