ATLAS: Format of the representations

Format information

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 various formats.

MeatAxe format

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 C_n(X). Among other things, these polynomials are irreducible of degree n, so by taking the polynomial ring modulo C_n(X) we obtain the field of order pⁿ. A polynomial a_n-1X^n-1 + ... is then stored as the non-negative integer a_n-1p^n-1 + ... .

Format 5

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.]

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.

GAP

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).

Magma

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.

Word programs

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.]

Shell scripts

These scripts make the above commands MeatAxe-compatible (with i above now corresponding to zi.)

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

We give more detailed information about word programs on a separate page here.

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.