replace BlockDecomposition by SymbolicWedderburn
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@ -4,23 +4,16 @@ authors = ["Marek Kaluba <kalmar@amu.edu.pl>"]
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version = "0.3.2"
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[deps]
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AbstractAlgebra = "c3fe647b-3220-5bb0-a1ea-a7954cac585d"
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Dates = "ade2ca70-3891-5945-98fb-dc099432e06a"
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GroupRings = "0befed6a-bd73-11e8-1e41-a1190947c2f5"
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Groups = "5d8bd718-bd84-11e8-3b40-ad14f4a32557"
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IntervalArithmetic = "d1acc4aa-44c8-5952-acd4-ba5d80a2a253"
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JLD = "4138dd39-2aa7-5051-a626-17a0bb65d9c8"
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JuMP = "4076af6c-e467-56ae-b986-b466b2749572"
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LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
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SCS = "c946c3f1-0d1f-5ce8-9dea-7daa1f7e2d13"
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SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
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SymbolicWedderburn = "858aa9a9-4c7c-4c62-b466-2421203962a2"
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[compat]
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AbstractAlgebra = "^0.10.0"
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GroupRings = "^0.3.2"
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Groups = "^0.5.0"
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IntervalArithmetic = "^0.16.0"
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JLD = "^0.9.0"
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JuMP = "^0.20.0"
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SCS = "^0.7.0"
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julia = "^1.3.0"
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@ -1,22 +1,16 @@
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__precompile__()
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module PropertyT
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using AbstractAlgebra
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using LinearAlgebra
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using SparseArrays
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using Dates
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using Groups
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using GroupRings
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using SymbolicWedderburn
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using JLD
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using JuMP
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import AbstractAlgebra: Group, NCRing
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include("laplacians.jl")
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include("RGprojections.jl")
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include("blockdecomposition.jl")
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include("sos_sdps.jl")
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include("checksolution.jl")
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@ -1,251 +0,0 @@
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module Projections
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using AbstractAlgebra
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using Groups
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using GroupRings
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export PermCharacter, DirectProdCharacter, rankOne_projections
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###############################################################################
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#
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# Characters of Symmetric Group and DirectProduct
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#
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###############################################################################
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abstract type AbstractCharacter end
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struct PermCharacter <: AbstractCharacter
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p::Generic.Partition
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end
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struct DirectProdCharacter{N, T<:AbstractCharacter} <: AbstractCharacter
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chars::NTuple{N, T}
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end
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function (chi::DirectProdCharacter)(g::DirectPowerGroupElem)
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res = 1
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for (χ, elt) in zip(chi.chars, g.elts)
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res *= χ(elt)
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end
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return res
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end
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function (chi::PermCharacter)(g::Generic.Perm)
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R = AbstractAlgebra.partitionseq(chi.p)
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p = Partition(Generic.permtype(g))
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return Int(Generic.MN1inner(R, p, 1, Generic._charvalsTable))
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end
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AbstractAlgebra.dim(χ::PermCharacter) = dim(YoungTableau(χ.p))
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for T in [PermCharacter, DirectProdCharacter]
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@eval begin
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function (chi::$T)(X::GroupRingElem)
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RG = parent(X)
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z = zero(eltype(X))
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result = z
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for i in 1:length(X.coeffs)
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if X.coeffs[i] != z
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result += chi(RG.basis[i])*X.coeffs[i]
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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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end
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end
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characters(G::Generic.SymmetricGroup) = (PermCharacter(p) for p in AllParts(G.n))
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function characters(G::DirectPowerGroup{N}) where N
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nfold_chars = Iterators.repeated(characters(G.group), N)
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return (DirectProdCharacter(idx) for idx in Iterators.product(nfold_chars...))
