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add G₂ script
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179
scripts/G₂_Adj.jl
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179
scripts/G₂_Adj.jl
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@ -0,0 +1,179 @@
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using LinearAlgebra
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BLAS.set_num_threads(1)
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ENV["OMP_NUM_THREADS"] = 4
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using MKL_jll
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include(joinpath(@__DIR__, "../test/optimizers.jl"))
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using Groups
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import Groups.MatrixGroups
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using PropertyT
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using SymbolicWedderburn
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using SymbolicWedderburn.StarAlgebras
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using PermutationGroups
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include(joinpath(@__DIR__, "G₂_gens.jl"))
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G, roots, Weyl = G₂_roots_weyl()
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const HALFRADIUS = 2
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const UPPER_BOUND = Inf
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RG, S, sizes = @time PropertyT.group_algebra(G, halfradius = HALFRADIUS)
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Δ = RG(length(S)) - sum(RG(s) for s in S)
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wd = let Σ = Weyl, RG = RG
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act = PropertyT.AlphabetPermutation{eltype(Σ),Int64}(
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Dict(g => PermutationGroups.perm(g) for g in Σ),
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)
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@time SymbolicWedderburn.WedderburnDecomposition(
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Float64,
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Σ,
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act,
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basis(RG),
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StarAlgebras.Basis{UInt16}(@view basis(RG)[1:sizes[HALFRADIUS]]),
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semisimple = false,
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)
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end
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elt = Δ^2
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unit = Δ
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@time model, varP = PropertyT.sos_problem_primal(
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elt,
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unit,
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wd;
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upper_bound = UPPER_BOUND,
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augmented = true,
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show_progress = true,
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)
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warm = nothing
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begin
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@time status, warm = PropertyT.solve(
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model,
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scs_optimizer(;
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linear_solver = SCS.MKLDirectSolver,
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eps = 1e-10,
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max_iters = 20_000,
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accel = 50,
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alpha = 1.95,
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),
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warm,
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)
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@info "reconstructing the solution"
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Q = @time begin
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wd = wd
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Ps = [JuMP.value.(P) for P in varP]
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if any(any(isnan, P) for P in Ps)
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throw("solver was probably interrupted, no valid solution available")
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end
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Qs = real.(sqrt.(Ps))
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PropertyT.reconstruct(Qs, wd)
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end
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P = Q' * Q
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@info "certifying the solution"
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@time certified, λ = PropertyT.certify_solution(
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elt,
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unit,
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JuMP.objective_value(model),
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Q;
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halfradius = HALFRADIUS,
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augmented = true,
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)
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end
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### grading below
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function desubscriptify(symbol::Symbol)
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digits = [
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Int(l) - 0x2080 for
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l in reverse(string(symbol)) if 0 ≤ Int(l) - 0x2080 ≤ 9
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]
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res = 0
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for (i, d) in enumerate(digits)
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res += 10^(i - 1) * d
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end
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return res
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end
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function PropertyT.grading(g::MatrixGroups.MatrixElt, roots = roots)
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id = desubscriptify(g.id)
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return roots[id]
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end
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Δs = PropertyT.laplacians(
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RG,
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S,
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x -> (gx = PropertyT.grading(x); Set([gx, -gx])),
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)
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elt = PropertyT.Adj(Δs)
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elt == Δ^2 - PropertyT.Sq(Δs)
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unit = Δ
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@time model, varP = PropertyT.sos_problem_primal(
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elt,
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unit,
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wd;
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upper_bound = UPPER_BOUND,
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augmented = true,
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)
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warm = nothing
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begin
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@time status, warm = PropertyT.solve(
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model,
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scs_optimizer(;
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linear_solver = SCS.MKLDirectSolver,
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eps = 1e-10,
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max_iters = 50_000,
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accel = 50,
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alpha = 1.95,
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),
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warm,
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)
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@info "reconstructing the solution"
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Q = @time begin
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wd = wd
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Ps = [JuMP.value.(P) for P in varP]
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if any(any(isnan, P) for P in Ps)
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throw("solver was probably interrupted, no valid solution available")
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end
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Qs = real.(sqrt.(Ps))
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PropertyT.reconstruct(Qs, wd)
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end
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P = Q' * Q
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@info "certifying the solution"
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@time certified, λ = PropertyT.certify_solution(
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elt,
