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