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end
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###############################################################################
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#
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# Projections
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#
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###############################################################################
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function central_projection(RG::GroupRing, chi::AbstractCharacter, T::Type=Rational{Int})
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result = RG(zeros(T, length(RG.basis)))
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dim = chi(one(RG.group))
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ord = Int(order(RG.group))
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for g in RG.basis
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result[g] = convert(T, (dim//ord)*chi(g))
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end
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return result
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end
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function alternating_emb(RG::GroupRing{Gr,T}, V::Vector{T}, S::Type=Rational{Int}) where {Gr<:AbstractAlgebra.AbstractPermutationGroup, T<:GroupElem}
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res = RG(S)
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for g in V
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res[g] += sign(g)
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end
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return res
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end
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function idempotents(RG::GroupRing{Generic.SymmetricGroup{S}}, T::Type=Rational{Int}) where S<:Integer
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if RG.group.n == 1
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return GroupRingElem{T}[one(RG,T)]
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elseif RG.group.n == 2
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Id = one(RG,T)
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transp = RG(perm"(1,2)", T)
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return GroupRingElem{T}[1//2*(Id + transp), 1//2*(Id - transp)]
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end
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projs = Vector{Vector{Generic.Perm{S}}}()
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for l in 2:RG.group.n
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u = RG.group([circshift([i for i in 1:l], -1); [i for i in l+1:RG.group.n]])
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i = 0
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while (l-1)*i <= RG.group.n
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v = RG.group(circshift(collect(1:RG.group.n), i))
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k = inv(v)*u*v
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push!(projs, generateGroup([k], RG.group.n))
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i += 1
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end
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end
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idems = Vector{GroupRingElem{T}}()
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for p in projs
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append!(idems, [RG(p, T)//length(p), alternating_emb(RG, p, T)//length(p)])
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end
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return unique(idems)
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end
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function rankOne_projection(chi::PermCharacter, idems::Vector{T}) where {T<:GroupRingElem}
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RG = parent(first(idems))
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S = eltype(first(idems))
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ids = [one(RG, S); idems]
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zzz = zero(S)
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for (i,j,k) in Base.product(ids, ids, ids)
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if chi(i) == zzz || chi(j) == zzz || chi(k) == zzz
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continue
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else
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elt = i*j*k
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if elt^2 != elt
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continue
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elseif chi(elt) == one(S)
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return elt
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# return (i,j,k)
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end
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end
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end
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throw("Couldn't find rank-one projection for $chi")
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end
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function rankOne_projections(RG::GroupRing{<:Generic.SymmetricGroup}, T::Type=Rational{Int})
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if RG.group.n == 1
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return [GroupRingElem([one(T)], RG)]
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end
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RGidems = idempotents(RG, T)
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min_projs = [central_projection(RG,chi)*rankOne_projection(chi,RGidems) for chi in characters(RG.group)]
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return min_projs
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end
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function ifelsetuple(a,b, k, n)
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x = [repeat([a], k); repeat([b], n-k)]
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return tuple(x...)
