diff --git a/AutF4.jl b/AutF4.jl deleted file mode 100644 index 32e15ab..0000000 --- a/AutF4.jl +++ /dev/null @@ -1,120 +0,0 @@ -using Combinatorics - -using JuMP -import SCS: SCSSolver -import Mosek: MosekSolver - -push!(LOAD_PATH, "./") -using SemiDirectProduct -using GroupAlgebras -include("property(T).jl") - -const N = 4 - -const VERBOSE = true - -function permutation_matrix(p::Vector{Int}) - n = length(p) - sort(p) == collect(1:n) || throw(ArgumentError("Input array must be a permutation of 1:n")) - A = eye(n) - return A[p,:] -end - -SymmetricGroup(n) = [nthperm(collect(1:n), k) for k in 1:factorial(n)] - -# const SymmetricGroup = [permutation_matrix(x) for x in SymmetricGroup_perms] - -function E(i, j; dim::Int=N) - @assert i≠j - k = eye(dim) - k[i,j] = 1 - return k -end - -function eltary_basis_vector(i; dim::Int=N) - result = zeros(dim) - if 0 < i ≤ dim - result[i] = 1 - end - return result -end - -v(i; dim=N) = eltary_basis_vector(i,dim=dim) - -ϱ(i,j::Int,n=N) = SemiDirectProductElement(E(i,j,dim=n), v(j,dim=n)) -λ(i,j::Int,n=N) = SemiDirectProductElement(E(i,j,dim=n), -v(j,dim=n)) - -function ɛ(i, n::Int=N) - result = eye(n) - result[i,i] = -1 - return SemiDirectProductElement(result) -end - -σ(permutation::Vector{Int}) = - SemiDirectProductElement(permutation_matrix(permutation)) - -# Standard generating set: 103 elements - -function generatingset_ofAutF(n::Int=N) - indexing = [[i,j] for i in 1:n for j in 1:n if i≠j] - ϱs = [ϱ(ij...) for ij in indexing] - λs = [λ(ij...) for ij in indexing] - ɛs = [ɛ(i) for i in 1:N] - σs = [σ(perm) for perm in SymmetricGroup(n)] - S = vcat(ϱs, λs, ɛs, σs); - S = unique(vcat(S, [inv(x) for x in S])); - return S -end - -#= -Note that the element - α(i,j,k) = ϱ(i,j)*ϱ(i,k)*inv(ϱ(i,j))*inv(ϱ(i,k)), -which surely belongs to ball of radius 4 in Aut(F₄) becomes trivial under the representation - Aut(F₄) → GL₄(ℤ)⋉ℤ⁴ → GL₅(ℂ). -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. - -We need a different approach! -=# - -const ID = eye(N+1) - -const S₁ = generatingset_ofAutF(N) - -matrix_S₁ = [matrix_repr(x) for x in S₁] - -const TOL=10.0^-7 - -matrix_S₁[1:10,:][:,1] - -Δ, cm = prepare_Laplacian_and_constraints(matrix_S₁) - -#solver = SCSSolver(eps=TOL, max_iters=ITERATIONS, verbose=true); -solver = MosekSolver(MSK_DPAR_INTPNT_CO_TOL_REL_GAP=TOL, -# MSK_DPAR_INTPNT_CO_TOL_PFEAS=1e-15, -# MSK_DPAR_INTPNT_CO_TOL_DFEAS=1e-15, -# MSK_IPAR_PRESOLVE_USE=0, - QUIET=!VERBOSE) - -# κ, A = solve_for_property_T(S₁, solver, verbose=VERBOSE) - -product_matrix = readdlm("SL₃Z.product_matrix", Int) -L = readdlm("SL₃Z.Δ.coefficients")[:, 1] -Δ = GroupAlgebraElement(L, product_matrix) - -A = readdlm("matrix.A.Mosek") -κ = readdlm("kappa.Mosek")[1] - -# @show eigvals(A) -@assert isapprox(eigvals(A), abs(eigvals(A)), atol=TOL) -@assert A == Symmetric(A) - - -const A_sqrt = real(sqrtm(A)) - -SOS_EOI_fp_L₁, Ω_fp_dist = check_solution(κ, A_sqrt, Δ) - -κ_rational = rationalize(BigInt, κ;) -A_sqrt_rational = rationalize(BigInt, A_sqrt) -Δ_rational = rationalize(BigInt, Δ) - -SOS_EOI_rat_L₁, Ω_rat_dist = check_solution(κ_rational, A_sqrt_rational, Δ_rational)