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

[Jon]

Dear Dima,

Thank you for the explanation on E. I should have guessed it to be the
case. Now with your confirmation, it is quite reassuring to me.

I know of a 2D irreducible representation to be real for S(4), but the 2D
irreducible representation returned by GAP is complex. I am wondering
whether there exists a unique transformation to bring all the complex
matrices pertinent to GAP to real matrices. Or, alternatively, for two
representations to be equivalent, should the two sets of matrices be
related to each other by a single unitrary transformation?

Sincerely,
Jon


On Sat, Aug 18, 2012 at 10:37 PM, Dima Pasechnik <dima at ntu.edu.sg> wrote:

> 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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