GroupsWithPropertyT/OrbitDecomposition.jl

push!(LOAD_PATH, "./")

using Nemo
using Groups
using WreathProducts

using GroupRings

import Nemo.elements

using JLD
using ProgressMeter

function products{T<:GroupElem}(X::AbstractVector{T}, Y::AbstractVector{T})
    result = Vector{T}()
    seen = Set{T}()
    @showprogress for x in X
        for y in Y
            z = x*y
            if !in(z, seen)
                push!(seen, z)
                push!(result, z)
            end
        end
    end
    return result
end

function generate_balls{T<:GroupElem}(S::Vector{T}, Id::T; radius=2)
    sizes = Vector{Int}()
    S = unshift!(S, Id)
    B = [Id]
    for i in 1:radius
        B = products(B, S);
        push!(sizes, length(B))
    end
    return B, sizes
end

function elements(F::Nemo.FqNmodFiniteField)
    deg = Int(degree(F))
    char = Int(characteristic(F))
    z = (gen(F)^i for i in 0:deg-1)
    crtsn_prd = Base.product([0:char-1 for i in 1:deg]...)

    function _it()
        for crt in crtsn_prd
            g = sum([b*a for (a,b) in zip(z,crt)])
            produce(g)
        end
    end
    return Task(_it)
end

function AutFG_emb(A::AutGroup, g::WreathProductElem)
    isa(A.objectGroup, FreeGroup) || throw("Not an Aut(FN)")
    parent(g).P.n == length(A.objectGroup.gens) || throw("No natural action of $(parent(g)) on $A")
    powers = [(elt == parent(elt)() ? 0: 1) for elt in g.n.elts]
    elt = reduce(*, [A(Groups.flip_autsymbol(i))^pow for (i,pow) in enumerate(powers)])
    Groups.r_multiply!(elt, [Groups.perm_autsymbol(g.p)])
    return elt
end

function (g::WreathProductElem)(a::AutGroupElem)
    g = AutFG_emb(parent(a),g)
    return g*a*g^-1
end

function orbit_decomposition(G::Nemo.Group, E::Vector, rdict=GroupRings.reverse_dict(E))

    elts = collect(elements(G))

    tovisit = trues(E);
    orbits = Vector{Set{Int}}()

    @showprogress "Orbit decomposition... " for i in 1:endof(E)
        if tovisit[i]
            orbit = Set{Int}()
            a = E[i]
            for g in elts
                idx = rdict[g(a)]
                tovisit[idx] = false
                push!(orbit,idx)
            end
            push!(orbits, orbit)
        end
    end
    return orbits
end

function matrix_repr(g::WreathProductElem, E, E_dict)
   rep_matrix = spzeros(Int, length(E), length(E))
   for (i,e) in enumerate(E)
      j = E_dict[g(e)]
      rep_matrix[i,j] = 1
   end
   return rep_matrix
end

function action_mreps(G::Nemo.Group, E, E_dict)
    result = @showprogress [matrix_repr(g, E, E_dict) for g in elements(G)]
    return result
end

function chars(G::PermutationGroup)
   permtype_unsorted(σ::Nemo.perm) = [length(c) for c in cycles(σ)]
   permtype(σ::Nemo.perm) = sort(permtype_unsorted(σ))

   χ_id(σ::Nemo.perm) = 1

   χ_sgn(σ::Nemo.perm) = (-1)^parity(σ)

   function χ_reg(σ::Nemo.perm)
      fixed_points = countnz([(x == y? 1 : 0) for (x,y) in enumerate(σ.d)])
      return fixed_points - 1
   end

   χ_regsgn(σ::Nemo.perm) = (-1)^parity(σ)*χ_reg(σ)

   function χ_regviaS3(σ::Nemo.perm)
      @assert parent(σ).n == 4
      t = permtype(σ)
         if t == [1,1,1,1]
            result = 2
         elseif t == [2,2]
            result = 2
         elseif t == [1,3]
            result = -1
         else
            result = 0
         end
      return result
   end

   chars = [χ_id, χ_sgn, χ_regviaS3, χ_reg, χ_regsgn]

   if G.n == 1
      return chars[1:1]
   elseif G.n == 2
      return chars[1:2]
   elseif G.n == 3
      return [chars[1:2]..., chars[4]]
   elseif G.n == 4
      return chars[1:5]
   else
      throw("Characters for $G unknown!")
   end
end

function epsilon(i, g::DirectProducts.DirectProductGroupElem)
    return reduce(*, 1, ((-1)^isone(g.elts[j]) for j in 1:i))
end

import Base.convert
convert(::Type{Int}, x::Nemo.fmpz) = x.d

function central_projection(RG::GroupRing, char::Function, T::Type=Rational{Int})
    result = RG(T)
    for g in RG.basis
        result[g] = char(g)
    end
    return convert(T, char(RG.group())//Int(order(RG.group))*result)
end

function Cstar_repr(x::GroupRingElem, matrix_reps)
    res = zeros(matrix_reps[1])
    for i in 1:length(parent(x).basis)
        res += x.coeffs[i]*matrix_reps[i]
    end
    return res
end

