GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it
1. INTRODUCTION TO CARAT
========================
CARAT is a compilation of various small programs written in C, which
can solve certain problems in crystallography. It is distributed via
Lehrstuhl B fuer Mathematik
RWTH-Aachen
Prof. Plesken
Templergraben 64
52064 Aachen
Germany
email: [email protected]
Every program should give some online-help if used with the option -h.
1.1 Standalones
===============
There are three categories of programs in CARAT, regarding their importance.
1.1.1 Most frequently used programs
===================================
Here a short list of the most important executables is given. For a
description in some more detail (syntax, option), please call them
with option -h.
Program/Synonyms Short description
================ =================
Aut_grp Calculates the automorphism group of
one or more quadratic forms.
Bravais_catalog/Datei Provides a list of all Bravais groups up
to degree 6.
Bravais_grp Calculates the Bravais group of a finite
unimodular group
Bravais_inclusions Outputs Bravais subgroups/supergroups for a
given Bravais group.
Bravais_type/Symbol Calculates the family symbol of a finite
unimodular group. Also calculates an equivalent
group in the catalog of Bravais groups.
Note that Bravais_type is nothing else then
Symbol -i
Extensions/Vector_systems Calculates all non-isomorphic extensions of
a finite unimodular group with a given lattice.
Form_space Calculates the space of invariant forms of
a unimodular group.
Graph Calculates the "graph of inclusions" for a
given geometric class.
Is_finite Decides finiteness of a given subgroup of
GL_n(Z). Calculates the order in case the group
is finite.
KSubgroups Calculates the maximal klassengleich subgroups
of a spacegroup for some prime-power index.
KSupergroups Calculates the maximal klassengleich
supergroups of a spacegroup for some
prime-power index.
Name Give a space group a name, ie. calculate
a string which describes the isomorphism
type uniquely, cf. Reverse_name.
Normalizer Calculates the Normalizer in GL_n(Z) of a given
finite unimodular group.
Orbit Fairly general implementation of the
orbit/stabilizer algorithm.
Order Calculates the order of a given finite subgroup
of GL_n(Q).
Q_catalog Provides a list of all Q_classes up
to degree 6.
QtoZ Splits a Q-class into Z-classes.
Reverse_name Constructs a space group with given name,
and check whether the name is valid, cf.
Name.
Same_generators Transforms the generators of a space group
to a prescribed linear part.
Torsionfree Decides whether a given space group is
torsion free. WARNING: The program assumes
the translation subgroup to be Z^n.
TSubgroups Calculates the maximal translationengleich
subgroups of a space group.
TSupergroups Calculates the minimal translationengleich
supergroups of a space group.
Z_equiv Decides whether two given finite unimodular
groups are conjugated in GL_n(Z).
1.1.2 Less frequently used programs
===================================
We continue with given the name of some additional functions which the
user might find useful.
Program/Synonyms Short description
================ =================
Bravais_equiv Decides whether the Bravais groups of two
given finite unimodular groups are conjugated
Conj_bravais Conjugates a Bravais group with a given matrix
Extract Standard_affine_form Tools to get from space groups to point groups
and vice versa. Note that Standard_affine_form
is just Extract -t
Idem Calculates (rational) central primitive
idempotents of the enveloping algebra of a
given matrix group.
Invar_space See Form_space. Is much faster than this,
but uses some random methods.
Isometry Calculates an isometry of with respect to
tuples of bilinear forms.
Long_solve Solves linear systems of equations using
multiple precision integers.
Mink_red The Minkowski reduction of bilinear forms.
Gives very good results, but use Pair_red
before.
Pair_red Pair reduction of bilinear forms. Very fast.
Presentation Calculates a presentation of a finite soluble
subgroup of GL_n(Z)
Red_gen Tries to reduce the number of elements of
a generating set of a finite matrix group.
Rein Purifies a lattice.
Rform Mostly used for finding a positive definite
G-invariant form or a finite unimodular group
G.
