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#############################################################################
##
#W crystcat.gd GAP library Volkmar Felsch
##
#Y Copyright (C) 1997, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany
##
## This file contains the declarations for fthe crystallographic groups
## library.
##
#############################################################################
##
## Some global variables.
##
#############################################################################
##
#A CrystCatRecord( <G> )
##
DeclareAttribute( "CrystCatRecord", IsGroup, "mutable" );
#############################################################################
##
#F CR_CharTableQClass( <CR parameter list> )
##
## 'CR_CharTableQClass' returns the character table of a representative
## group of the specified Q-class.
##
## This is an internal function which does not check the arguments. The
## corresponding public function is 'CharTableQClass'.
##
DeclareGlobalFunction("CR_CharTableQClass");
#############################################################################
##
#F CR_DisplayQClass( <CR parameter list> )
##
## 'CR_DisplayQClass' displays for the specified Q-class the following
## information:
## - the size of the groups in the Q-class,
## - the isomorphism type of the groups in the Q-class,
## - the Hurley pattern,
## - the rational constituents,
## - the number of Z-classes in the Q-class, and
## - the number of space-group types in the Q-class.
##
## This is an internal function which does not check the arguments. The
## corresponding public function is 'DisplayQClass'.
##
DeclareGlobalFunction("CR_DisplayQClass");
#############################################################################
##
#F CR_DisplaySpaceGroupGenerators( <CR parameter list> )
##
## 'CR_DisplaySpaceGroupGenerators' displays the non-translation generators
## of the space group specified by the given parameters.
##
## This is an internal function which does not check the arguments. The
## corresponding public function is 'DisplaySpaceGroupGenerators'.
##
DeclareGlobalFunction("CR_DisplaySpaceGroupGenerators");
#############################################################################
##
#F CR_DisplaySpaceGroupType( <CR parameter list> )
##
## 'CR_DisplaySpaceGroupType' displays for the specified space-group type
## the following information:
## - the orbit size associated with the space-group type,
## - the IT number (only in case dim = 2 or dim = 3), and
## - the Hermann-Mauguin symbol (only in case dim = 2 or dim = 3).
##
## This is an internal function which does not check the arguments. The
## corresponding public function is 'DisplaySpaceGroupType'.
##
DeclareGlobalFunction("CR_DisplaySpaceGroupType");
#############################################################################
##
#F CR_DisplayZClass( <CR parameter list> )
##
## 'CR_DisplayZClass' displays for the specified Z-class the following
## information:
## - the Hermann-Mauguin symbol of a representative space-group type which
## belongs to the Z-class (only in case dim = 2 or dim = 3),
## - the Bravais type,
## - some decomposability information,
## - the number of space-group types belonging to the Z-class, and
## - the size of the associated cohomology group.
##
## This is an internal function which does not check the arguments. The
## corresponding public function is 'DisplayZClass'.
##
DeclareGlobalFunction("CR_DisplayZClass");
#############################################################################
##
#F CR_FpGroupQClass( <CR parameter list> )
##
## 'CR_FpGroupQClass' returns a f. p. group isomorphic to the groups in the
## specified Q-class.
##
## This is an internal function which does not check the arguments. The
## corresponding public function is 'FpGroupQClass'.
##
DeclareGlobalFunction("CR_FpGroupQClass");
#############################################################################
##
#F CR_GeneratorsSpaceGroup( <CR parameter list> )
##
## 'CR_GeneratorsSpaceGroup' returns the non-translation generators of the
## space group specified by the given parameters.
##
DeclareGlobalFunction("CR_GeneratorsSpaceGroup");
#############################################################################
##
#F CR_GeneratorsZClass( <dim>, <zclass> )
