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183
AutFN.jl
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183
AutFN.jl
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@ -0,0 +1,183 @@
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using JLD
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using JuMP
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import SCS: SCSSolver
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import Mosek: MosekSolver
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using Groups
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using ProgressMeter
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#=
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Note that the element
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α(i,j,k) = ϱ(i,j)*ϱ(i,k)*inv(ϱ(i,j))*inv(ϱ(i,k)),
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which surely belongs to ball of radius 4 in Aut(F₄) becomes trivial under the representation
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Aut(F₄) → GL₄(ℤ)⋉ℤ⁴ → GL₅(ℂ).
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Moreover, due to work of Potapchik and Rapinchuk [1] every real representation of Aut(Fₙ) into GLₘ(ℂ) (for m ≤ 2n-2) factors through GLₙ(ℤ)⋉ℤⁿ, so will have the same problem.
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We need a different approach: Here we actually compute in Aut(𝔽₄)
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=#
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import Combinatorics.nthperm
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SymmetricGroup(n) = [nthperm(collect(1:n), k) for k in 1:factorial(n)]
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function generating_set_of_AutF(N::Int)
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indexing = [[i,j] for i in 1:N for j in 1:N if i≠j]
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σs = [symmetric_AutSymbol(perm) for perm in SymmetricGroup(N)[2:end]];
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ϱs = [rmul_AutSymbol(i,j) for (i,j) in indexing]
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λs = [lmul_AutSymbol(i,j) for (i,j) in indexing]
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ɛs = [flip_AutSymbol(i) for i in 1:N];
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S = vcat(ϱs,λs)
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S = vcat(S..., σs..., ɛs)
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S = vcat(S..., [inv(g) for g in S])
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return Vector{AutWord}(unique(S))
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end
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function generating_set_of_OutF(N::Int)
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indexing = [[i,j] for i in 1:N for j in 1:N if i≠j]
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ϱs = [rmul_AutSymbol(i,j) for (i,j) in indexing]
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λs = [lmul_AutSymbol(i,j) for (i,j) in indexing]
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ɛs = [flip_AutSymbol(i) for i in 1:N];
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S = ϱs
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push!(S, λs..., ɛs...)
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push!(S,[inv(g) for g in S]...)
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return Vector{AutWord}(unique(S))
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end
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function generating_set_of_Sym(N::Int)
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σs = [symmetric_AutSymbol(perm) for perm in SymmetricGroup(N)[2:end]];
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S = σs
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push!(S, [inv(s) for s in S]...)
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return Vector{AutWord}(unique(S))
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end
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function products(S1::Vector{AutWord}, S2::Vector{AutWord})
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result = Vector{AutWord}()
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seen = Set{Vector{FGWord}}()
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n = length(S1)
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p = Progress(n, 1, "Computing complete products...", 50)
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for (i,x) in enumerate(S1)
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for y in S2
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z::AutWord = x*y
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v::Vector{FGWord} = z(domain)
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if !in(v, seen)
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push!(seen, v)
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push!(result, z)
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end
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end
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next!(p)
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end
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return result
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end
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function products_images(S1::Vector{AutWord}, S2::Vector{AutWord})
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result = Vector{Vector{FGWord}}()
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seen = Set{Vector{FGWord}}()
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n = length(S1)
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p = Progress(n, 1, "Computing images of elts in B₄...", 50)
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for (i,x) in enumerate(S1)
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z = x(domain)
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for y in S2
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v = y(z)
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if !in(v, seen)
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push!(seen, v)
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push!(result, v)
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end
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end
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next!(p)
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end
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return result
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end
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function hashed_product{T}(image::T, B, images_dict::Dict{T, Int})
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n = size(B,1)
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column = zeros(Int,n)
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Threads.@threads for j in 1:n
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w = (B[j])(image)
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k = images_dict[w]
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k ≠ 0 || throw(ArgumentError(
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"($i,$j): $(x^-1)*$y don't seem to be supported on basis!"))
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column[j] = k
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end
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return column
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end
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function create_product_matrix(basis::Vector{AutWord}, images)
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n = length(basis)
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product_matrix = zeros(Int, (n, n));
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print("Creating hashtable of images...")