Scpr Calculates scalar products w.r.t a given form.
Short Calculates short vectors of a given positive
definite symmetric form.
Shortest Shortest vectors of a given positive definite
symmetric quadratic form.
Signature Sylvester type of a quadratic form. In
particular it decides whether a given form
is positive definite.
Sublattices ZZprog Find G-invariant sublattices of Z^n. Note
that this is a dualisation of finding
centerings.
Tr_bravais Transposes a finite unimodular group.
Zass_main Calculates H^1(G,Q^n/Z^n) for a given finite
unimodular group.
1.1.3 Programs seldom used and those for debugging
==================================================
The remaining functions are merely of debugging and processing the results,
nevertheless an experienced user might calculate relevant data with them.
Program/Synonyms Short description
================ ================= Add Adds matrices
Con Conjugates matrices
Conjugated Decides whether two groups are conjugate
under third group.
Conv Converts CARAT input-file (matrix_TYP)
into GAP and Maple format.
Elt An elementary divisors algorithm.
First_perfect Find G-perfect forms.
Form_elt Elementary divisors of the trace bilinear form
of a finite unimodular group. Useful for
distinguishing Bravais groups.
Formtovec Writes a given form as linear combination
of others.
Full Outputs given matrices in a full form, which
might be easier to edit.
Gauss An implementation of Gauss's algorithm.
Inv Inverts matrices.
Kron Kronecker product of matrices.
Ltm Inverse to Mtl.
Minpol Minimal polynomial of integral matrices.
Modp Takes all entries of a matrix mod p a prime.
Mtl Writes matrices in lines.
Mul Multiplies matrices.
Normalizer_in_N Calculates the normalizer of a finite group
in a second one.
Normlin Calculates for each matrix A in file2 a matrix
X with the property that
\sum_j X_{i,j} F_j = A^{tr} F_j A
with F_j in 'file1'
P_lse_solve Solves a system of equations modularly.
Pdet Determinant of a matrix mod p.
Perfect_neighbours Gives the perfect neighbours of a given
G-perfect form.
Polyeder
Rest_short
Scalarmul Multiplies matrices with rational number.
Short_reduce
Simplify_mat Divides all entries of a matrix by their
greatest common divisor.
Tr Transposes matrices.
Trace Trace of matrices.
Trbifo Trace bilinear form of a finite unimodular
group.
Vectoform Calculates a linear combintion of forms.
Vor_vertices
1.2 Files for in/output
========================
In principle CARAT does know two different file formats in which the
in/output takes place. The first and most basic one is matrix_TYP, cf.
1.2.1 below and the second and most frequent one is bravais_TYP, cf.
1.2.2 below.
1.2.1 matrix_TYP
================
The format of a single matrix for CARAT is a preceding line
NxM % comment
telling the programs to read a matrix with N lines and M columns. Spaces,
tabs and so on are ignored, and so is everything behind % in the
same line.
Following this line the program will read N*M integers, which represent
the matrix ROW BY ROW, regardless of spaces, cr, tabs and so on.
Therefore all the following examples stand for the same matrix.
3x4 % most natural way to put it
1 2 3 4
5 6 7 8
9 10 11 12
3x4 % even this
1 2 3 4 5 6 7 8 9 10 11 12
3x4
1 2 3 4 5 6
7 8 9 10 11 12
Furthermore there are some abbreviations allowed, which deal with
square matrices and those having symmetries.
In the header line of a matrix N is equivalent to NxN. The following
examples describe the same matrix:
2x2
1 2
3 4
2
1 2
3 4
Again, formating characters are ignored. Coming to matrices which obey
symmetries CARAT follows the konvention that Nx0 means an symmetric N by
N matrix, of which program just will read the lower triangular.
Note that all the following examples have the same meaning:
2
1 2
2 1
2x0
1
2 1
2x0
1 2 1
The last abbreviation are meant for diagonal matrices, which are Nd1
for a N by N diagonal matrix, of which program will read N diagonal entries,
and Nd0 for a N by N scalar matrix, of which only the defining scalar is read.
