ATLAS: Format of the representations
As a general rule, matrices over finite fields and permutations are stored in
three formats, namely MeatAxe,
MeatAxe binary and
GAP. Some other information, such as
presentations and representations in characteristic 0, is given in a variety
of formats, mainly Magma. The programs
that produce words for subgroups, conjugacy classes, etc., are shell scripts
that are designed to run in conjunction with the
MeatAxe [but you will have to write a
number of little shell scripts so that lines such as mu 1 3 4 are
interpreted as zmu z1 z3 z4, but this shouldn't be too much
trouble]. Eventually, we hope to provide automatic translation between the
- Each file starts with a header line which contains some
information about the content of the file. This can be ignored if you are
not using the Meataxe.
- A permutation p is stored in image format, which means
that the file consists of a list of the images p(1), p(2), ... of
the points 1, 2, ...
- A matrix is written in the usual way, reading across
each row in turn. There are line breaks at the end of each row, and after each
80 characters (usually) in each row. There may or may not be space(s) between
the individual entries.
- The entries in a matrix come from a finite field,
of order pn say. If this is a prime field, then
the elements are just the integers 0, 1, 2, ...p-1.
- If the field is not prime, then the labelling of the elements is
done by reference to the so-called Conway polynomials
Among other things, these polynomials are irreducible of degree n,
so by taking the polynomial ring modulo Cn(X)
we obtain the field of order pn. A polynomial
an-1Xn-1 + ... is then stored as the
non-negative integer an-1pn-1 + ... .
These are integer matrices intended for reduction modulo p where
p is a prime. So replace the p in the header line [and an
appropriate number of spaces to the left of it] by your desired prime. The last
digit of your prime should be the 8th character of the header line. [The header
line is supposedly in free format, but this doesn't seem to be the case for our
Meataxe binaries - for advanced users only
Most representations are also stored as Meataxe
binaries. They are almost always significantly smaller than the
ASCII files, but as they may be machine and meataxeversion
dependent, we cannot guarantee that you will be able to read them. We [mostly]
used the small field version of MeatAxe 1.5, and the conversions were [usually]
carried out on a SUN or ALPHA workstation. As a result, there are inconsistencies
with `endianism', which can be overcome by using the Meataxe program `conv'.
[Since we used the small field version of the
MeatAxe, there are no representations over finite fields of order greater than
256 in this ATLAS.]
NB: If you view the binaries in Netscape for example, it will often appear that
there is nothing there. However, there is something there, and the usual `save
as' button on the browser will [apparently] save the binary properly.
Most of the GAP representations are only
in two files since they were made with the MeatAxe using zpr -g. Of
course, both matrices/permutations can be placed in one file, but do remember
to change the identifiers at the same time [otherwise all the matrices will
still be called matrix and the permutations will still be called
bin1 or z1]. In the matrix representations, Z(q)
denotes the primitive element of GF(q) as defined by its Conway
polynomial. One quirk of GAP is that
matrices over prime fields are not what they initially appear to be because
they have a *Z(p) at the end of them. Thus
[[3,0],[0,3]]*Z(5) denotes the 2 × 2 identity matrix over GF(5).
These files can be loaded directly into
Magma. We have usually called the
groups G and the generators of the groups x and
y. The group generators will also [usually] be given an identifier
which corresponds to the names of the standard generators as given on the
HTML page. We have called our fields F, and our notation for the algebraic irrationalities of those fields follows the notation as given in
the ATLAS, with the exception that we have used w
instead of z3 to denote a primitive cube root of unity. For
representations in characteristic 0, we intend to give generators for the
vector spaces of all symmetric, antisymmetric and Hermitian forms preserved
by G (these are labelled B1, B2, etc.). We also
intend to give vector space generators for the centraliser algebra of
G labelled C1, C2, etc., with C1 being
the identity matrix; this may be accompanied by further information giving
a corresponence between the Ci and elements of a suitable
field/division ring. Where appropriate, we shall try to indicate which of our
symmetric/Hermitian forms are positive definite (finite groups will always
have at least one such form), but in general this problem seems hard. Any
indication of Schur index means the Schur index of an appropriate absolutely
irreducible representation over Q.
Though the irrationalities may seem better presented in the
NumberField format, calculations will [usually] proceed much faster
if QuadraticField or CyclotomicField is used instead. [This
is because NumberField is a general construction, whereas the latter
two are specialised constructions.] By a theorem of Brauer, all (finite
dimensional) characteristic 0 representations of all finite groups can be
expressed over an appropriate cyclotomic field . . . which is just as well.
These have command lines like pwr 4 3 7.
- mu i j k: multiply i by j and place the result [ij] in k.
- iv i j: invert i and place the result in j.
- pwr i j k: take ith power of j and place the result in k.
- cjr i j: conjugate i by j and place the result back in i.
[We don't intend to use this, but it may still be present in some of the programs.]
These scripts make the above commands MeatAxe-compatible (with i above
now corresponding to zi.)
We give more detailed information about word programs on a separate page
- For mu:
- zmu z$1 z$2 z$3
- For iv:
- ziv z$1 z$2
- For pwr:
- zsm pwr$1 z$2 z$3
Go to main ATLAS (version 2.0) page.
Anonymous ftp access is also available.
See here for details.
Version 2.0 created on 4th May 1999.
Last updated 05.02.02 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.