[GAP Forum] first example towards GAP, need some explanation

[Dima Pasechnik]

Dear Jon,

> The first example I tried on GAP is about the symmetric group of 4
> elements(?). I tried to get its irreducible matrix representation. The
> outcome I got from GAP is
>
> gap> List(g,g->g^reps[3]);
> [ [ [ 1, 0 ], [ 0, 1 ] ], [ [ 0, E(3) ], [ E(3)^2, 0 ] ], [ [ E(3)^2, 0 ],
> ....
>
> My question is:
>
> (1) How do I know which matrix corresponds to which group element?

It's a bit hard to see what you're doing. How did you get all the
irreducible representations?
Your List(g,g->g^reps[3]); probably means List(g,x->x^reps[3]);
In this case you can simply do
List(g,x->[x,x^reps[3]]);
to get the pairs [group element, its representation].

> (2) What does E(3) mean?
Try doing
gap> ?E
(well, this is to point out the convenient GAP help facility :-))

E(3) is an algebraic number.
But in fact, if you group is indeed a symmetric group, it's a bit
sub-optimal to work with algebraic numbers, as all the complex
irreducible matrix representations of symmetric groups can be
written using only rational numbers.
As well, please note that the symmetric group of 4 points does not
have a faithful irreducible representation of dimension 2.
So you have probably constructed a representation with the kernel
of order 4.

> (3) There can be different representations which has all matrix elements
> real, how can I find a similarity transformation which can do this?

This is, in principle, easy linear algebra, but I don't know of a specific
GAP command for this purpose. (By the way, if two representations are
related by a similarity that aren't even considered "different").

> (4) Can the output be set in a way that these 24 matrices can be read in
> directly by say Fortran?
Hmm, do you want to call Fortran from GAP directly? This can be done.
If you just want to write out a text file which can be then read in
by another program, PrintTo and AppendTo are GAP commands you can do
this. (You'd need to write a loop, I suppose).

Best,
Dmitrii

CONFIDENTIALITY:This email is intended solely for the person(s) named and may be confidential and/or privileged.If you are not the intended recipient,please delete it,notify us and do not copy,use,or disclose its content.

Towards A Sustainable Earth:Print Only When Necessary.Thank you.