Initial AutF4 work (naive basis, etc)

AutF4.jl

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+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::Int) = [nthperm(collect(1:n), k) for k in 1:factorial(n)]
+
+# const SymmetricGroup = [permutation_matrix(x) for x in SymmetricGroup_perms]
+
+function E(i::Int, j::Int; dim::Int=N)
+    @assert i≠j
+    k = eye(dim)
+    k[i,j] = 1
+    return k
+end
+
+function eltary_basis_vector(i::Int; dim::Int=N)
+    result = zeros(dim)
+    if 0 < i ≤ dim
+        result[i] = 1
+    end
+    return result
+end
+
+v(i::Int; dim=N) = eltary_basis_vector(i,dim=dim)
+
+ϱ(i::Int,j::Int,n=N) = SemiDirectProductElement(E(i,j,dim=n), v(j,dim=n))
+λ(i::Int,j::Int,n=N) = SemiDirectProductElement(E(i,j,dim=n), -v(j,dim=n))
+
+function ɛ(i::Int, n::Int=N)
+    result = eye(n)
+    result[i,i] = -1
+    return SemiDirectProductElement(result)
+end
+
+σ(permutation::Vector{Int}) =
+     SemiDirectProductElement(permutation_matrix(permutation))
+
+function AutF_generating_set(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
+
+const ID = eye(N+1)
+
+const S₁ = AutF_generating_set(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)