118 lines
4.0 KiB
Julia
118 lines
4.0 KiB
Julia
using Groups
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@testset "FreeGroup algebra" begin
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New.star(g::Groups.New.GroupElement) = inv(g)
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F = Groups.New.FreeGroup(4)
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S = [gens(F); inv.(gens(F))]
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ID = one(F)
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RADIUS=3
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@time E_R, sizes = Groups.wlmetric_ball(S, ID, radius=2*RADIUS);
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@test sizes == [9, 65, 457, 3201, 22409, 156865]
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b = New.Basis{UInt32}(E_R)
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@testset "MTables" begin
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mstr = New.MTable{false}(b, table_size=(sizes[RADIUS], sizes[RADIUS]))
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@test mstr isa New.MTable{UInt32, false}
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@test all(mstr[i,i]≠1 for i in 2:size(mstr, 1))
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@test all(mstr[1,i]==i for i in 1:size(mstr, 2))
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@test all(mstr[i,1]==i for i in 1:size(mstr, 1))
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tmstr = New.MTable{true}(b, table_size=(sizes[RADIUS], sizes[RADIUS]))
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@test tmstr isa New.MTable{UInt32, true}
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@test all(tmstr[i,i]==1 for i in 1:size(tmstr, 1))
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@test all(tmstr[1,i]==i for i in 1:size(tmstr, 2))
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@test all(tmstr[i,1]≠ i for i in 2:size(tmstr, 1))
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end
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tmstr = New.MTable{true}(b, table_size=(sizes[RADIUS], sizes[RADIUS]))
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RG = New.StarAlgebra(F, b, tmstr)
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g, h, k, l = S[1:4]
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length(b)
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G = (one(RG)-RG(g))
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G
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@test G^2 == New.mul!(zero(G), G, G) == 2one(RG) - RG(g) - New.star(RG(g))
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@test New.star(G*G) == G*G
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@testset "Sums of hermitian squares" begin
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∗ = New.star
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𝕀 = one(RG)
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G = (𝕀 - RG(g))
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H = (𝕀 - RG(h))
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K = (𝕀 - RG(k))
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L = (𝕀 - RG(l))
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GH = (𝕀 - RG(g*h))
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KL = (𝕀 - RG(k*l))
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X = (2𝕀 - ∗(RG(g)) - RG(h))
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Y = (2𝕀 - ∗(RG(g*h)) - RG(k))
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@test -(2𝕀 - RG(g*h) - ∗(RG(g*h))) + 2G^2 + 2H^2 == X^2
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@test (2𝕀 - RG(g*h) - ∗(RG(g*h))) == GH^2
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@test -(2𝕀 - RG(g*h*k) - ∗(RG(g*h*k))) + 2GH^2 + 2K^2 == Y^2
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@test -(2𝕀 - RG(g*h*k) - ∗(RG(g*h*k))) +
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2(GH^2 - 2G^2 - 2H^2) +
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4G^2 + 4H^2 + 2K^2 ==
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Y^2
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@test GH^2 - 2G^2 - 2H^2 == - X^2
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@test -(2𝕀 - RG(g*h*k) - ∗(RG(g*h*k))) + 4G^2 + 4H^2 + 2K^2 == 2X^2 + Y^2
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@test GH^2 == 2G^2 + 2H^2 - (2𝕀 - ∗(RG(g)) - RG(h))^2
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@test KL^2 == 2K^2 + 2L^2 - (2𝕀 - ∗(RG(k)) - RG(l))^2
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@test -(2𝕀 - ∗(RG(g*h*k*l)) - RG(g*h*k*l)) + 2*GH^2 + 2*KL^2 ==
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(2𝕀 - ∗(RG(g*h)) - RG(k*l))^2
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@test -(2𝕀 - ∗(RG(g*h*k*l)) - RG(g*h*k*l)) +
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2(2G^2 + 2H^2 - (2𝕀 - ∗(RG(g)) - RG(h))^2) +
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2(2K^2 + 2L^2 - (2𝕀 - ∗(RG(k)) - RG(l))^2) ==
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(2𝕀 - ∗(RG(g*h)) - RG(k*l))^2
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@test -(2𝕀 - ∗(RG(g*h*k*l)) - RG(g*h*k*l)) +
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2(2G^2 + 2H^2) +
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2(2K^2 + 2L^2) ==
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(2𝕀 - ∗(RG(g*h)) - RG(k*l))^2 +
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2(2𝕀 - ∗(RG(g)) - RG(h))^2 +
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2(2𝕀 - ∗(RG(k)) - RG(l))^2
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@test -(2𝕀 - ∗(RG(g*h*k*l)) - RG(g*h*k*l)) +
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2(2𝕀 - ∗(RG(g*h*k)) - RG(g*h*k)) + 2L^2 ==
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(2𝕀 - ∗(RG(g*h*k)) - RG(l))^2
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@test 2𝕀 - ∗(RG(g*h*k)) - RG(g*h*k) ==
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2GH^2 + 2K^2 - (2𝕀 - ∗(RG(g*h)) - RG(k))^2
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@test -(2𝕀 - ∗(RG(g*h*k*l)) - RG(g*h*k*l)) +
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2(2GH^2 + 2K^2 - (2𝕀 - ∗(RG(g*h)) - RG(k))^2) + 2L^2 ==
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(2𝕀 - ∗(RG(g*h*k)) - RG(l))^2
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@test -(2𝕀 - ∗(RG(g*h*k*l)) - RG(g*h*k*l)) +
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2(2GH^2 + 2K^2) + 2L^2 ==
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(2𝕀 - ∗(RG(g*h*k)) - RG(l))^2 +
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2(2𝕀 - ∗(RG(g*h)) - RG(k))^2
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@test -(2𝕀 - ∗(RG(g*h*k*l)) - RG(g*h*k*l)) +
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8G^2 + 8H^2 + 4K^2 + 2L^2 ==
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(2𝕀 - ∗(RG(g*h*k)) - RG(l))^2 + 2(2𝕀 - ∗(RG(g*h)) - RG(k))^2 + 4(2𝕀 - ∗(RG(g)) - RG(h))^2
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@test -(2𝕀 - ∗(RG(g*h*k*l)) - RG(g*h*k*l)) +
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2GH^2 + 2KL^2 == (2𝕀 - ∗(RG(g*h)) - RG(k*l))^2
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@test -(2𝕀 - ∗(RG(g*h*k*l)) - RG(g*h*k*l)) + 2(2G^2 + 2H^2) + 2(2K^2 + 2L^2) ==
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(2𝕀 - ∗(RG(g*h)) - RG(k*l))^2 + 2(2𝕀 - ∗(RG(k)) - RG(l))^2 + 2(2𝕀 - ∗(RG(g)) - RG(h))^2
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end
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end
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