GroupsWithPropertyT/OrbitDecomposition.jl

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push!(LOAD_PATH, "./")
using Nemo
using Groups
using WreathProducts
using GroupRings
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using PropertyT
import Nemo.elements
using JLD
include("Projections.jl")
###############################################################################
#
# Iterator protocol for Nemo.FinField
#
###############################################################################
type FFEltsIter{T<:Nemo.FinField}
all::Int
field::T
function FFEltsIter(F::T)
return new(Int(characteristic(F)^degree(F)), F)
end
end
FFEltsIter{T<:Nemo.FinField}(F::T) = FFEltsIter{T}(F)
import Base: start, next, done, eltype, length
Base.start(A::FFEltsIter) = (zero(A.field), 0)
Base.next(A::FFEltsIter, state) = next_ffelem(state...)
Base.done(A::FFEltsIter, state) = state[2] >= A.all
Base.eltype(::Type{FFEltsIter}) = elem_type(A.field)
Base.length(A::FFEltsIter) = A.all
function next_ffelem(f::Nemo.FinFieldElem, c::Int)
if c == 0
return (f, (f, 1))
elseif c == 1
f = one(parent(f))
return (f, (f, 2))
else
f = gen(parent(f))*f
return (f, (f, c+1))
end
end
import Nemo.elements
elements(F::Nemo.FinField) = FFEltsIter(F)
###############################################################################
#
# Action of Premutations on Nemo.MatElem
#
###############################################################################
function (p::Nemo.perm)(A::Nemo.MatElem)
length(p.d) == A.r == A.c || throw("Can't act via $p on matrix of size ($(A.r), $(A.c))")
R = parent(A)
inv_p = inv(p)
return R(Nemo.matrix_repr(p))*A*R(Nemo.matrix_repr(inv_p))
end
###############################################################################
#
# Orbit stuff
#
###############################################################################
function orbit_decomposition(G::Nemo.Group, E::Vector, rdict=GroupRings.reverse_dict(E))
elts = collect(elements(G))
tovisit = trues(E);
orbits = Vector{Set{Int}}()
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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 orbit_spvector(vect::AbstractVector, orbits)
orb_vector = spzeros(length(orbits))
for (i,o) in enumerate(orbits)
k = vect[collect(o)]
val = k[1]
@assert all(k .== val)
orb_vector[i] = val
end
return orb_vector
end
function orbit_constraint(cnstrs::Vector{Vector{Vector{Int64}}}, n)
result = spzeros(n,n)
for cnstr in cnstrs
for p in cnstr
result[p[1],p[2]] += 1.0
end
end
return 1/length(cnstrs)*result
end
###############################################################################
#
# Matrix- and C*-representations
#
###############################################################################
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
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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 orthSVD(M::AbstractMatrix)
M = full(M)
fact = svdfact(M)
sings = fact[:S]
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}}()
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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 compute_orbit_data{T<:GroupElem}(logger, name::String, G::Group, S::Vector{T}, AutS; radius=2)
isdir(name) || mkdir(name)
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info(logger, "Generating ball of radius $(2*radius)")
if isa(G, Nemo.Ring)
Id = one(G)
else
Id = G()
end
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@time E4, sizes = Groups.generate_balls(S, Id, radius=2*radius);
info(logger, "Balls of sizes $sizes.")
info(logger, "Reverse dict")
@time E_dict = GroupRings.reverse_dict(E4)
info(logger, "Product matrix")
@time pm = GroupRings.create_pm(E4, E_dict, sizes[radius], twisted=true)
RG = GroupRing(G, E4, E_dict, pm)
Δ = PropertyT.splaplacian(RG, S)
@assert GroupRings.augmentation(Δ) == 0
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save(joinpath(name, "delta.jld"), "Δ", Δ.coeffs)
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save(joinpath(name, "pm.jld"), "pm", pm)
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info(logger, "Decomposing E into orbits of $(AutS)")
@time orbs = orbit_decomposition(AutS, E4, E_dict)
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@assert sum(length(o) for o in orbs) == length(E4)
save(joinpath(name, "orbits.jld"), "orbits", orbs)
info(logger, "Action matrices")
E2 = E4[1:sizes[radius]]
@time AutS_matrixreps = [matrix_repr(g, E2, E_dict) for g in elements(AutS)]
info(logger, "Projections")
@time AutS_mps = rankOne_projections(AutS);
@time π_E_projections = [Cstar_repr(p, AutS_matrixreps) for p in AutS_mps]
info(logger, "Uπs...")
@time Uπs = Uπ_matrices(π_E_projections);
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multiplicities = [size(U,2) for U in Uπs];
info(logger, "multiplicities = $multiplicities")
dimensions = [Int(p[AutS()]*Int(order(AutS))) for p in AutS_mps];
info(logger, "dimensions = $dimensions")
@assert dot(multiplicities, dimensions) == sizes[radius]
save(joinpath(name, "U_pis.jld"), "Uπs", Uπs, "dims", dimensions)
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return 0
end