##
## 'CR_GeneratorsZClass' returns a set of matrix generators for a
## representative of the specified Z-class. These generators are chosen such
## that they satisfy the defining relators which are returned by the
## 'CR_FpGroupQClass' function for the representative of the corresponding
## Q-class.
##
DeclareGlobalFunction("CR_GeneratorsZClass");
#############################################################################
##
#F CR_InitializeRelators( <CR catalogue> )
##
## 'CR_InitializeRelators' initializes the relator words list of the
## crystallographic goups catalogue.
##
DeclareGlobalFunction("CR_InitializeRelators");
#############################################################################
##
#F CR_MatGroupZClass( <CR parameter list> )
##
## 'CR_MatGroupZClass' returns a representative group of the specified
## Z-class. The generators of the resulting matrix group are chosen such
## that they satisfy the defining relators which are returned by the
## 'CR_FpGroupQClass' function for the representative of the corresponding
## Q-class.
##
## This is an internal function which does not check the arguments. The
## corresponding public function is 'MatGroupZClass'.
##
DeclareGlobalFunction("CR_MatGroupZClass");
#############################################################################
##
#F CR_Name( <string>, <CR parameter list>, <nparms> )
##
## 'CR_Name' returns the "name" of the specified object which may be a
## Z-class representative, a Q-class representative or its character table,
## or a space group. The resulting name is a string which consists of the
## given string followed by the relevant parameters which are separated by
## commas and enclosed in parentheses.
##
DeclareGlobalFunction("CR_Name");
#############################################################################
##
#F CR_NormalizerZClass( <CR parameter list> )
##
## 'CR_NormalizerZClass' returns the normalizer in GL(dim,Z) of the
## specified Z-class representative matrix group. If the order of the
## normalizer is finite, then the group record components "crZClass" and
## "crConjugator" will be set properly.
##
## This is an internal function which does not check the arguments. The
## corresponding public function is 'NormalizerZClass'.
##
DeclareGlobalFunction("CR_NormalizerZClass");
#############################################################################
##
#F CR_Parameters( [ <dim>, <system>, <qclass>, <zclass>, <sgtype> ], nparms)
#F CR_Parameters( [ <dim>, <system>, <qclass>, <zclass> ], nparms )
#F CR_Parameters( [ <dim>, <system>, <qclass> ], nparms )
#F CR_Parameters( [ <dim>, <IT number> ], nparms )
#F CR_Parameters( [ <Hermann-Mauguin symbol> ], nparms )
##
## Valid argument lists are
##
## [ dim, sys, qcl, zcl, sgt ], 5
## [ dim, sys, qcl, zcl ], 4
## [ dim, sys, qcl ], 3
## [ 3, it ], n
## [ 2, it ], n
## [ symbol ], n
##
## where
##
## dim = dimension,
## sys = crystal system number with respect to a given dimension,
## qcl = Q-class number with respect to given dimension and crystal system,
## zcl = Z-class number with respect to given dimension, crystal system, and
## Q-class,
## sgt = space-group type with respect to given dimension, crystal system,
## Q-class, and Z-class,
## it = corresponding number in the International Tables (only for
## dimensions 2 and 3),
## n = 3 or 4 or 5,
## symbol = Hermann-Mauguin symbol (only for dimensions 2 or 3).
##
## 'CR_Parameters' checks the given arguments to be consistent and in range,
## and returns them in form of an "internal CR parameter list"
##
## [ dim, sys, qcl, zcl, sgt ]
##
## which contains the "local parameters" of the respective object. The
## following "global parameters" of the same object are used as local
## variables.
##
## q = Q-class number with respect to the list of all Q-classes of the
## current dimension,
## z = Z-class number with respect to the list of all Z-classes of the
## current dimension,
## t = space-group type with respect to the list of all space-group types
## of the current dimension,
## CR = catalogue record CrystGroupsCatalogue[dim] for the current
## dimension dim.
##
DeclareGlobalFunction("CR_Parameters");
#############################################################################
##
#F CR_PcGroupQClass(<CR parameter list>,<warning>)
