Merge pull request #964 from oscar-system/Reps

add stuff to compute representations for PcGroups

experimental/GModule/Cohomology.jl

@@ -5,21 +5,25 @@ import Oscar:action
 import AbstractAlgebra: Group, Module
 import Base: parent
 
-#TODO: rename into GModule
 @attributes mutable struct GModule{gT,mT}
   G::gT
   M::mT
   ac::Vector{Map} # automorphisms of M, one for each generator of G
 
-  function GModule(G::T, ac::Vector{<:Map}) where {T <: Oscar.GAPGroup}
-    r = new{T,typeof(domain(ac[1]))}()
+  function GModule(M, G::T, ac::Vector{<:Map}) where {T <: Oscar.GAPGroup}
+    r = new{T,typeof(M)}()
     r.G = G
     r.ac = ac
-    r.M = domain(ac[1])
+    r.M = M
     @assert all(x -> domain(x) == codomain(x) == r.M, ac)
     return r
   end
 
+
+  function GModule(G::T, ac::Vector{<:Map}) where {T <: Oscar.GAPGroup}
+    return GModule(domain(ac[1]), G, ac)
+  end
+
   F::Group # G as an Fp-group (if set)
   mF::GAPGroupHomomorphism  # F -> G, maps F[i] to G[i]
 
@@ -62,6 +66,21 @@ function Oscar.relations(F::FPGroup)
   return [(x, z) for x = R]
 end
 
+function Oscar.relations(G::Oscar.GAPGroup)
+   f = GAP.Globals.IsomorphismFpGroupByGenerators(G.X, GAP.Globals.GeneratorsOfGroup(G.X))
+   f !=GAP.Globals.fail || throw(ArgumentError("Could not convert group into a group of type FPGroup"))
+   H = FPGroup(GAP.Globals.Image(f))
+   return relations(H)
+end
+
+function Oscar.relations(G::PcGroup)
+   f = GAP.Globals.IsomorphismFpGroupByPcgs(GAP.Globals.FamilyPcgs(G.X), GAP.julia_to_gap("g"))
+   f !=GAP.Globals.fail || throw(ArgumentError("Could not convert group into a group of type FPGroup"))
+   H = FPGroup(GAP.Globals.Image(f))
+   return relations(H)
+end
+
+
 AbstractAlgebra.Group(C::GModule) = C.G
 AbstractAlgebra.Module(C::GModule) = C.M
 action(C::GModule) = C.ac
@@ -183,6 +202,22 @@ function H_zero(C::GModule)
   return k, z
 end
 
+#= TODO
+ - break out coboundaries and cochains
+ - depending on the module type:
+   - intersect yields an embedding (Z-module) or not GrpAb
+   - make sure that image/ kernel are consisten
+   - preimage 
+   - issubset yields (for GrpAb) only true/ false, not the map
+   - issubgroup has the "wrong" order of arguments (and cannot apply
+     to modules)
+   - quo does ONLY work if B is a direct submodule of A (Z-modules)
+   - mat or matrix is used to get "the matrix" from a hom
+   - zero_hom/ zero_obj/ identity_hom is missing
+   - Janko-Module-Homs have different types, they probably need to
+     come under a common abstract type or be more selective
+=#
+
 function H_one(C::GModule)
   z = get_attribute(C, :H_one)
   if z !== nothing
@@ -197,8 +232,8 @@ function H_one(C::GModule)
   X(r_i) corresponds to some map phi_i : M^r -> M
   phi_i = oplus h_j M for some homs h_j coming from the word in r
   so, a kernel computation again
-
   =#
+
   G = Group(C)
   n = ngens(G)
   M = Module(C)
@@ -407,6 +442,11 @@ function Oscar.mat(M::FreeModuleHom{FreeMod{QabElem}, FreeMod{QabElem}})
   return M.matrix
 end
 
+#= Hulpke-Dietrich:
+UNIVERSAL COVERS OF FINITE GROUPS
+https://arxiv.org/pdf/1910.11453.pdf
+almost the same as Holt
+=#
 function H_two(C::GModule)
   z = get_attribute(C, :H_two)
   if z !== nothing
@@ -693,7 +733,7 @@ function H_two(C::GModule)
 end
 
 """
-For a cohomology module `C` compute the `i`-th cohomology group
+For a gmodule `C` compute the `i`-th cohomology group
   where `i` can be `0`, `1` or `2`.
 Together with the abstract module, a map is provided that will 
   produce explicit cochains.
@@ -817,6 +857,9 @@ Note: we do not check that this defined indeed a `ZZ[H]` module.
 function gmodule(H::Oscar.GAPGroup, ac::Vector{<:Map})
   return GModule(H, ac)
 end
+function gmodule(M, H::Oscar.GAPGroup, ac::Vector{<:Map})
+  return GModule(M, H, ac)
+end
 
 function _gmodule(k::AnticNumberField, H::PermGroup, mu::Map{GrpAbFinGen, FacElemMon{AnticNumberField}})
   u = domain(mu)
@@ -927,7 +970,7 @@ Oscar.group(C::GModule) = C.G
 