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@time images_dict = Dict{Vector{FGWord}, Int}(x => i
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for (i,x) in enumerate(images))
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p = Progress(n, 1, "Computing product matrix in basis...", 50)
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for i in 1:n
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z = (inv(basis[i]))(domain)
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product_matrix[i,:] = hashed_product(z, basis, images_dict)
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next!(p)
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end
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return product_matrix
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end
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function ΔandSDPconstraints(identity::AutWord, S::Vector{AutWord})
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println("Generating Balls of increasing radius...")
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@time B₁ = vcat([identity], S)
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@time B₂ = products(B₁,B₁);
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@show length(B₂)
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if length(B₂) != length(B₁)
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@time B₃ = products(B₁, B₂)
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@show length(B₃)
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if length(B₃) != length(B₂)
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@time B₄_images = products_images(B₁, B₃)
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else
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B₄_images = unique([f(domain) for f in B₃])
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end
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else
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B₃ = B₂
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B₄ = B₂
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B₄_images = unique([f(domain) for f in B₃])
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end
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@show length(B₄_images)
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# @assert length(B₄_images) == 3425657
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println("Creating product matrix...")
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@time pm = create_product_matrix(B₂, B₄_images)
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println("Creating sdp_constratints...")
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@time sdp_constraints = constraints_from_pm(pm)
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L_coeff = splaplacian_coeff(S, B₂, length(B₄_images))
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Δ = GroupAlgebraElement(L_coeff, Array{Int,2}(pm))
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return Δ, sdp_constraints
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end
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@everywhere push!(LOAD_PATH, "./")
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using GroupAlgebras
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include("property(T).jl")
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const symbols = [FGSymbol("x₁",1), FGSymbol("x₂",1), FGSymbol("x₃",1), FGSymbol("x₄",1), FGSymbol("x₅",1), FGSymbol("x₆",1)]
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const TOL=1e-8
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const N = 4
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const domain = Vector{FGWord}(symbols[1:N])
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const ID = one(AutWord)
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# const name = "SYM$N"
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# const upper_bound=factorial(N)-TOL^(1/5)
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# S() = generating_set_of_Sym(N)
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# name = "AutF$N"
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# S() = generating_set_of_AutF(N)
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name = "OutF$N"
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S() = generating_set_of_OutF(N)
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const upper_bound=0.05
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BLAS.set_num_threads(4)
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@time check_property_T(name, ID, S; verbose=true, tol=TOL, upper_bound=upper_bound)
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133
GroupAlgebras.jl
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133
GroupAlgebras.jl
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@ -0,0 +1,133 @@
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module GroupAlgebras
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import Base: convert, show, isequal, ==
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import Base: +, -, *, //
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import Base: size, length, norm, rationalize
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export GroupAlgebraElement
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immutable GroupAlgebraElement{T<:Number}
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coefficients::AbstractVector{T}
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product_matrix::Array{Int,2}
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# basis::Array{Any,1}
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function GroupAlgebraElement(coefficients::AbstractVector,
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product_matrix::Array{Int,2})
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size(product_matrix, 1) == size(product_matrix, 2) ||
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throw(ArgumentError("Product matrix has to be square"))
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new(coefficients, product_matrix)
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end
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end
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# GroupAlgebraElement(c,pm,b) = GroupAlgebraElement(c,pm)
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GroupAlgebraElement{T}(c::AbstractVector{T},pm) = GroupAlgebraElement{T}(c,pm)
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convert{T<:Number}(::Type{T}, X::GroupAlgebraElement) =
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GroupAlgebraElement(convert(AbstractVector{T}, X.coefficients), X.product_matrix)
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show{T}(io::IO, X::GroupAlgebraElement{T}) = print(io,
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"Element of Group Algebra over $T of length $(length(X)):\n $(X.coefficients)")
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function isequal{T, S}(X::GroupAlgebraElement{T}, Y::GroupAlgebraElement{S})
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if T != S
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warn("Comparing elements with different coefficients Rings!")