function rankOne_projections(G::PermutationGroup, T::Type=Rational{Int})
   RG = GroupRing(G)
   projections = [central_projection(RG, χ, T) for χ in chars(G)]

   if G.n == 1 || G.n == 2
      return  projections
   elseif G.n == 3
      rankone_projs = [
         projections[1],
         projections[2],
         1//2*(one(RG) - RG(RG.group([2,1,3])))*projections[3]
         ]
      return rankone_projs
   elseif G.n == 4
      rankone_projs = [
         projections[1],
         projections[2],
         1//2*(one(RG) - RG(RG.group([2,1,3,4])))*projections[3],
         1//2*(one(RG) - RG(RG.group([2,1,3,4])))*projections[4],
         1//2*(one(RG) + RG(RG.group([2,1,3,4])))*projections[5]]
      return rankone_projs
   else
      throw("Rank-one projections for $G unknown!")
   end

end

function rankOne_projections(BN::WreathProducts.WreathProduct, T::Type=Rational{Int})

   N = BN.P.n
    # projections as elements of the group rings RSₙ
   SNprojs_nc = [rankOne_projections(PermutationGroup(i), T) for i in 1:N]

   # embedding into group ring of BN
   RBN = GroupRing(BN)
   RFFFF_projs = [central_projection(GroupRing(BN.N), g->epsilon(i,g), T)
        for i in 0:BN.P.n]
   Qs = [RBN(q, g -> BN(g)) for q in RFFFF_projs]

   function incl(k::Int, g::perm, WP::WreathProduct=BN)
      @assert length(g.d) + k <= WP.P.n
      arr = [1:k; g.d .+ k; (length(g.d)+k+1):WP.P.n]
      return WP(WP.P(arr))
   end

    all_projs=[Qs[1]*RBN(p, g-> incl(0,g)) for p in SNprojs_nc[N]]

   for i in 1:N-1
        Sk_first = [RBN(p, g->incl(0,g)) for p in SNprojs_nc[i]]
        Sk_last  = [RBN(p, g->incl(i,g)) for p in SNprojs_nc[N-i]]
        append!(all_projs, [Qs[i+1]*p1*p2
                for (p1,p2) in Base.product(Sk_first,Sk_last)])
   end

   append!(all_projs, [Qs[N+1]*RBN(p, g-> incl(0,g)) for p in SNprojs_nc[N]])

   return all_projs
end

function orthSVD(M::AbstractMatrix)
    M = full(M)
#     matrixRank = rank(M)
    fact = svdfact(M)
    sings = fact[:S]
#     @show sings[sings.>1e-14]
    M_rank = sum(fact[:S] .> maximum(size(M))*eps(eltype(fact[:S])))
    Ufactor = fact[:U]
    return Ufactor[:,1:M_rank]
end

function Uπ_matrices(P_matrices; orth=orthSVD)
    U_p_matrices = Vector{Array{Float64,2}}()
    @showprogress "Computing Uπ mats..." for (i,p_mat) in enumerate(P_matrices)
        U_p = orth(p_mat)
        push!(U_p_matrices, U_p)
    end
    return U_p_matrices
end

function main(N::Int)
   name = "test"
   # SOutF$(N)_E4

   isdir(name) || mkdir(name)

   SOutFN = AutGroup(FreeGroup(N), special=true, outer=true)
   S = generators(SOutFN);
   S = [S; [inv(s) for s in S]]
   @show S

   info("Generating ball of radius 4")
   @time E4, sizes = generate_balls(S, SOutFN(), radius=4);
   info("Reverse dict")
   @time E_dict = GroupRings.reverse_dict(E4)


   BN = WreathProduct(Nemo.FiniteField(2,1, "a")[1], PermutationGroup(N))

   if !isfile(joinpath(name, "orbits.jld"))
      info("Decomposing E into orbits of B$(N)")
      @time orbs = orbit_decomposition(BN, E4, E_dict)
      @assert sum(length(o) for o in orbs) == length(E4)
      save(joinpath(name, "orbits.jld"), "orbits", orbs)
   end

   info("Action matrices")

   pm = GroupRings.create_pm(E4, E_dict, sizes[2], twisted=true)

   RSOutFN = GroupRing(SOutFN, E4, E_dict, pm)

   PropertyT.splaplacian(RSOutFN, S)

   E2 = E4[1:sizes[2]]
   @time BNactionE_mreps = action_mreps(BN, E2, E_dict)

   save(joinpath(name, "matrix_reps.jld"), "matrix_reps", BNactionE_mreps)

   info("Projections")
   @time BN_mps = rankOne_projections(BN);

   @time π_E_projections = @showprogress [Cstar_repr(p, BNactionE_mreps) for p in BN_mps]

   info("Uπs...")
   @time Uπs = Uπ_matrices(π_E_projections);

   @show multiplicities = [size(U,2) for U in Uπs];
   @show dimensions = [Int(p[BN()]*Int(order(BN))) for p in BN_mps];
   @assert dot(multiplicities, dimensions) == sizes[2]

   save(joinpath(name, "U_pis.jld"), "Uπs", Uπs, "dims", dimensions)
end

@time main(4)