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end
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function orbit_selector(n::Integer, k::Integer,
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chi::AbstractCharacter, psi::AbstractCharacter)
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return Projections.DirectProdCharacter(ifelsetuple(chi, psi, k, n))
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end
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function rankOne_projections(RBn::GroupRing{G}, T::Type=Rational{Int}) where {G<:WreathProduct}
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Bn = RBn.group
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N = Bn.P.n
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# projections as elements of the group rings RSₙ
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Sn_rankOnePr = [rankOne_projections(
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GroupRing(SymmetricGroup(i), collect(SymmetricGroup(i))))
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for i in typeof(N)(1):N]
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# embedding into group ring of BN
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RN = GroupRing(Bn.N, collect(Bn.N))
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sign, id = collect(characters(Bn.N.group))
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# Bn.N = (Z/2Z)ⁿ characters corresponding to the first k coordinates:
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BnN_orbits = Dict(i => orbit_selector(N, i, sign, id) for i in 0:N)
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Q = Dict(i => RBn(g -> Bn(g), central_projection(RN, BnN_orbits[i], T)) for i in 0:N)
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Q = Dict(key => GroupRings.dense(val) for (key, val) in Q)
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all_projs = [Q[0]*RBn(g->Bn(g), p) for p in Sn_rankOnePr[N]]
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r = collect(1:N)
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for i in 1:N-1
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first_emb = g->Bn(Generic.emb!(one(Bn.P), g, view(r, 1:i)))
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last_emb = g->Bn(Generic.emb!(one(Bn.P), g, view(r, (i+1):N)))
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Sk_first = (RBn(first_emb, p) for p in Sn_rankOnePr[i])
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Sk_last = (RBn(last_emb, p) for p in Sn_rankOnePr[N-i])
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append!(all_projs,
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[Q[i]*p1*p2 for (p1,p2) in Base.product(Sk_first,Sk_last)])
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end
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append!(all_projs, [Q[N]*RBn(g->Bn(g), p) for p in Sn_rankOnePr[N]])
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return all_projs
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end
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##############################################################################
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#
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# General Groups Misc
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#
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##############################################################################
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"""
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products(X::Vector{GroupElem}, Y::Vector{GroupElem}[, op=*])
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Return a vector of all possible products (or `op(x,y)`), where `x ∈ X` and
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`y ∈ Y`. You may change which operation is used by specifying `op` argument.
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"""
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function products(X::AbstractVector{T}, Y::AbstractVector{T}, op=*) where {T<:GroupElem}
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result = Vector{T}()
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seen = Set{T}()
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sizehint!(result, length(X)*length(Y))
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sizehint!(seen, length(X)*length(Y))
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for x in X
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for y in Y
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z = op(x,y)
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if !in(z, seen)
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push!(seen, z)
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push!(result, z)
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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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generateGroup(gens::Vector{GroupElem}, r, [op=*])
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Produce all elements of a group generated by elements in `gens` in ball of
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radius `r` (word-length metric induced by `gens`). The binary operation can
