This chapter introduces the access functions to the catalog of 3 and 4dimensional crystallographic groups. This catalog is an electronic version of the classification obtained in KD.
The following three main functions are available to access the library of almost crystallographic groups as rational matrix groups.
AlmostCrystallographicGroup(
dim,
type,
parameters )
AlmostCrystallographicDim3(
type,
parameters )
AlmostCrystallographicDim4(
type,
parameters )
dim is the dimension of the required group. Thus dim must be either 3 or 4. The inputs type and parameters are used to define the desired group as described in KD. We outline the possible choices for type and parameters here briefly. A more extended description is given later in Section More about almost crystallographic groups or can be obtained from KD.
type specifies the type of the required group. There are 17 types
in dimension 3 and 95 types in dimension 4. The input type can either
be an integer defining the position of the desired type among all types;
that is, in this case type is a number in [1..17] in dimension 3 or a
number in [1..95] in dimension 4. Alternatively, type can be a string
defining the desired type. In dimension 3 the possible strings are
"01"
, "02"
, …, "17"
. In dimension 4 the possible strings
are listed in the list ACDim4Types
and thus can be accessed from GAP.
parameters is a list of integers. Its length depends on the type of
the chosen group. The lists ACDim3Param
and ACDim4Param
contain
at position i the length of the parameter list for the type number i.
Every list of integers of this length is a valid parameter input.
Alternatively, one can input false
instead of a parameter list. Then
GAP will chose a random parameter list of suitable length.
gap> G := AlmostCrystallographicGroup( 4, 50, [ 1, 4, 1, 2 ] ); <matrix group of size infinity with 5 generators> gap> DimensionOfMatrixGroup( G ); 5 gap> FieldOfMatrixGroup( G ); Rationals gap> GeneratorsOfGroup( G ); [ [ [ 1, 0, 1/2, 0, 0 ], [ 0, 1, 0, 0, 1 ], [ 0, 0, 1, 0, 0 ], [ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ] ], [ [ 1, 1/2, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 1 ], [ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ] ], [ [ 1, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ], [ 0, 0, 0, 1, 1 ], [ 0, 0, 0, 0, 1 ] ], [ [ 1, 0, 0, 0, 1 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ], [ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ] ], [ [ 1, 4, 1, 0, 1/2 ], [ 0, 0, 1, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 0, 1, 1/4 ], [ 0, 0, 0, 0, 1 ] ] ] gap> G.1; [ [ 1, 0, 1/2, 0, 0 ], [ 0, 1, 0, 0, 1 ], [ 0, 0, 1, 0, 0 ], [ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ] ] gap> ACDim4Types[50]; "076" gap> ACDim4Param[50]; 4
All the almost crystallographic groups considered in this package are polycyclic. Hence they have a polycyclic presentation and this can be used to facilitate efficient computations with the groups. To obtain the polycyclic presentation of an almost crystallographic group we supply the following functions. Note that the share package Polycyclic must be installed to use these functions.
AlmostCrystallographicPcpGroup(
dim,
type,
parameters )
AlmostCrystallographicPcpDim3(
type,
parameters )
AlmostCrystallographicPcpDim4(
type,
parameters )
The input is the same as for the corresponding matrix group functions. The output is a pcp group isomorphic to the corresponding matrix group. An explicit isomorphism from an almost crystallographic matrix group to the corresponding pcp group can be obtained by the following function.
IsomorphismPcpGroup(
G )
We can use the polycyclic presentations of almost crystallographic groups to exhibit structure information on these groups. For example, we can determine their Fitting subgroup and ask grouptheoretic questions about this nilpotent group. The factor G / Fit(G) of an almost crystallographic group G is called holonomy group. We provide access to this factor of a pcp group via the following functions. Let G be an almost crystallographic pcp group.
HolonomyGroup(
G )
NaturalHomomorphismOnHolonomyGroup(
G )
The following example shows applications of these functions.
gap> G := AlmostCrystallographicPcpGroup( 4, 50, [ 1, 4, 1, 2 ] ); Pcpgroup with orders [ 4, 0, 0, 0, 0 ] gap> Cgs(G); [ g1, g2, g3, g4, g5 ] gap> F := FittingSubgroup( G ); Pcpgroup with orders [ 0, 0, 0, 0 ] gap> Centre(F); Pcpgroup with orders [ 0, 0 ] gap> LowerCentralSeries(F); [ Pcpgroup with orders [ 0, 0, 0, 0 ], Pcpgroup with orders [ 0 ], Pcpgroup with orders [ ] ] gap> UpperCentralSeries(F); [ Pcpgroup with orders [ 0, 0, 0, 0 ], Pcpgroup with orders [ 0, 0 ], Pcpgroup with orders [ ] ] gap> MinimalGeneratingSet(F); [ g2, g3, g4 ] gap> H := HolonomyGroup( G ); Pcpgroup with orders [ 4 ] gap> hom := NaturalHomomorphismOnHolonomyGroup( G ); [ g1, g2, g3, g4, g5 ] > [ g1, identity, identity, identity, identity ] gap> U := Subgroup( H, [Pcp(H)[1]^2] ); Pcpgroup with orders [ 2 ] gap> PreImage( hom, U ); Pcpgroup with orders [ 2, 0, 0, 0, 0 ]
Each group from this library knows that it is almost crystallographic and, additionally, it knows its type and defining parameters.