 #= TODO
   for Z, Z/nZ, F_p and F_q moduln -> find Fp-presentation
-  #done: for GrpAbFinGen            -> find Fp-presentation
+  #done: for GrpAbFinGen          -> find Fp-presentation
   #done: for a in H^2 find Fp-presentation
   for a in H^2 find (low degree) perm group using the perm group we have?
   Magma's DistinctExtensions
@@ -938,9 +981,6 @@ Sort:
  - move (and fix) the ModuleHom stuff
  - add proper quo for Modules
 
-  group better: the GModule should
-   - cache the H^i's that are computed
-
   features   
    - inflation, restriction, long exact sequence  
 
@@ -954,7 +994,6 @@ Sort:
    - understand Derek Holt and use BSGS for large perm groups
      rather than the RWS (or use BSGS to get an RWS?)
 
-
   GModule for 
     - local field (add (trivial) and mult)
     - (S-)units

experimental/GModule/GModule.jl

@@ -28,8 +28,14 @@ function irreducible_modules(G::Oscar.GAPGroup)
   K = abelian_closure(QQ)[1]
   for m in im
     z = map(x->matrix(map(y->map(K, y), m(x.X))), gens(G))
-    F = free_module(K, nrows(z[1]))
-    push!(IM, gmodule(G, map(x->hom(F, F, x), z)))
+    if ngens(G) == 0
+      F = free_module(K, 0)
+      zz = typeof(hom(F, F, elem_type(F)[]))[]
+    else
+      F = free_module(K, nrows(z[1]))
+      zz = map(x->hom(F, F, x), z)
+    end
+    push!(IM, gmodule(F, G, zz))
   end
   return IM
 end
@@ -116,9 +122,13 @@ function gmodule(::typeof(CyclotomicField), C::GModule)
   for g = C.ac
     l = lcm(l, lcm(collect(map_entries(x->Hecke.iscyclotomic_type(parent(x.data))[2], mat(g)))))
   end
-  K = cyclotomic_field(l, cached = false)[1]
+  K = cyclotomic_field(base_ring(C), l)[1]
   F = free_module(K, dim(C))
-  return gmodule(group(C), [hom(F, F, map_entries(x->K(x.data), mat(x))) for x = C.ac])
+  if d == 0 
+    h = hom(F, F, elem_type(F)[])
+    return gmodule(F, group(C), typeof(h)[hom(F, F, map_entries(x->K(x.data), mat(x))) for x = C.ac])
+  end
+  return gmodule(F, group(C), [hom(F, F, map_entries(x->K(x.data), mat(x))) for x = C.ac])
 end
 
 import Base: ^
@@ -133,12 +143,12 @@ end
 
 function ^(C::GModule{<:Any, Generic.FreeModule{QabElem}}, phi::Map{QabField, QabField})
   F = free_module(codomain(phi), dim(C))
-  return GModule(group(C), [hom(F, F, map_entries(phi, mat(x))) for x = C.ac])
+  return GModule(F, group(C), [hom(F, F, map_entries(phi, mat(x))) for x = C.ac])
 end
 
 function gmodule(::FlintRationalField, C::GModule{<:Any, Generic.FreeModule{nf_elem}})
   F = free_module(QQ, dim(C)*degree(base_ring(C)))
-  return GModule(group(C), [hom(F, F, hvcat(dim(C), [representation_matrix(x) for x = transpose(mat(y))]...)) for y = C.ac])
+  return GModule(F, group(C), [hom(F, F, hvcat(dim(C), [representation_matrix(x) for x = transpose(mat(y))]...)) for y = C.ac])
 end
 
 function gmodule(k::Nemo.GaloisField, C::GModule{<:Any, Generic.FreeModule{fmpq}})
@@ -245,6 +255,9 @@ function hom_base(C::_T, D::_T) where _T <: GModule{<:Any, <:Generic.FreeModule{
       push!(t, hom_base(z1[i], z2[i]))
     end
     tt = [Hecke.modular_lift([t[i][j] for i=1:length(z1)], me) for j=1:length(t[1])]
+    if length(tt) == 0
+      return []
+    end
     @assert base_ring(tt[1]) == k
     if isone(pp)
       pp = fmpz(p)
@@ -321,6 +334,39 @@ function hom_base(C::_T, D::_T) where _T <: GModule{<:Any, <:Generic.FreeModule{
   end
 end
 
