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

[Jon]

Dear Dima,

Thanks for your detailed explanation. However, algebraic number is a new
concept to me. I googled a bit and know the basic idea about it, but still
cannot follow the notation of E(3). I used ?E, but it gives a lot series of
entries, and I don't know how to check a specific one to look into. I thus
wonder whether you could explain a bit more about it, or just give me the
mathematical expression for it.

By the way, in physics, S(4) is isomorphic to Td point group. In Td, it is
known that there exists a 2D vector which can be used to generate all the
irreducible matrices to be used (matrix elements only involving real
numbers), I thus wonder whether the matrix representation given by GAP is
of this type or not, if not, how can I possibly find a similarity
transformation to go from GAP result to another form (say results used in
physics). I understand that two representations related with a similarity
transformation cannot be said to be distinct representation, but I hope you
know what I mean here.

Thanks again,

Sincerely,
Jon


On Fri, Aug 17, 2012 at 11:09 PM, Dima Pasechnik <dima at ntu.edu.sg> wrote:

> 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
>
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