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