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be optionally specified.
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"""
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function generateGroup(gens::Vector{T}, r, op=*) where {T<:GroupElem}
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n = 0
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R = 1
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elts = [one(first(gens)); gens]
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while n ≠ length(elts) && R < r
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# @show elts
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R += 1
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n = length(elts)
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elts = products(gens, elts, op)
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end
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return elts
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end
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end # of module Projections
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@ -1,287 +0,0 @@
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###############################################################################
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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)
|
||||
nzeros = findall(!iszero, x.coeffs)
|
||||
return sum(x[i].*mreps[parent(x).basis[i]] for i in nzeros)
|
||||
end
|
||||
|
||||
function matrix_reps(preps::Dict{T,Generic.Perm{I}}) where {T<:GroupElem, I<:Integer}
|
||||
kk = collect(keys(preps))
|
||||
mreps = Vector{SparseMatrixCSC{Float64, Int}}(undef, length(kk))
|
||||
Threads.@threads for i in 1:length(kk)
|
||||
mreps[i] = AbstractAlgebra.matrix_repr(preps[kk[i]])
|
||||
end
|
||||
return Dict(kk[i] => mreps[i] for i in 1:length(kk))
|
||||
end
|
||||
|
||||
###############################################################################
|
||||
#
|
||||
# actions
|
||||
#
|
||||
###############################################################################
|
||||
|
||||
function Base.:^(y::GroupRingElem, g::GroupRingElem, op = ^)
|
||||
res = parent(y)()
|
||||
for elt in GroupRings.supp(g)
|
||||
res += g[elt] * ^(y, elt, op)
|
||||
end
|
||||
return res
|
||||
end
|
||||
|
||||
function Base.:^(y::GroupRingElem, g::GroupElem, op = ^)
|
||||
RG = parent(y)
|
||||
result = zero(RG, eltype(y.coeffs))
|
||||
|
||||
for (idx, c) in enumerate(y.coeffs)
|
||||
if !iszero(c)
|
||||
result[op(RG.basis[idx], g)] = c
|
||||
end
|
||||
end
|
||||
return result
|
||||
end
|
||||
|
||||
function Base.:^(
|
||||
y::GroupRingElem{T,<:SparseVector},
|
||||
g::GroupElem,
|
||||
op = ^,
|
||||
) where {T}
|
||||
RG = parent(y)
|
||||
index = [RG.basis_dict[op(RG.basis[idx], g)] for idx in y.coeffs.nzind]
|
||||
|
||||
result = GroupRingElem(sparsevec(index, y.coeffs.nzval, y.coeffs.n), RG)
|
||||
|
||||
return result
|
||||
end
|
||||
|
||||
###############################################################################
|
||||
#
|
||||
# perm && WreathProductElems actions: MatAlgElem
|
||||
#
|
||||
###############################################################################
|
||||
|
||||
function Base.:^(A::MatAlgElem, p::Generic.Perm)
|
||||
length(p.d) == size(A, 1) == size(A, 2) ||
|
||||
throw("Can't act via $p on matrix of size $(size(A))")
|
||||
result = similar(A)
|
||||
@inbounds for i = 1:size(A, 1)
|
||||
for j = 1:size(A, 2)
|
||||
result[p[i], p[j]] = A[i, j] # action by permuting rows and colums/conjugation
|
||||
end
|
||||
end
|
||||
return result
|
||||
end
|
||||
|
||||
function Base.:^(A::MatAlgElem, g::WreathProductElem{N}) where {N}
|
||||
# @assert N == size(A,1) == size(A,2)
|
||||
flips = ntuple(i -> (g.n[i].d[1] == 1 && g.n[i].d[2] == 2 ? 1 : -1), N)
|
||||
result = similar(A)
|
||||
R = base_ring(parent(A))
|
||||
tmp = R(1)
|
||||
|
||||
@inbounds for i = 1:size(A, 1)
|
||||
for j = 1:size(A, 2)
|
||||
x = A[i, j]
|
||||
if flips[i] * flips[j] == 1
|
||||
result[g.p[i], g.p[j]] = x
|
||||
else
|
||||
result[g.p[i], g.p[j]] = -x
|
||||
end
|
||||
end
|
||||
end
|
||||
return result
|
||||
end
|
||||
|
||||
###############################################################################
|
||||
#
|
||||
# perm && WreathProductElems actions: Automorphism
|
||||
#
|
||||
###############################################################################
|
||||
|
||||
function Base.:^(a::Automorphism, g::GroupElem)
|
||||
Ag = parent(a)(g)
|
||||
Ag_inv = inv(Ag)
|
||||
res = append!(Ag, a, Ag_inv)
|
||||
return Groups.freereduce!(res)
|
||||
end
|
||||
|
||||
(A::AutGroup)(p::Generic.Perm) = A(Groups.AutSymbol(p))
|
||||
|
||||
function (A::AutGroup)(g::WreathProductElem)
|
||||
isa(A.objectGroup, FreeGroup) || throw("Not an Aut(Fₙ)")
|
||||
parent(g).P.n == length(A.objectGroup.gens) ||
|
||||
throw("No natural embedding of $(parent(g)) into $A")
|
||||
elt = one(A)
|
||||
Id = one(parent(g.n.elts[1]))
|
||||
for i = 1:length(g.p.d)
|
||||
if g.n.elts[i] != Id
|
||||
push!(elt, Groups.flip(i))
|
||||
end
|
||||
end
|
||||
push!(elt, Groups.AutSymbol(g.p))
|
||||
return elt
|
||||
end
|
||||
|
||||
|
||||
# fallback:
|
||||
Base.one(p::Generic.Perm) = Perm(length(p.d))
|
||||
Loading…
Reference in New Issue