+function gmodule(K::AnticNumberField, M::GModule{<:Any, <:Generic.FreeModule{nf_elem}})
+  F = free_module(K, dim(M))
+  return gmodule(F, group(M), [hom(F, F, map_entries(K, mat(x))) for x = M.ac])
+end
+
+function (K::QabField)(a::nf_elem)
+  fl, f = Hecke.iscyclotomic_type(parent(a))
+  @assert fl
+  return QabElem(a, f)
+end
+
+function hom_base(C::_T, D::_T) where _T <: GModule{<:Any, <:Generic.FreeModule{QabElem}}
+  C1 = gmodule(CyclotomicField, C)
+  D1 = gmodule(CyclotomicField, D)
+  fl, Cf = Hecke.iscyclotomic_type(base_ring(C1))
+  @assert fl
+  fl, Df = Hecke.iscyclotomic_type(base_ring(D1))
+  @assert fl
+  l = lcm(Cf, Df)
+  K, _ = cyclotomic_field(base_ring(C), l)
+  if l != Cf
+    C1 = gmodule(K, C1)
+  end
+  if l != Df
+    D1 = gmodule(K, D1)
+  end
+  h = hom_base(C1, D1)
+  if length(h) == 0
+    return h
+  end
+  return map(x->map_entries(base_ring(C), x), h)
+end
+
 function gmodule(::FlintRationalField, C::GModule{<:Any, <:Generic.FreeModule{fmpz}})
   F = free_module(QQ, dim(C))
   return GModule(group(C), [hom(F, F, map_entries(QQ, mat(x))) for x = C.ac])
@@ -518,3 +564,96 @@ end #module GModuleFromGap
 using .GModuleFromGap
 
 export irreducible_modules, isabsolutely_irreducible, isdecomposable
+
+module RepPc
+using Oscar
+
+Base.pairs(M::MatElem) = Base.pairs(IndexCartesian(), M)
+Base.pairs(::IndexCartesian, M::MatElem) = Base.Pairs(M, CartesianIndices(axes(M)))
+
+function Hecke.roots(a::fq_nmod, i::Int)
+  kx, x = PolynomialRing(parent(a), cached = false)
+  return roots(x^i-a)
+end
+
+#=TODO
+ - construct characters along the way as well?
+ - compare characters rather than the hom_base
+ - maybe reason from theory what reps are going to be new?
+ - conjugate to smallest field?
+ - allow trivial stuff
+=# 
+
+function reps(K, G::PcGroup)
+  s, ms = sub(G, [gens(G)[end]])
+  o = Int(order(s))
+  @assert isprime(o)
+  z = roots(K(1), o)
+  F = free_module(K, 1)
+  R = [gmodule(F, s, [hom(F, F, [r*F[1]])]) for r = z]
+
+  for i=ngens(G)-1:-1:1
+    h = G[i]
+    ns, mns = sub(G, gens(G)[i:end])
+    p = Int(divexact(order(ns), order(s)))
+    @assert isprime(p)
+    new_R = []
+    for r = R
+      F = r.M
+      rh = gmodule(group(r), [action(r, preimage(ms, x^h)) for x = gens(s)])
+      l = Oscar.GModuleFromGap.hom_base(r, rh)
+      @assert length(l) <= 1
+      nr = []
+      Y = mat(action(r, preimage(ms, h^p)))
+      if length(l) == 1
+        X = l[1]
+        Xp = X^p
+        #Brueckner: C*Xp == Y for some scalar C
+        i = findfirst(x->!iszero(x), Xp)
+        @assert !iszero(Y[i])
+        C = divexact(Y[i], Xp[i])
+        @assert C*Xp == Y
+        # I think they should always be roots of one here.
+        rt = roots(C, p)
+        Y = r.ac
+        for x = rt
+          nw = gmodule(F, ns,  vcat([hom(F, F, x*X)], Y))
+          push!(nr, nw)
+        end
+      else #need to extend dim
+        n = dim(r)
+        F = free_module(K, dim(r)*p)
+        z = zero_matrix(K, dim(F), dim(F))
+        z[(p-1)*n+1:end, 1:n] = Y
+        for i=1:p-1
+          z[(i-1)*n+1:i*n, i*n+1:(i+1)*n] = identity_matrix(K, n)
+        end
+        md = [hom(F, F, z)]
+        for g = gens(s)
+          z = zero_matrix(K, dim(F), dim(F))
+          for j=1:p
+            Y = action(r, g)
+            z[(j-1)*n+1:j*n, (j-1)*n+1:j*n] = mat(Y)
+            g = preimage(ms, g^h)
+          end
+          push!(md, hom(F, F, z))
+        end
+        push!(nr, gmodule(F, ns, md))
+      end
+      for nw = nr
+        if any(x->length(Oscar.GModuleFromGap.hom_base(x, nw))>0, new_R)
+          continue
+        else
+          push!(new_R, nw)
+        end
+      end
+    end
+    s, ms = ns, mns
+    R = new_R
+  end
+  return R
+end
+
+end #module RepPc
+
+using .RepPc

src/Rings/AbelianClosure.jl

@@ -119,7 +119,7 @@ _variable(K::QabField) = K.s
 
 _variable(b::QabElem) = Expr(:call, Symbol(_variable(_Qab)), b.c)
 
-function cyclotomic_field(K::QabField, c::Int)
+function Hecke.cyclotomic_field(K::QabField, c::Int)
   if haskey(K.fields, c)
     k = K.fields[c]
     return k, gen(k)