##
## 'CR_PcGroupQClass' returns a pc group isomorphic to the groups in the
## specified Q-class. If <warning> = true, then a warning will be displayed
## in case that the given presentation is not a prime order pcgs.
##
## This is an internal function which does not check the arguments. The
## corresponding public function is 'PcGroupQClass'.
##
DeclareGlobalFunction("CR_PcGroupQClass");
#############################################################################
##
#F CR_SpaceGroup( <CR parameter list> )
##
## 'CR_SpaceGroup' returns a representative matrix group (of dimension
## dim+1) of the specified space-group type.
##
## This is an internal function which does not check the arguments. The
## corresponding public function is 'SpaceGroup'.
##
## In particular, the function expects that, whenever the order of the point
## group is not a multiple of 60, the given point group presentation is a
## polycyclic power commutator presentation containing a list of n power
## relators and n*(n-1)/2 commutator relators in some prescribed order,
## where n is the number of its generators.
##
DeclareGlobalFunction("CR_SpaceGroup");
#############################################################################
##
#F CR_ZClassRepsDadeGroup( <CR parameter list>, <d> )
##
## 'CR_ZClassRepsDadeGroup' returns a list of representatives of those
## conjugacy classes of subgroups of the given Dade group which consist of
## groups belonging to the given Z-class.
##
## This is an internal function which does not check the arguments. The
## corresponding public function is 'ZClassRepsDadeGroup'.
##
DeclareGlobalFunction("CR_ZClassRepsDadeGroup");
#############################################################################
##
#F CharTableQClass( <dim>, <system>, <qclass> )
#F CharTableQClass( <dim>, <IT number> )
#F CharTableQClass( <Hermann-Mauguin symbol> )
##
## 'CharTableQClass' returns the character table of a representative group
## of the specified Q-class.
##
DeclareGlobalFunction("CharTableQClass");
#############################################################################
##
#F DadeGroup( <dim>, <n> )
##
## 'DadeGroup' returns the n-th Dade group of dimension dim.
##
DeclareGlobalFunction("DadeGroup");
#############################################################################
##
#F DadeGroupNumbersZClass( <dim>, <system>, <qclass>, <zclass> )
#F DadeGroupNumbersZClass( <dim>, <IT number> )
#F DadeGroupNumbersZClass( <Hermann-Mauguin symbol> )
##
## 'DadeGroupNumbersZClass' returns a list of the numbers of those Dade
## groups which contain groups from the given Z-class.
##
DeclareGlobalFunction("DadeGroupNumbersZClass");
#############################################################################
##
#F DisplayCrystalFamily( <dim>, <family> )
##
## 'DisplayCrystalFamily' displays for the specified crystal family the
## following information:
## - the family name,
## - the number of parameters,
## - the common rational decomposition pattern,
## - the common real decomposition pattern,
## - the number of crystal systems in the family, and
## - the number of Bravais flocks in the family.
##
DeclareGlobalFunction("DisplayCrystalFamily");
#############################################################################
##
#F DisplayCrystalSystem( <dim>, <system> )
##
## 'DisplayCrystalSystem' displays for the specified crystal system the
## following information:
## - the number of Q-classes in the crystal system, and
## - the triple (dim, sys, qcl) of parameters of the Q-class which is the
## holohedry of the crystal system.
##
DeclareGlobalFunction("DisplayCrystalSystem");
#############################################################################
##
#F DisplayQClass( <dim>, <system>, <qclass> )
#F DisplayQClass( <dim>, <IT number> )
#F DisplayQClass( <Hermann-Mauguin symbol> )
##
## 'DisplayQClass' displays for the specified Q-class the following
## information:
## - the size of the groups in the Q-class,
## - the isomorphism type of the groups in the Q-class,
## - the Hurley pattern,
## - the rational constituents,
## - the number of Z-classes in the Q-class, and
## - the number of space-group types in the Q-class.
##
DeclareGlobalFunction("DisplayQClass");
#############################################################################
##
#F DisplaySpaceGroupGenerators( <dim>, <system>,<qclass>,<zclass>,<sgtype> )
#F DisplaySpaceGroupGenerators( <dim>, <IT number> )
#F DisplaySpaceGroupGenerators( <Hermann-Mauguin symbol> )
