Dear Stefan, Marc and Max, Thanks very much for your help.
Best wishes, Jerry. On Sun, Jun 18, 2017 at 4:16 PM, Max Horn <m...@quendi.de> wrote: > Dear Jerry, > > > On 18 Jun 2017, at 15:05, Jerry Swan <dr.jerry.s...@gmail.com> wrote: > > > > Dear all, > > > > For some element g of a group G for which irr := > > IrreducibleRepresentations(G) have been obtained, is it possible to > recover > > g from images := List(irr,r->Image(r,g)) ? > > The representations are homomorphisms, and as such, you can compute > preimages -- which in general are of course not unique, but rather cosets > of the kernel. But for a finite group, the intersection of the kernel of > all irreducible representations is trivial, so you can recover g like this: > > pre:=Intersection(List([1..Length(irr)], i -> PreImagesElm(irr[i], > images[i]))); > > Applied to a concrete example: > > gap> G:=SymmetricGroup(5);; > gap> irr:=IrreducibleRepresentations(G);; > gap> g:=Random(G); > (1,4,3) > gap> images := List(irr,r->Image(r,g));; > gap> pre:=Intersection(List([1..Length(irr)], i -> PreImagesElm(irr[i], > images[i]))); > [ (1,4,3) ] > > > In a later email, you clarified that your group G is always a symmetric > group S_n. In that case, at least for n>=5, most irreducible > representations are actually faithful, the exception being the trivial and > the sign representation. In that case, you can simply take a preimage of > one of the faithful representations, like so: > > gap> List(irr, IsInjective); > [ false, true, true, true, true, true, false ] > gap> irr[1]; # this is the sign representation > [ (1,2,3,4,5), (1,2) ] -> [ [ [ 1 ] ], [ [ -1 ] ] ] > gap> irr[7]; # this is the trivial representation > [ (1,2,3,4,5), (1,2) ] -> [ [ [ 1 ] ], [ [ 1 ] ] ] > gap> PreImage(irr[2], images[2]); > (1,4,3) > gap> PreImage(irr[3], images[3]); > (1,4,3) > > > Hope that helps, > Max _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum
