mirror of
https://github.com/kalmarek/PropertyT.jl.git
synced 2024-07-12 01:35:29 +02:00
288 lines
8.2 KiB
Julia
288 lines
8.2 KiB
Julia
###############################################################################
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#
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# BlockDecomposition
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#
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###############################################################################
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struct BlockDecomposition{T<:AbstractArray{Float64, 2}, GEl<:GroupElem, P<:Generic.Perm}
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orbits::Vector{Vector{Int}}
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preps::Dict{GEl, P}
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Uπs::Vector{T}
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dims::Vector{Int}
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end
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function BlockDecomposition(RG::GroupRing, autS::Group; verbose=true)
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verbose && @info "Decomposing basis of RG into orbits of" autS
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@time orbs = orbit_decomposition(autS, RG.basis, RG.basis_dict)
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@assert sum(length(o) for o in orbs) == length(RG.basis)
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verbose && @info "The action has $(length(orbs)) orbits"
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verbose && @info "Finding projections in the Group Ring of" autS
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@time autS_mps = Projections.rankOne_projections(GroupRing(autS, collect(autS)))
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verbose && @info "Finding AutS-action matrix representation"
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@time preps = perm_reps(autS, RG.basis[1:size(RG.pm,1)], RG.basis_dict)
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@time mreps = matrix_reps(preps)
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verbose && @info "Computing the projection matrices Uπs"
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@time Uπs = [orthSVD(matrix_repr(p, mreps)) for p in autS_mps]
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multiplicities = size.(Uπs,2)
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dimensions = [Int(p[one(autS)]*Int(order(autS))) for p in autS_mps]
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if verbose
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info_strs = ["",
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lpad("multiplicities", 14) * " =" * join(lpad.(multiplicities, 4), ""),
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lpad("dimensions", 14) * " =" * join(lpad.(dimensions, 4), "")
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]
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@info join(info_strs, "\n")
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end
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@assert dot(multiplicities, dimensions) == size(RG.pm,1)
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return BlockDecomposition(orbs, preps, Uπs, dimensions)
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end
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function decimate(od::BlockDecomposition; verbose=true)
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nzros = [i for i in 1:length(od.Uπs) if !isempty(od.Uπs[i])]
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Us = sparsify!.(od.Uπs, eps(Float64) * 1e4, verbose = verbose)[nzros]
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#dimensions of the corresponding Uπs:
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dims = od.dims[nzros]
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return BlockDecomposition(od.orbits, od.preps, Array{Float64}.(Us), dims)
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end
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function orthSVD(M::AbstractMatrix{T}) where {T<:AbstractFloat}
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fact = svd(convert(Matrix{T}, M))
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M_rank = sum(fact.S .> maximum(size(M)) * eps(T))
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return fact.U[:, 1:M_rank]
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end
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orbit_decomposition(
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G::Group,
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E::AbstractVector,
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rdict = GroupRings.reverse_dict(E);
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op = ^,
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) = orbit_decomposition(collect(G), E, rdict; op=op)
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function orbit_decomposition(elts::AbstractVector{<:GroupElem}, E::AbstractVector, rdict=GroupRings.reverse_dict(E); op=^)
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tovisit = trues(size(E));
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orbits = Vector{Vector{Int}}()
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orbit = zeros(Int, length(elts))
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for i in eachindex(E)
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if tovisit[i]
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g = E[i]
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Threads.@threads for j in eachindex(elts)
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orbit[j] = rdict[op(g, elts[j])]
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end
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tovisit[orbit] .= false
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push!(orbits, unique(orbit))
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end
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end
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return orbits
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end
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###############################################################################
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#
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# Sparsification
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#
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###############################################################################
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dens(M::SparseMatrixCSC) = nnz(M)/length(M)
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dens(M::AbstractArray) = count(!iszero, M)/length(M)
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function sparsify!(M::SparseMatrixCSC{Tv,Ti}, tol=eps(Tv); verbose=false) where {Tv,Ti}
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densM = dens(M)
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droptol!(M, tol)
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verbose && @info(
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"Sparsified density:",
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rpad(densM, 20),
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" → ",
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rpad(dens(M), 20),
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" ($(nnz(M)) non-zeros)"
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)
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return M
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end
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function sparsify!(M::AbstractArray{T}, tol=eps(T); verbose=false) where T
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densM = dens(M)
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clamp_small!(M, tol)
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if verbose
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@info("Sparsifying $(size(M))-matrix... \n $(rpad(densM, 20)) → $(rpad(dens(M),20))), ($(count(!iszero, M)) non-zeros)")
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end
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return sparse(M)
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end
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function clamp_small!(M::AbstractArray{T}, tol=eps(T)) where T
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for n in eachindex(M)
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if abs(M[n]) < tol
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M[n] = zero(T)
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end
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end
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return M
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end
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function sparsify(U::AbstractArray{T}, tol=eps(T); verbose=false) where T
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return sparsify!(deepcopy(U), tol, verbose=verbose)
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end
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###############################################################################
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#
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# perm-, matrix-, representations
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#
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###############################################################################
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function perm_repr(g::GroupElem, E::Vector, E_dict)