##
## 'DisplaySpaceGroupGenerators' displays the non-translation generators of
## the space group specified by the given parameters.
##
DeclareGlobalFunction("DisplaySpaceGroupGenerators");
#############################################################################
##
#F DisplaySpaceGroupType( <dim>, <system>, <qclass>, <zclass>, <sgtype> )
#F DisplaySpaceGroupType( <dim>, <IT number> )
#F DisplaySpaceGroupType( <Hermann-Mauguin symbol> )
##
## 'DisplaySpaceGroupType' displays for the specified space-group type the
## following information:
## - the orbit size associated with the space-group type,
## - the IT number (only in case dim = 2 or dim = 3), and
## - the Hermann-Mauguin symbol (only in case dim = 2 or dim = 3).
##
DeclareGlobalFunction("DisplaySpaceGroupType");
#############################################################################
##
#F DisplayZClass( <dim>, <system>, <qclass>, <zclass> )
#F DisplayZClass( <dim>, <IT number> )
#F DisplayZClass( <Hermann-Mauguin symbol> )
##
## 'DisplayZClass' displays for the specified Z-class the following
## information:
## - the Hermann-Mauguin symbol of a representative space-group type which
## belongs to the Z-class (only in case dim = 2 or dim = 3),
## - the Bravais type,
## - some decomposability information,
## - the number of space-group types belonging to the Z-class, and
## - the size of the associated cohomology group.
##
DeclareGlobalFunction("DisplayZClass");
#############################################################################
##
#F FpGroupQClass( <dim>, <system>, <qclass> )
#F FpGroupQClass( <dim>, <IT number> )
#F FpGroupQClass( <Hermann-Mauguin symbol> )
##
## 'FpGroupQClass' returns a f. p. group isomorphic to the groups in the
## specified Q-class.
##
DeclareGlobalFunction("FpGroupQClass");
#############################################################################
##
#F MatGroupZClass( <dim>, <system>, <qclass>, <zclass> )
#F MatGroupZClass( <dim>, <IT number> )
#F MatGroupZClass( <Hermann-Mauguin symbol> )
##
## 'MatGroupZClass' returns a representative group of the specified Z-class.
## The generators of the resulting matrix group are chosen such that they
## satisfy the defining relators which are returned by the
## 'FpGroupQClass' function for the representative of the corresponding
## Q-class.
##
DeclareGlobalFunction("MatGroupZClass");
#############################################################################
##
#F NormalizerZClass( <dim>, <system>, <qclass>, <zclass> )
#F NormalizerZClass( <dim>, <IT number> )
#F NormalizerZClass( <Hermann-Mauguin symbol> )
##
## 'NormalizerZClass' returns the normalizer in GL(dim,Z) of the specified
## Z-class representative matrix group. If the order of the normalizer is
## finite, then the group record components "crZClass" and "crConjugator"
## will be set properly.
##
DeclareGlobalFunction("NormalizerZClass");
#############################################################################
##
#F NrCrystalFamilies( <dim> )
##
## 'NrCrystalFamilies' returns the number of crystal families of the given
## dimension.
##
DeclareGlobalFunction("NrCrystalFamilies");
#############################################################################
##
#F NrCrystalSystems( <dim> )
##
## 'NrCrystalSystems' returns the number of crystal systems of the given
## dimension.
##
DeclareGlobalFunction("NrCrystalSystems");
#############################################################################
##
#F NrDadeGroups( <dim> )
##
## 'NrDadeGroups' returns the number of Dade groups of the given dimension.
##
DeclareGlobalFunction("NrDadeGroups");
#############################################################################
##
#F NrQClassesCrystalSystem( <dim>, <system> )
##
## 'NrQClassesCrystalSystem' returns the number of Q-classes in the given
## crystal system.
##
DeclareGlobalFunction("NrQClassesCrystalSystem");
#############################################################################
##
#F NrSpaceGroupTypesZClass( <dim>, <system>, <qclass>, <zclass> )
#F NrSpaceGroupTypesZClass( <dim>, <IT number> )
#F NrSpaceGroupTypesZClass( <Hermann-Mauguin symbol> )