Again a couple of outputs meaning the same thing should make it clear.
3x3
2 0 0
0 2 0
0 0 2
3d1
2 2 2
3d0
2
Most programs will read more than one matrix. Therefore a matrix_TYP
normaly constits of a preceding line of the form #A , where A is
the number of matrices to be read.
In the next example we give a matrix_TYP consisting of 2 matrices
(which generate a group isomorphic to S_4, the permutation group
on four letters).
#2
3 % presentation for a transposition
0 1 0
1 0 0
0 0 1
3 % presentation of a 4-cycle
0 1 0
0 0 1
-1 -1 -1
1.2.1.2 rational matrices
=========================
The way CARAT presents rational matrices is to divide the whole thing
by an integer:
3/2 % divide the whole matrix by 2
1 2 3
4 5 6
7 8 9
1.2.1.3 A matrix discribing a presentation
=========================================
This is a slight abuse of notation, but nevertheless a matrix_TYP in
CARAT can discribe a finitely presented group.
A single line of this matrix will present a relation fullfilled by the
generators of the group, and the biggest entry in modulus will be the
number of generators.
Words in the free group translate in the obvious way to a line
of a matrix, therefore we just give a couple of ways of presenting
the group V_4 = C_2 X C_2. To make the matrix rectangular, fill
the shorter rows with zeroes.
3x4 % we will need 3 relations, the longest of which will have 4 entries
1 1 0 0
2 2 0 0
1 2 -1 -2
The three lines read: x_1*x_1 = 1, x_2*x_2 = 1, x_1*x_2*x_1^(-1)*x_2^(-1) = 1.
Of course there are various ways to put it, like
3x4
1 1 0 0
2 2 0 0
1 2 1 2
or
3x4
1 1 2 2
1 1 0 0
1 2 1 2
1.2.2 bravais_TYP
=================
A bravais_TYP in CARAT is used to decribe a group generated by matrices
together with additional information like their normalizers and
a basis for the space of invariant forms.
The bravais_TYP consists of a header line, which tells the program how
many matrices to be read, and how to interpret them.
This header line takes the following form:
#gA fB ZC nD cE % just a comment
where A, B, C, D and E are natural numbers. It advises the program to read
A + B + C + D + E matrices, where A matrices are meant to generate the group,
the next B matrices form an integral basis of the space of fixed forms,
followed by C matrices giving so called "centerings". The program proceeds
in reading D matrices which generate the normalizer of the group (modulo
the group generated by the group and its centralizer), and E matrices
which generate the centralizer of the group.
Note: It is possible to ommit any of the records which discribe generators,
the space of forms and so on, but it is NOT possible to switch components.
The next example gives a bravais_TYP generated by the matrices given in
1.2.1:
#g2 f1 n3 % group with complete normalizer
3 % generator
0 1 0
1 0 0
0 0 1
3 % generator
0 1 0
0 0 1
-1 -1 -1
3x0 % invariant form
2
1 2
1 1 2
3 % generator of normalizer
1 1 1
0 -1 0
-1 0 0
3 % generator of normalizer
-1 0 0
0 0 -1
1 1 1
3x0 % generator of normalizer
1
0 0
0 1 0
2^3 * 3^1 = 24 % order of the group
Note that the order of the group is given at the end, and that it
is factorized.
(NB: This is the output of Normalizer if run on a file just containing the
matrices of 1.2.1)
As the reader already saw, the bravais_TYP closes with a tail line which
states the order of the group. The only programs using this lines are the
programs Sublattices (or ZZprog), QtoZ Bravais_inclusions and Symbol
(or Bravais_type) if used with the option -B. These programs assume the
order given to be right.
2. BUGS
=======
If you find any Bug in CARAT, we are please to here from you. Please
send us a copy of the file you produced the error with, and a log
from the things you did with it.
A short explanation why you encounter the result (if you got any) to
be wrong would be helpful as well.
Please send it to: [email protected]