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

[Dima Pasechnik]

On Sun, Aug 19, 2012 at 02:13:06AM +0800, Jon wrote:
> 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.

OK, so you do 
gap> ?E
and you get a list like
[1] ....
[2] ...
...
If it doesn't fit on one screen, there will be a message at the bottom,
saying "hit q to quit, space for next page", etc
So, hit q, this gives you GAP prompt back.
Now, do
gap> ?1
to list the entry number 1 (or ?2 to list entry number 2, etc)

By the way, if you're more confortable using a web browser, you can 
read GAP manual online:
http://www.gap-system.org/Manuals/doc/ref/chap0.html#contents
It has index, too:
http://www.gap-system.org/Manuals/doc/ref/chapInd.html
which would lead you to the explanation on E quite easily:
http://www.gap-system.org/Manuals/doc/ref/chap18.html#X8631458886314588

E(n), specifically, is a primitiive nth root of unity, i.e. 
exp(2\pi i/n).

Hope this helps,
Dmitrii



> 
> 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<mailto: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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