AlmostCrystallographicInfo(
G )
This attribute is set for groups from the library only. It is not possible at current to determine the type and the defining parameters for an arbitrary almost crystallographic groups which is not defined by the library access functions.
gap> G := AlmostCrystallographicGroup( 4, 70, false ); <matrix group of size infinity with 5 generators> gap> IsAlmostCrystallographic(G); true gap> AlmostCrystallographicInfo(G); rec( dim := 4, type := 70, param := [ 1, 4, 1, 2, 3 ] )
gap> G := AlmostCrystallographicPcpGroup( 4, 70, false ); Pcpgroup with orders [ 6, 0, 0, 0, 0 ] gap> IsAlmostCrystallographic(G); true gap> AlmostCrystallographicInfo(G); rec( dim := 4, type := 70, param := [ 3, 2, 5, 1, 0 ] )
We consider the types of almost crystallographic groups in more detail. The almost crystallographic groups in dimensions 3 and 4 fall into three families
These families are split up further into subfamilies in KD and to
each subfamily is assigned a type; that is, a string which is used to
identify the subfamily. As mentioned above, for the 3dimensional almost
crystallographic groups the type is a string representing the numbers from
1 to 17, i.e. the available types are "01"
, "02"
, …, "17"
.
For the 4dimensional almost crystallographic groups with a Fitting subgroup
of class 2 the type is a string of 3 or 4 characters. In general, a string of
3 characters representing
the number of the table entry in KD is used. So possible types are
"001"
, "002"
, …. The reader is warned however that not all
possible numbers are used, e.g. there are no groups of type "016"
. Also,
the types do not appear in their natural order in KD. Moreover, for
certain numbers there is more than one family of groups listed in KD.
For example, the 3 families of groups corresponding to number 19 on pages
179180 of KD have types "019"
, "019b"
and "019c"
(the order is
the one given in KD).
For the last category of groups, the 4dimensional almost crystallographic
groups with a Fitting subgroup of class 3, the type is a string of 2 or 3
characters, where the first character is always the letter "B"
. This "B"
is followed by the number of the table entry as found in KD,
eventually followed by a "b"
or "c"
as in the previous case.
For each type of almost crystallographic group contained in the library
there exists a function taking a parameter list as input and returning
the desired matrix or pcp group. These functions can be accessed
from GAP using the lists ACDim3Funcs
, ACDim4Funcs
, ACPcpDim3Funcs
and ACPcpDim4Funcs
which consist of the corresponding functions.
Although we include these direct access functions here for completeness, we note that the user should in general use the higherlevel functions introduced above to obtain almost crystallographic groups from the library. In particular, these lowlevel access functions return matrix or pcp groups, but the almost crystallographic info flags will not be attached to them.
gap> ACDim3Funcs[15]; function( k1, k2, k3, k4 ) ... end gap> ACDim3Funcs[15](1,1,1,1); <matrix group with 5 generators> gap> ACPcpDim3Funcs[1](1); Pcpgroup with orders [ 0, 0, 0 ]
The package aclib
can be considered as the electronic version of
Chapter 7 of KD. In this section we outline the relationship
between the library presented in this manual and the printed version
in KD. First we consider an example. At page 175 of KD,
we find the following groups in the table starting with entry ``13''.
13. Q=P2/c




∀k > 0, k ≡ 0 mod 2, (k,0,1,0,1,0)
The number ``13'' at the beginning of this entry is the type of the almost crystallographic group in this library. This family of groups with type 13 depends on 6 parameters k_{1}, k_{2}, …, k_{6} and these are the parameters list in this library. The rational matrix representation in GAP corresponds exactly to the printed version in KD where it is named λ. In the example below, we consider the group with parameters (k_{1},k_{2},k_{3},k_{4},k_{5},k_{6})=(8,0,1,0,1,0).
gap> G:=AlmostCrystallographicDim4("013",[8,0,1,0,1,0]); <matrix group with 6 generators> gap> G.5; [ [ 1, 4, 0, 0, 1/2 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ], [ 0, 0, 0, 1, 1/2 ], [ 0, 0, 0, 0, 1 ] ] gap> G.6; [ [ 1, 8, 0, 0, 1/2 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ], [ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ] ]
For a 4dimensional almost crystallographic group the matrix group is built up such that { a, b, c, d, α, β, γ} as described in KD forms the defining generating set of G. For certain types the elements α, β or γ may not be present. Similarly, for a 3dimensional group we have the generating set { a, b, c, α, β} and α and β may be absent.
To obtain a polycyclic generating sequence from the defining generators of the matrix group we have to order the elements in the generating set suitably. For this purpose we take the subsequence of (γ, β, α, a, b, c, d) of those generators which are present in the defining generating set of the matrix group. This new ordering of the generators is then used to define a polycyclic presentation of the given almost crystallographic group.
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