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p = Vector{Int}(undef, length(E))
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for (i,elt) in enumerate(E)
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p[i] = E_dict[elt^g]
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end
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return p
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end
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function perm_reps(G::Group, E::Vector, E_rdict=GroupRings.reverse_dict(E))
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elts = collect(G)
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l = length(elts)
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preps = Vector{Generic.Perm}(undef, l)
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permG = SymmetricGroup(length(E))
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Threads.@threads for i in 1:l
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preps[i] = permG(PropertyT.perm_repr(elts[i], E, E_rdict), false)
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end
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return Dict(elts[i]=>preps[i] for i in 1:l)
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end
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function matrix_repr(x::GroupRingElem, mreps::Dict)
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nzeros = findall(!iszero, x.coeffs)
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return sum(x[i].*mreps[parent(x).basis[i]] for i in nzeros)
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end
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function matrix_reps(preps::Dict{T,Generic.Perm{I}}) where {T<:GroupElem, I<:Integer}
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kk = collect(keys(preps))
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mreps = Vector{SparseMatrixCSC{Float64, Int}}(undef, length(kk))
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Threads.@threads for i in 1:length(kk)
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mreps[i] = AbstractAlgebra.matrix_repr(preps[kk[i]])
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end
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return Dict(kk[i] => mreps[i] for i in 1:length(kk))
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end
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###############################################################################
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#
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# actions
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#
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###############################################################################
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function Base.:^(y::GroupRingElem, g::GroupRingElem, op = ^)
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res = parent(y)()
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for elt in GroupRings.supp(g)
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res += g[elt] * ^(y, elt, op)
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end
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return res
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end
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function Base.:^(y::GroupRingElem, g::GroupElem, op = ^)
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RG = parent(y)
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result = zero(RG, eltype(y.coeffs))
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for (idx, c) in enumerate(y.coeffs)
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if !iszero(c)
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result[op(RG.basis[idx], g)] = c
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end
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end
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return result
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end
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function Base.:^(
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y::GroupRingElem{T,<:SparseVector},
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g::GroupElem,
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op = ^,
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) where {T}
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RG = parent(y)
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index = [RG.basis_dict[op(RG.basis[idx], g)] for idx in y.coeffs.nzind]
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result = GroupRingElem(sparsevec(index, y.coeffs.nzval, y.coeffs.n), RG)
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return result
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end
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###############################################################################
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#
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# perm && WreathProductElems actions: MatAlgElem
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#
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###############################################################################
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function Base.:^(A::MatAlgElem, p::Generic.Perm)
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length(p.d) == size(A, 1) == size(A, 2) ||
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throw("Can't act via $p on matrix of size $(size(A))")
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result = similar(A)
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@inbounds for i = 1:size(A, 1)
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for j = 1:size(A, 2)
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result[p[i], p[j]] = A[i, j] # action by permuting rows and colums/conjugation
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end
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end
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return result
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end
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function Base.:^(A::MatAlgElem, g::WreathProductElem{N}) where {N}
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# @assert N == size(A,1) == size(A,2)
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flips = ntuple(i -> (g.n[i].d[1] == 1 && g.n[i].d[2] == 2 ? 1 : -1), N)
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result = similar(A)
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R = base_ring(parent(A))
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tmp = R(1)
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@inbounds for i = 1:size(A, 1)
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for j = 1:size(A, 2)
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x = A[i, j]
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if flips[i] * flips[j] == 1
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result[g.p[i], g.p[j]] = x
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else
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result[g.p[i], g.p[j]] = -x
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end
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end
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end
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return result
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end
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###############################################################################
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#
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# perm && WreathProductElems actions: Automorphism
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#
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###############################################################################
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function Base.:^(a::Automorphism, g::GroupElem)
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Ag = parent(a)(g)
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Ag_inv = inv(Ag)
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res = append!(Ag, a, Ag_inv)
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return Groups.freereduce!(res)
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end
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(A::AutGroup)(p::Generic.Perm) = A(Groups.AutSymbol(p))
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function (A::AutGroup)(g::WreathProductElem)
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isa(A.objectGroup, FreeGroup) || throw("Not an Aut(Fₙ)")
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parent(g).P.n == length(A.objectGroup.gens) ||
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throw("No natural embedding of $(parent(g)) into $A")
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elt = one(A)
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Id = one(parent(g.n.elts[1]))
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for i = 1:length(g.p.d)
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if g.n.elts[i] != Id
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push!(elt, Groups.flip(i))
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end
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end
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push!(elt, Groups.AutSymbol(g.p))
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return elt
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end
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# fallback:
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Base.one(p::Generic.Perm) = Perm(length(p.d))
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