Quelle aclib.tex Sprache: Latech

\Chapter{The catalog of almost crystallographic groups}

This chapter introduces the access functions to the catalog of
3- and 4-dimensional crystallographic groups. This catalog is an
electronic version of the classification obtained in \cite{KD}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Rational matrix groups}

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 \cite{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 \cite{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"', $\ldots$, `"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.

\beginexample
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
\endexample

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Polycyclically presented groups}

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 {\sf 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 group-theoretic
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.

\beginexample
gap> G := AlmostCrystallographicPcpGroup( 4, 50, [ 1, -4, 1, 2 ] );
Pcp-group with orders [ 4, 0, 0, 0, 0 ]
gap> Cgs(G);
[ g1, g2, g3, g4, g5 ]

gap> F := FittingSubgroup( G );
Pcp-group with orders [ 0, 0, 0, 0 ]
gap> Centre(F);
Pcp-group with orders [ 0, 0 ]
gap> LowerCentralSeries(F);
[ Pcp-group with orders [ 0, 0, 0, 0 ], Pcp-group with orders [ 0 ],
  Pcp-group with orders [  ] ]
gap> UpperCentralSeries(F);
[ Pcp-group with orders [ 0, 0, 0, 0 ], Pcp-group with orders [ 0, 0 ],
  Pcp-group with orders [  ] ]
gap> MinimalGeneratingSet(F);
[ g2, g3, g4 ]

gap> H := HolonomyGroup( G );
Pcp-group 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] );
Pcp-group with orders [ 2 ]
gap> PreImage( hom, U );
Pcp-group with orders [ 2, 0, 0, 0, 0 ]
\endexample

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{More about the type and the defining parameters}

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.

\beginexample
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 ] )
\endexample

\beginexample
gap> G := AlmostCrystallographicPcpGroup( 4, 70, false );
Pcp-group 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 ] )
\endexample

We consider the types of almost crystallographic groups in more detail. The
almost crystallographic groups in dimensions 3 and 4 fall into three families

\beginlist
\item{(1)} 3-dimensional almost crystallographic groups.
\item{(2)} 4-dimensional almost crystallographic groups with a
           Fitting subgroup of class 2.
\item{(3)} 4-dimensional almost crystallographic groups with a
           Fitting subgroup of class 3.
\endlist

These families are split up further into subfamilies in \cite{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 3-dimensional almost
crystallographic groups the type is a string representing the numbers from
1 to 17, i.e. the available types are `"01"', `"02"', $\ldots$, `"17"'.

For the 4-dimensional 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 \cite{KD} is used. So possible types are
`"001"', `"002"', $\ldots$. 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 \cite{KD}. Moreover, for
certain numbers there is more than one family of groups listed in \cite{KD}.
For example, the 3 families of groups corresponding to number 19 on pages
179-180 of \cite{KD} have types `"019"', `"019b"' and `"019c"' (the order is
the one given in \cite{KD}).

For the last category of groups, the 4-dimensional 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 \cite{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 higher-level functions
introduced above to obtain almost crystallographic groups from the
library. In particular, these low-level access functions return matrix
or pcp groups, but the almost crystallographic info flags will not be
attached to them.

\beginexample
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);
Pcp-group with orders [ 0, 0, 0 ]
\endexample

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{The electronic versus the printed library}

The package `aclib' can be considered as the electronic version of
Chapter 7 of \cite{KD}. In this section we outline the relationship
between the library presented in this manual and the printed version
in \cite{KD}. First we consider an example. At page 175 of \cite{KD},
we find the following groups in the table starting with entry ``13''.

\medskip
13. $Q=P2/c$
$$
\matrix{ E:\;\langle a,b,c,d,\alpha,\beta\;\|\; &  {\,[b,a]=1}\hskip 1.61cm
       {[d,a]=1}
\hfill  & \rangle \cr
               &    \matrix{ {[c,a]=d^{2 k_1}}\hfill  & {[d,b]=1}\hfill \cr
                    {[c,b]=1}\hfill  & {[d,c]=1}\hfill \cr
\alpha a=a^{-1}\alpha d^{k_{2}}\hfill   & \alpha^2=d^{k_3}\hfill \cr
\alpha b=b\alpha \hfill  & \alpha d= d \alpha\hfill \cr
\alpha c=c^{-1}\alpha d^{-2 k_6}\hfill   & \cr
\beta a=a^{-1}\beta d^{k_1+k_2} \hfill  & \beta^2=d^{k_5}\hfill \cr
\beta b=b^{-1}\beta d^{k_4}\hfill  & \beta d= d \beta\hfill \cr
\beta c=c^{-1}\beta d^{-2 k_6}\hfill  & \alpha \beta=c\beta\alpha d^{k_6}
\hfill } & }
$$

$$\lambda(\alpha)=\left(\matrix{
1& \frac{k_1}{2}+k_2 & 0 & -2 k_6 & \frac{k_3}{2}+\frac{k_6}{2} \cr
0& -1 & 0 & 0 & 0 \cr
0 & 0& 1  &  0 & 0 \cr
0 & 0 & 0 & -1 & \frac12\cr
0 & 0 & 0 & 0 & 1
}\right)
\;\;\lambda(\beta)=\left(\matrix{
1& k_1+k_2 & k_4 & -2 k_6 & \frac{k_5}{2} \cr
0& -1 & 0 & 0 & 0 \cr
0 & 0& -1  &  0 & 0 \cr
0 & 0 & 0 & -1 & 0\cr
0 & 0 & 0 & 0 & 1
    }\right)
$$

$$H^2(Q,\Z{})=\Z{}\oplus(\Z{}_2)^4=\Z^{6}/A,$$
$$A=\{(k_1,\ldots,k_6)\|
k_1=0,\;k_2,\ldots, k_5\in 2\Z{},\;k_6\in\Z\}$$
AB-groups:

$\forall k>0,\;k\equiv 0\bmod 2,\;<(k,0,1,0,1,0)>$

\medskip

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 $k_1, k_2, \ldots, k_6$ and these
are the <parameters> list in this library. The rational matrix
representation in \GAP\ corresponds exactly to the printed version in
\cite{KD} where it is named $\lambda$.  In the example below, we consider
the group with parameters $(k_1,k_2,k_3,k_4,k_5,k_6)=(8,0,1,0,1,0)$.

\beginexample
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 ] ]
\endexample

For a 4-dimensional almost crystallographic group the matrix group is
built up such that $\{ a, b, c, d, \alpha, \beta, \gamma \}$ as described
in \cite{KD} forms the defining generating set of $G$. For certain types
the elements $\alpha$, $\beta$ or $\gamma$ may not be present.
Similarly, for a 3-dimensional group we have the generating set $\{ a, b,
c, \alpha, \beta \}$ and $\alpha$ and $\beta$ may be absent.

\bigskip

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 $(\gamma, \beta,
\alpha, a, b, c, d)$ 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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