##
## 'NrSpaceGroupTypesZClass' returns the number of space-group types in the
## given Z-class.
##
DeclareGlobalFunction("NrSpaceGroupTypesZClass");
#############################################################################
##
#F NrZClassesQClass( <dim>, <system>, <qclass> )
#F NrZClassesQClass( <dim>, <IT number> )
#F NrZClassesQClass( <Hermann-Mauguin symbol> )
##
## 'NrZClassesQClass' returns the number of Z-classes in the given Q-class.
##
DeclareGlobalFunction("NrZClassesQClass");
#############################################################################
##
#F PcGroupQClass( <dim>, <system>, <qclass> )
#F PcGroupQClass( <dim>, <IT number> )
#F PcGroupQClass( <Hermann-Mauguin symbol> )
##
## 'PcGroupQClass' returns an ag group isomorphic to the groups in the
## specified Q-class.
##
DeclareGlobalFunction("PcGroupQClass");
#############################################################################
##
#F SpaceGroupOnLeftBBNWZ( <dim>, <system>, <qclass>, <zclass>, <sgtype> )
#F SpaceGroupOnLeftBBNWZ( <dim>, <IT number> )
#F SpaceGroupOnLeftBBNWZ( <Hermann-Mauguin symbol> )
##
## 'SpaceGroupOnLeftBBNWZ' returns a representative matrix group
## (of dimension dim+1) of the specified space-group type.
##
DeclareGlobalFunction("SpaceGroupOnLeftBBNWZ");
#############################################################################
##
#F SpaceGroupOnRightBBNWZ( <dim>, <system>, <qclass>, <zclass>, <sgtype> )
#F SpaceGroupOnRightBBNWZ( <dim>, <IT number> )
#F SpaceGroupOnRightBBNWZ( <Hermann-Mauguin symbol> )
#F SpaceGroupOnRightBBNWZ( S )
##
## 'SpaceGroupOnRightBBNWZ' returns the transposed matrix group of
## the given or specified space group.
##
## The reason is the following. Each space group is presented in the form of
## a group of augmented matrices (of dimension dim+1) of the following form:
##
## [ M | t ]
## [-----+---] Here, M is the `linear part' and
## [ 0 | 1 ] t is the `translational part'.
##
## Therefore, the natural action of a space group in this form is from the
## left. This collides with the convention in GAP to have all actions from
## the right. This function does the necessary conversions. In fact, it does
## not only transpose the matrices, but it also adapts the relators given in
## CrystCatRecord(S).fpGroup to the new generators.
##
DeclareGlobalFunction("SpaceGroupOnRightBBNWZ");
#############################################################################
##
#F SpaceGroupBBNWZ( <dim>, <system>, <qclass>, <zclass>, <sgtype> ) .
#F SpaceGroupBBNWZ( <dim>, <IT number> ) . . . . . . . . . . . . . .
#F SpaceGroupBBNWZ( <Hermann-Mauguin symbol> ) . . . . . . . . . . .
##
## Calls either `SpaceGroupOnRightBBNWZ' or `SpaceGroupOnLeftBBNWZ'
## depending on the value of `CrystGroupDefaultAction'
##
DeclareGlobalFunction( "SpaceGroupBBNWZ" );
#############################################################################
##
#F ZClassRepsDadeGroup( <dim>, <system>, <qclass>, <zclass>, <n> )
#F ZClassRepsDadeGroup( <dim>, <IT number>, <n> )
#F ZClassRepsDadeGroup( <Hermann-Mauguin symbol>, <n> )
##
## 'ZClassRepsDadeGroup' returns a list of representatives of those
## conjugacy classes of subgroups of the given Dade group which consist of
## groups belonging to the given Z-class.
##
DeclareGlobalFunction("ZClassRepsDadeGroup");
#############################################################################
##
#F FpGroupSpaceGroupBBNWZ( <S> ) . . FpGroup isomorphic to BBNWZ space group
##
DeclareGlobalFunction( "FpGroupSpaceGroupBBNWZ" );
#############################################################################
##
#E cryst.gd . . . . . . . . . . . . . . . . . . . . . . . . . . . ends here
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