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end
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X.product_matrix == Y.product_matrix || return false
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X.coefficients == Y.coefficients || return false
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return true
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end
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(==)(X::GroupAlgebraElement, Y::GroupAlgebraElement) = isequal(X,Y)
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function add{T<:Number}(X::GroupAlgebraElement{T}, Y::GroupAlgebraElement{T})
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X.product_matrix == Y.product_matrix || throw(ArgumentError(
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"Elements don't seem to belong to the same Group Algebra!"))
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return GroupAlgebraElement(X.coefficients+Y.coefficients, X.product_matrix)
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end
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function add{T<:Number, S<:Number}(X::GroupAlgebraElement{T},
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Y::GroupAlgebraElement{S})
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warn("Adding elements with different base rings!")
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return GroupAlgebraElement(+(promote(X.coefficients, Y.coefficients)...),
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X.product_matrix)
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end
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(+)(X::GroupAlgebraElement, Y::GroupAlgebraElement) = add(X,Y)
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(-)(X::GroupAlgebraElement) = GroupAlgebraElement(-X.coefficients, X.product_matrix)
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(-)(X::GroupAlgebraElement, Y::GroupAlgebraElement) = add(X,-Y)
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function algebra_multiplication{T<:Number}(X::AbstractVector{T}, Y::AbstractVector{T}, pm::Array{Int,2})
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result = zeros(X)
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for (j,y) in enumerate(Y)
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if y != zero(T)
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for (i, index) in enumerate(pm[:,j])
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if X[i] != zero(T)
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index == 0 && throw(ArgumentError("The product don't seem to belong to the span of basis!"))
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result[index] += X[i]*y
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end
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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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function group_star_multiplication{T<:Number}(X::GroupAlgebraElement{T},
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Y::GroupAlgebraElement{T})
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X.product_matrix == Y.product_matrix || ArgumentError(
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"Elements don't seem to belong to the same Group Algebra!")
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result = algebra_multiplication(X.coefficients, Y.coefficients, X.product_matrix)
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return GroupAlgebraElement(result, X.product_matrix)
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end
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function group_star_multiplication{T<:Number, S<:Number}(
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X::GroupAlgebraElement{T},
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Y::GroupAlgebraElement{S})
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S == T || warn("Multiplying elements with different base rings!")
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return group_star_multiplication(promote(X,Y)...)
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end
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(*){T<:Number, S<:Number}(X::GroupAlgebraElement{T},
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Y::GroupAlgebraElement{S}) = group_star_multiplication(X,Y);
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(*){T<:Number}(a::T, X::GroupAlgebraElement{T}) = GroupAlgebraElement(
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a*X.coefficients, X.product_matrix)
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function scalar_multiplication{T<:Number, S<:Number}(a::T,
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X::GroupAlgebraElement{S})
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promote_type(T,S) == S || warn("Scalar and coefficients are in different rings! Promoting result to $(promote_type(T,S))")
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return GroupAlgebraElement(a*X.coefficients, X.product_matrix)
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end
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(*){T<:Number}(a::T,X::GroupAlgebraElement) = scalar_multiplication(a, X)
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//{T<:Rational, S<:Rational}(X::GroupAlgebraElement{T}, a::S) =
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GroupAlgebraElement(X.coefficients//a, X.product_matrix)
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//{T<:Rational, S<:Integer}(X::GroupAlgebraElement{T}, a::S) =
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X//convert(T,a)
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length(X::GroupAlgebraElement) = length(X.coefficients)
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size(X::GroupAlgebraElement) = size(X.coefficients)
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function norm(X::GroupAlgebraElement, p=2)
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if p == 1
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return sum(abs(X.coefficients))
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elseif p == Inf
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return max(abs(X.coefficients))
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else
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return norm(X.coefficients, p)
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end
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end
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ɛ(X::GroupAlgebraElement) = sum(X.coefficients)
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function rationalize{T<:Integer, S<:Number}(
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::Type{T}, X::GroupAlgebraElement{S}; tol=eps(S))
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v = rationalize(T, X.coefficients, tol=tol)
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return GroupAlgebraElement(v, X.product_matrix)
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end
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end
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142
SL3Z.jl
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142
SL3Z.jl
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@ -0,0 +1,142 @@
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using JLD
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using JuMP
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import Primes: isprime
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import SCS: SCSSolver
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import Mosek: MosekSolver
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using Mods
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using Groups
