121 lines
3.2 KiB
Julia
121 lines
3.2 KiB
Julia
using Combinatorics
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using JuMP
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import SCS: SCSSolver
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import Mosek: MosekSolver
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push!(LOAD_PATH, "./")
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using SemiDirectProduct
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using GroupAlgebras
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include("property(T).jl")
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const N = 4
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const VERBOSE = true
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function permutation_matrix(p::Vector{Int})
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n = length(p)
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sort(p) == collect(1:n) || throw(ArgumentError("Input array must be a permutation of 1:n"))
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A = eye(n)
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return A[p,:]
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end
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SymmetricGroup(n) = [nthperm(collect(1:n), k) for k in 1:factorial(n)]
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# const SymmetricGroup = [permutation_matrix(x) for x in SymmetricGroup_perms]
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function E(i, j; dim::Int=N)
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@assert i≠j
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k = eye(dim)
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k[i,j] = 1
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return k
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end
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function eltary_basis_vector(i; dim::Int=N)
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result = zeros(dim)
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if 0 < i ≤ dim
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result[i] = 1
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end
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return result
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end
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v(i; dim=N) = eltary_basis_vector(i,dim=dim)
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ϱ(i,j::Int,n=N) = SemiDirectProductElement(E(i,j,dim=n), v(j,dim=n))
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λ(i,j::Int,n=N) = SemiDirectProductElement(E(i,j,dim=n), -v(j,dim=n))
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function ɛ(i, n::Int=N)
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result = eye(n)
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result[i,i] = -1
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return SemiDirectProductElement(result)
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end
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σ(permutation::Vector{Int}) =
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SemiDirectProductElement(permutation_matrix(permutation))
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# Standard generating set: 103 elements
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function generatingset_ofAutF(n::Int=N)
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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 = [ϱ(ij...) for ij in indexing]
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λs = [λ(ij...) for ij in indexing]
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ɛs = [ɛ(i) for i in 1:N]
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σs = [σ(perm) for perm in SymmetricGroup(n)]
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S = vcat(ϱs, λs, ɛs, σs);
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S = unique(vcat(S, [inv(x) for x in S]));
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return S
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end
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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!
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=#
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const ID = eye(N+1)
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const S₁ = generatingset_ofAutF(N)
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matrix_S₁ = [matrix_repr(x) for x in S₁]
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const TOL=10.0^-7
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matrix_S₁[1:10,:][:,1]
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Δ, cm = prepare_Laplacian_and_constraints(matrix_S₁)
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#solver = SCSSolver(eps=TOL, max_iters=ITERATIONS, verbose=true);
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solver = MosekSolver(MSK_DPAR_INTPNT_CO_TOL_REL_GAP=TOL,
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# MSK_DPAR_INTPNT_CO_TOL_PFEAS=1e-15,
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# MSK_DPAR_INTPNT_CO_TOL_DFEAS=1e-15,
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# MSK_IPAR_PRESOLVE_USE=0,
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QUIET=!VERBOSE)
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# κ, A = solve_for_property_T(S₁, solver, verbose=VERBOSE)
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product_matrix = readdlm("SL₃Z.product_matrix", Int)
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L = readdlm("SL₃Z.Δ.coefficients")[:, 1]
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Δ = GroupAlgebraElement(L, product_matrix)
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A = readdlm("matrix.A.Mosek")
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κ = readdlm("kappa.Mosek")[1]
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# @show eigvals(A)
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@assert isapprox(eigvals(A), abs(eigvals(A)), atol=TOL)
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@assert A == Symmetric(A)
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const A_sqrt = real(sqrtm(A))
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SOS_EOI_fp_L₁, Ω_fp_dist = check_solution(κ, A_sqrt, Δ)
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κ_rational = rationalize(BigInt, κ;)
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A_sqrt_rational = rationalize(BigInt, A_sqrt)
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Δ_rational = rationalize(BigInt, Δ)
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SOS_EOI_rat_L₁, Ω_rat_dist = check_solution(κ_rational, A_sqrt_rational, Δ_rational)
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