Dear GAP Forum, I need to calculate the image of a homomorphism of a matrix group:
gap> m1 := [ [0,1,0],[-1,0,0],[0,0,1] ];; gap> m2 := [ [0,1,0],[0,0,1],[1,0,0] ];; gap> O := Group(m1,m2);; # one of the two 3-dim irrep of SymmetricGroup(4) (S_4) # compute the five conjugacy classes and combine them into one list # skip some code... gap> ccall := Concatenation(cc1,cc2,cc3,cc4,cc5); I want to calculate the matrices corresponding to the two dimensional irrep of S_4. Call this W. Examining the character table (and some computation) shows that W is a representation of the quotient of S_4 by the group generated by the elements of the fifth conjugacy class (which has character 2 in the W representation). gap> H := Group(cc5);; gap> IsNormal(O,H); true gap> OmodH := FactorGroup(O,H); Group([ f1, f2^2 ]) gap> hom := NaturalHomomorphismByNormalSubgroup(O,H); CompositionMapping( [ (1,2,6,5), (1,2,3)(4,6,5) ] -> [ f1, f2^2 ], <action isomorphism> ) Compute the images of elements of O: gap> imghom := List([1..Size(O)], i -> Image(hom,ccall[i])); [ <identity> of ..., f1*f2, f1*f2, f1*f2^2, f1*f2^2, f1, f1, f2^2, f2, f2, f2^2, f2, f2^2, f2, f2^2, f1, f1, f1*f2, f1*f2, f1*f2^2, f1*f2^2, <identity> of ..., <identity> of ..., <identity> of ... ] Now, what I would like to have are the 2x2 matrices corresponding to each of the elements of "imghom". I can guess what f1 and f2 are in this example, but would be wary of doing so for a larger group. Is there a systematic way of getting the matrices f1 and f2? Thanks, Ravi _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum