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unit,
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JuMP.objective_value(model),
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Q;
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halfradius = HALFRADIUS,
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augmented = true,
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)
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end
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# Δ² - 1 / 1 · Sq → -0.8818044647162608
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# Δ² - 2 / 3 · Sq → -0.1031738
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# Δ² - 1 / 2 · Sq → 0.228296213895906
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# Δ² - 1 / 3 · Sq → 0.520
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# Δ² - 0 / 1 · Sq → 0.9676851592000731
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# Sq → 0.333423
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# vals = [
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# 1.0 -0.8818
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# 2/3 -0.1032
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# 1/2 0.2282
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# 1/3 0.520
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# 0 0.9677
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# ]
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308
scripts/G₂_gens.jl
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308
scripts/G₂_gens.jl
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@ -0,0 +1,308 @@
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#= GAP code to generate matrices
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alg := SimpleLieAlgebra("G", 2, Rationals);
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root_sys := RootSystem(alg);
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pos_gens := PositiveRootVectors(root_sys);
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pos_rts := PositiveRoots(root_sys);
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neg_gens := NegativeRootVectors(root_sys);
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neg_rts := NegativeRoots(root_sys);
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alg_gens := ShallowCopy(pos_gens);;
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Append(alg_gens, neg_gens);
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grading := ShallowCopy(pos_rts);
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Append(grading, neg_rts);
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mats := List(alg_gens, x->AdjointMatrix(Basis(alg), x));
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W := WeylGroup(root_sys);
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PW := Action(W, grading, OnRight);
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=#
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using LinearAlgebra
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function matrix_exp(M::AbstractMatrix{<:Integer})
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res = zeros(Rational{eltype(M)}, size(M))
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res += I
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k = 0
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expM = one(M)
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while !iszero(expM)
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k += 1
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expM *= M
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@. res += 1 // factorial(k) * expM
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if k == 20
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@warn "matrix exponential did not converge" norm(expM - exp(M))
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break
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end
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end
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@debug "matrix_exp converged after $k iterations"
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return res
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end
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const gap_adj_mats = [
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[
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 1],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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],
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[
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, -2],
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[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0],
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],
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[
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[0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1],
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[2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0],
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],
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[
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[0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0],
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[3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0],
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],
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[
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[0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 1],
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[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0],
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],
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[
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0],
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],
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[
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
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[0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -1],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
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],
|
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[
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[0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
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[0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0],
|
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 2],
|
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[0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0],
|
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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],
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[
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[0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[-3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1],
|
||||
[0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0],
|
||||
[0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
],
|
||||
[
|
||||
[0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[-2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0],
|
||||
[0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
],
|
||||
[
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, -1],
|
||||
[0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
],
|
||||
[
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1],
|
||||
[0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
],
|
||||
]
|
||||
|
||||
function G₂_matrices_roots()
|
||||
adj_mats = map(gap_adj_mats) do m
|
||||
return hcat(m...)
|
||||
end
|
||||
adj_mats = filter!(!isdiag, adj_mats) # remove the ones from center
|
||||
|
||||
gens_mats = [convert(Matrix{Int}, matrix_exp(m')) for m in adj_mats]
|
||||
|
||||
#=
|
||||
The roots from
|
||||
|
||||
G₂roots_gap = [
|
||||
[2, -1], # α = e₁ - e₂
|
||||
[-3, 2], # A = -α + β = -e₁ + 2e₂ - e₃
|
||||
[-1, 1], # β = e₂ - e₃
|
||||
[1, 0], # α + β = e₁ - e₃
|
||||
[3, -1], # B = 2α + β = 2e₁ - e₂ - e₃
|
||||
[0, 1], # A + B = α + 2β = e₁ + e₂ - 2e₃
|
||||
[-2, 1], # -α
|
||||
[3, -2], # -A
|
||||
[1, -1], # -β
|
||||
[-1, 0], # -α - β
|
||||
[-3, 1], # -B
|
||||
[0, -1], # -A - B
|
||||
]
|
||||
|
||||
G₂roots_gap are the ones from cartan matrix. To obtain the standard
|
||||
(hexagonal) picture map them by `T` defined as follows:
|
||||
```julia
|
||||
cartan = hcat(G₂roots_gap[1:2]...)
|
||||
rot(α) = [cos(α) -sin(α); sin(α) cos(α)]
|
||||
|
||||
c₁ = [√2, 0]
|
||||
c₂ = rot(5π / 6) * [√2, 0] * √3 # (= 1/2[√6, 1])
|
||||
|
||||
T = hcat(c₁, c₂) * inv(cartan)
|
||||
```
|
||||
By plotting one against the others (or by blind calculation) one can see
|
||||
the following assignment. Here `⟨α, β⟩_ℤ = A₂` and `⟨A, B⟩_ℤ ≅ √3/√2 A₂`.
|
||||
=#
|
||||
e₁ = PropertyT.Roots.𝕖(3, 1)
|
||||
e₂ = PropertyT.Roots.𝕖(3, 2)
|
||||
e₃ = PropertyT.Roots.𝕖(3, 3)
|
||||
|
||||
α = e₁ - e₂
|
||||
β = e₂ - e₃
|
||||
A = -α + β
|
||||
B = α + (α + β)
|
||||
|
||||
roots = [α, A, β, α + β, B, A + B, -α, -A, -β, -α - β, -B, -A - B]
|
||||
|
||||
return gens_mats, roots
|
||||
end
|
||||
|
||||
function G₂_roots_weyl()
|
||||
(mats, roots) = G₂_matrices_roots()
|
||||
d = size(first(mats), 1)
|
||||
G₂ = Groups.MatrixGroup{d}(mats)
|
||||
|
||||
m = Groups.gens(G₂)
|
||||
|
||||
σ = let w = m[1] * inv(m[7]) * m[1], m = union(m, inv.(m))
|
||||
PermutationGroups.Perm([findfirst(==(inv(w) * x * w), m) for x in m])
|
||||
end
|
||||
|
||||
τ = let w = m[2] * inv(m[8]) * m[2], m = union(m, inv.(m))
|
||||
PermutationGroups.Perm([findfirst(==(inv(w) * x * w), m) for x in m])
|
||||
end
|
||||
|
||||
W = PermGroup(σ, τ)
|
||||
|
||||
return G₂, roots, W
|
||||
end
|
||||
Loading…
Reference in New Issue
Block a user