Missing hom method between WeylGroup and GAPGroup

[TWiedemann]

Similar to #4460, but for WeylGroup.

Setup:

julia> G = symmetric_group(3)
Sym(3)

julia> W = weyl_group(:A, 2)
Weyl group for root system defined by Cartan matrix [2 -1; -1 2]

julia> gen_G = [ cperm(G, [1,2]), cperm(G, [2,3]) ]
2-element Vector{PermGroupElem}:
 (1,2)
 (2,3)

julia> gen_W = gens(W)
2-element Vector{WeylGroupElem}:
 s1
 s2

It should be possible to define a homomorphism between G and W by lists of generators, but currently it is not:

julia> hom(W, G, gen_W, gen_G)
ERROR: MethodError: no method matching hom(::WeylGroup, ::PermGroup, ::Vector{WeylGroupElem}, ::Vector{PermGroupElem})
The function `hom` exists, but no method is defined for this combination of argument types.

Closest candidates are:
  hom(::Oscar.GAPGroup, ::Oscar.GAPGroup, ::Vector, ::Vector; check)
   @ Oscar ~/.julia/packages/Oscar/5kSAT/src/Groups/homomorphisms.jl:120
  hom(::AbstractVariety, ::AbstractVariety, ::Vector, ::Any; inclusion, symbol)
   @ Oscar ~/.julia/packages/Oscar/5kSAT/experimental/IntersectionTheory/src/Main.jl:278
  hom(::Oscar.MPolyLocRing, ::NCRing, ::Any, ::Vector; check)
   @ Oscar ~/.julia/packages/Oscar/5kSAT/src/AlgebraicGeometry/Schemes/AffineSchemes/Morphisms/Methods.jl:226
  ...

Stacktrace:
 [1] top-level scope
   @ REPL[7]:1

julia> hom(G, W, gen_G, gen_W)
ERROR: MethodError: no method matching hom(::PermGroup, ::WeylGroup, ::Vector{PermGroupElem}, ::Vector{WeylGroupElem})
The function `hom` exists, but no method is defined for this combination of argument types.

Closest candidates are:
  hom(::Oscar.GAPGroup, ::Oscar.GAPGroup, ::Vector, ::Vector; check)
   @ Oscar ~/.julia/packages/Oscar/5kSAT/src/Groups/homomorphisms.jl:120
  hom(::AbstractVariety, ::AbstractVariety, ::Vector, ::Any; inclusion, symbol)
   @ Oscar ~/.julia/packages/Oscar/5kSAT/experimental/IntersectionTheory/src/Main.jl:278
  hom(::Oscar.MPolyLocRing, ::NCRing, ::Any, ::Vector; check)
   @ Oscar ~/.julia/packages/Oscar/5kSAT/src/AlgebraicGeometry/Schemes/AffineSchemes/Morphisms/Methods.jl:226
  ...

Stacktrace:
 [1] top-level scope
   @ REPL[8]:1

[fingolfin]

One way to deal with this now that we have isomorphisms into GAP fp and perm groups would be to implement hom(G::WeylGroup, H::Group, ...) and hom(H::Group, G::WeylGroup, ...) by first constructing an isomorphism between G and a GAP group G2, then use hom(G2,H) resp. hom(H,G2), and compose that suitably with the iso.

I hope (but did not test) that this will handle images and preimages of elements; hopefully also kernel can be computed, isinjective / issurjective tested, images and preimages of subgroups () and more.

But this is a bit of a bandaid, on the long run we need to revise how we implement group homs in OSCAR and perhaps switch to something a bit closer to what GAP does.

[fingolfin]

Aha, see also issue #4474