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# Computing decompositions
of representations
## Overview
The main
function of this package is
to compute the decomposition
of a
representation ρ,
of a finite group G over the complex numbers into
irreducibles, as well as the corresponding decomposition
of the
centraliser
of R.
While methods
for doing this were well known before this package was
written (cf. Serre
's book "Linear Representations of Finite Groups"),
there were no open-source computer programs that implemented these
methods, nor were details
on how
to achieve good performance
of such
an implementation published.
This package is useful
in,
for example, semidefinite programming
and
optimisation/feasibility problems involving coding theory, graph
theory, algebraic geometry, combinatorics
and more (see
https:
//arxiv.org/abs/1007.2905 for more examples of possible
applications). Specifically, it allows
to achieve substantial
reductions
in the dimension
of these problems; potentially known
results, e.g.
on upper bounds
on sizes
of nonlinear codes, could be
improved
with the functions implemented
in this package.
This package was written as part
of my (Kaashif Hymabaccus
's) Master's
degree
at the University
of Oxford, supervised
by Dmitrii Pasechnik.
## Installation
Make sure that, when you install GAP, you installed all
of the
packages! Our package uses GRAPE
and IO
for some functions.
### Latest version included
with GAP
If you have version
4.
11.
0 or later
of the
[GAP system](https:
//www.gap-system.org/Download/index.html)
installed, you
do not have
to install RepnDecomp manually since it is
already distributed
with GAP.
### Latest released version
If you would like the latest released version
of RepnDecomp,
and there
has
not yet been a
release of GAP including it,
then you can download
the latest
release
[here](https:
//gap-packages.github.io/RepnDecomp/).
Create the directory `~/.gap/pkg`, which will contain your local
packages
and extract the archive you downloaded into it.
For example:
$ mkdir -
p ~/.gap/pkg
$ tar -C ~/.gap/pkg -xvzf RepnDecomp-
1.
1.
0.tar.gz
### Development version
If you would like
to install the latest code directly
from git master
(unsuitable
for anything but development
of this package),
then you
can clone this repo directly into `~/.gap/pkg`:
$ mkdir -
p ~/.gap/pkg
$
cd ~/.gap/pkg
$ git clone https:
//github.com/gap-packages/RepnDecomp.git
### Post-installation
Now, you can run GAP however you like, load the package
and use the
functions provided:
```
$ gap
<some output>
gap> LoadPackage(
"RepnDecomp");
───────────────────────────────────────────────────────────────────────────────
Loading GRAPE
4.
8.
2 (GRaph Algorithms using PErmutation groups)
by Leonard H. Soicher (http:
//www.maths.qmul.ac.uk/~lsoicher/).
Homepage: https:
//gap-packages.github.io/grape
Report issues
at https:
//github.com/gap-packages/grape/issues
───────────────────────────────────────────────────────────────────────────────
───────────────────────────────────────────────────────────────────────────────
Loading RepnDecomp
0.
1 (Decompose representations
of finite groups into irreducibles)
by Kaashif Hymabaccus (https:
//kaashif.co.uk).
with contributions
by:
Dmitrii Pasechnik.
Homepage: http:
//gitlab.com/kaashif/decomp/
───────────────────────────────────────────────────────────────────────────────
true
gap> A := IdentityMat(
5);
[ [
1,
0,
0,
0,
0 ], [
0,
1,
0,
0,
0 ], [
0,
0,
1,
0,
0 ], [
0,
0,
0,
1,
0 ],
[
0,
0,
0,
0,
1 ] ]
gap> B := LDLDecomposition(A);
rec( D := [
1,
1,
1,
1,
1 ],
L := [ [
1,
0,
0,
0,
0 ], [
0,
1,
0,
0,
0 ], [
0,
0,
1,
0,
0 ],
[
0,
0,
0,
1,
0 ], [
0,
0,
0,
0,
1 ] ] )
```
Where `LDLDecomposition` is a
function provided
by this package.
## Testing
Make sure your current directory is this repo,
for example:
$ git clone https:
//github.com/gap-packages/RepnDecomp.git
$
cd RepnDecomp
First, generate the tests
from the documentation:
$ gap -
q < makedoc.g
To run the tests, make sure you have all needed packages installed
(GRAPE
and also IO
if you want
to compute
in parallel).
$ gap tst/testall.g
This will run all tests
and (hopefully) pass.
In order
for the tests
to be as useful as possible
to me i.e. catch as many bugs as possible,
there is a lot
of randomness
in them. This means that sometimes the
tests pick a pathologically bad example
to decompose which causes the
tests
to hang forever. The tests will be fixed
to be more
deterministic.
## Documentation
There
's a GAPDoc documentation book hosted
[here](https:
//gap-packages.github.io/RepnDecomp/doc/chap0.html). This
is generated
from the source files
and comments
in the `lib/`
directory
of this repo, so you can also look there
for the same
information.
There are also some examples
in the `examples` directory, which are
embedded
in the manual, but the most complete examples are
in the
`old_tst` directory. These are the old, poorly documented, but
essentially complete tests. They are being converted into nicer
looking tests that can be embedded into the manual, but this could
take some
time.
## Paper
This package is described
in a freely available [paper](https:
//joss.theoj.org/papers/10.21105/joss.01835) in
[Journal
of Open Source Software](https:
//joss.theoj.org/).
## Contributing
Open a pull request
or issue
in this repository. You can also email me
directly, I
'll be more likely to notice an email.