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function SL_generatingset(n::Int)
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indexing = [(i,j) for i in 1:n for j in 1:n if i≠j]
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S = [E(i,j,N=n) for (i,j) in indexing];
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S = vcat(S, [convert(Array{Int,2},x') for x in S]);
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S = vcat(S, [convert(Array{Int,2},inv(x)) for x in S]);
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return unique(S)
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end
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function E(i::Int, j::Int; val=1, N::Int=3, mod=Inf)
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@assert i≠j
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m = eye(Int, N)
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m[i,j] = val
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if mod == Inf
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return m
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else
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return [Mod(x,mod) for x in m]
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end
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end
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function cofactor(i,j,M)
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z1 = ones(Bool,size(M,1))
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z1[i] = false
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z2 = ones(Bool,size(M,2))
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z2[j] = false
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return M[z1,z2]
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end
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import Base.LinAlg.det
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function det(M::Array{Mod,2})
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if size(M,1) ≠ size(M,2)
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d = Mod(0,M[1,1].mod)
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elseif size(M,1) == 2
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d = M[1,1]*M[2,2] - M[1,2]*M[2,1]
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else
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d = zero(eltype(M))
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for i in 1:size(M,1)
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d += (-1)^(i+1)*M[i,1]*det(cofactor(i,1,M))
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end
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end
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# @show (M, d)
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return d
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end
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function adjugate(M)
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K = similar(M)
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for i in 1:size(M,1), j in 1:size(M,2)
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K[j,i] = (-1)^(i+j)*det(cofactor(i,j,M))
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end
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return K
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end
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import Base: inv, one, zero, *
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one(::Type{Mod}) = 1
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zero(::Type{Mod}) = 0
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zero(x::Mod) = Mod(x.mod)
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function inv(M::Array{Mod,2})
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d = det(M)
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d ≠ 0*d || thow(ArgumentError("Matrix is not invertible!"))
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return inv(det(M))*adjugate(M)
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return adjugate(M)
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end
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function SL_generatingset(n::Int, p::Int)
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(p > 1 && n > 1) || throw(ArgumentError("Both n and p should be integers!"))
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isprime(p) || throw(ArgumentError("p should be a prime number!"))
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indexing = [(i,j) for i in 1:n for j in 1:n if i≠j]
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S = [E(i,j, N=n, mod=p) for (i,j) in indexing]
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S = vcat(S, [inv(s) for s in S])
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S = vcat(S, [permutedims(x, [2,1]) for x in S]);
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return unique(S)
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end
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function products{T}(U::AbstractVector{T}, V::AbstractVector{T})
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result = Vector{T}()
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for u in U
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for v in V
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push!(result, u*v)
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end
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end
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return unique(result)
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end
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function ΔandSDPconstraints(identity, S)
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B₁ = vcat([identity], S)
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B₂ = products(B₁, B₁);
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B₃ = products(B₁, B₂);
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B₄ = products(B₁, B₃);
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@assert B₄[1:length(B₂)] == B₂
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product_matrix = create_product_matrix(B₄,length(B₂));
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sdp_constraints = constraints_from_pm(product_matrix, length(B₄))
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L_coeff = splaplacian_coeff(S, B₂, length(B₄));
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Δ = GroupAlgebraElement(L_coeff, product_matrix)
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return Δ, sdp_constraints
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end
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@everywhere push!(LOAD_PATH, "./")
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using GroupAlgebras
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include("property(T).jl")
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const N = 3
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const name = "SL$(N)Z"
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const ID = eye(Int, N)
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S() = SL_generatingset(N)
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const upper_bound=0.27
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# const p = 7
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# const upper_bound=0.738 # (N,p) = (3,7)
|
||||
|
||||
# const name = "SL($N,$p)"
|
||||
# const ID = [Mod(x,p) for x in eye(Int,N)]
|
||||
# S() = SL_generatingset(N, p)
|
||||
|
||||
BLAS.set_num_threads(4)
|
||||
@time check_property_T(name, ID, S; verbose=true, tol=1e-10, upper_bound=upper_bound)
|
||||
Loading…
Reference in New Issue
Block a user