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 > <d...@ntu.edu.sg<mailto:d...@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 > > 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. > _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum
