mirror of
https://github.com/kalmarek/Groups.jl.git
synced 2024-11-19 14:35:28 +01:00
279 lines
9.2 KiB
Julia
279 lines
9.2 KiB
Julia
@testset "Automorphisms" begin
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G = SymmetricGroup(Int8(4))
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@testset "AutSymbol" begin
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@test_throws MethodError Groups.AutSymbol(:a)
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@test_throws MethodError Groups.AutSymbol(:a, 1)
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f = Groups.AutSymbol(:a, 1, Groups.FlipAut(2))
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@test f isa Groups.GSymbol
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@test f isa Groups.AutSymbol
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@test Groups.AutSymbol(perm"(4)") isa Groups.AutSymbol
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@test Groups.AutSymbol(perm"(1,2,3,4)") isa Groups.AutSymbol
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@test Groups.transvection_R(1,2) isa Groups.AutSymbol
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@test Groups.transvection_R(3,4) isa Groups.AutSymbol
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@test Groups.flip(3) isa Groups.AutSymbol
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@test Groups.id_autsymbol() isa Groups.AutSymbol
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@test inv(Groups.id_autsymbol()) isa Groups.AutSymbol
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@test inv(Groups.id_autsymbol()) == Groups.id_autsymbol()
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end
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a,b,c,d = gens(FreeGroup(4))
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D = NTuple{4,FreeGroupElem}([a,b,c,d])
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@testset "flip correctness" begin
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@test Groups.flip(1)(deepcopy(D)) == (a^-1, b,c,d)
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@test Groups.flip(2)(deepcopy(D)) == (a, b^-1,c,d)
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@test Groups.flip(3)(deepcopy(D)) == (a, b,c^-1,d)
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@test Groups.flip(4)(deepcopy(D)) == (a, b,c,d^-1)
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@test inv(Groups.flip(1))(deepcopy(D)) == (a^-1, b,c,d)
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@test inv(Groups.flip(2))(deepcopy(D)) == (a, b^-1,c,d)
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@test inv(Groups.flip(3))(deepcopy(D)) == (a, b,c^-1,d)
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@test inv(Groups.flip(4))(deepcopy(D)) == (a, b,c,d^-1)
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end
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@testset "perm correctness" begin
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σ = Groups.AutSymbol(perm"(4)")
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@test σ(deepcopy(D)) == deepcopy(D)
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@test inv(σ)(deepcopy(D)) == deepcopy(D)
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σ = Groups.AutSymbol(perm"(1,2,3,4)")
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@test σ(deepcopy(D)) == (b, c, d, a)
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@test inv(σ)(deepcopy(D)) == (d, a, b, c)
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σ = Groups.AutSymbol(perm"(1,2)(4,3)")
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@test σ(deepcopy(D)) == (b, a, d, c)
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@test inv(σ)(deepcopy(D)) == (b, a, d, c)
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σ = Groups.AutSymbol(perm"(1,2,3)(4)")
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@test σ(deepcopy(D)) == (b, c, a, d)
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@test inv(σ)(deepcopy(D)) == (c, a, b, d)
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end
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@testset "rmul/transvection_R correctness" begin
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i,j = 1,2
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r = Groups.transvection_R(i,j)
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l = Groups.transvection_L(i,j)
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@test r(deepcopy(D)) == (a*b, b, c, d)
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@test inv(r)(deepcopy(D)) == (a*b^-1,b, c, d)
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@test l(deepcopy(D)) == (b*a, b, c, d)
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@test inv(l)(deepcopy(D)) == (b^-1*a,b, c, d)
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i,j = 3,1
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r = Groups.transvection_R(i,j)
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l = Groups.transvection_L(i,j)
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@test r(deepcopy(D)) == (a, b, c*a, d)
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@test inv(r)(deepcopy(D)) == (a, b, c*a^-1,d)
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@test l(deepcopy(D)) == (a, b, a*c, d)
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@test inv(l)(deepcopy(D)) == (a, b, a^-1*c,d)
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i,j = 4,3
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r = Groups.transvection_R(i,j)
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l = Groups.transvection_L(i,j)
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@test r(deepcopy(D)) == (a, b, c, d*c)
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@test inv(r)(deepcopy(D)) == (a, b, c, d*c^-1)
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@test l(deepcopy(D)) == (a, b, c, c*d)
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@test inv(l)(deepcopy(D)) == (a, b, c, c^-1*d)
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i,j = 2,4
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r = Groups.transvection_R(i,j)
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l = Groups.transvection_L(i,j)
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@test r(deepcopy(D)) == (a, b*d, c, d)
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@test inv(r)(deepcopy(D)) == (a, b*d^-1,c, d)
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@test l(deepcopy(D)) == (a, d*b, c, d)
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@test inv(l)(deepcopy(D)) == (a, d^-1*b,c, d)
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end
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@testset "AutGroup/Automorphism constructors" begin
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f = Groups.AutSymbol(:a, 1, Groups.FlipAut(1))
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@test isa(Automorphism{3}(f), Groups.GWord)
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@test isa(Automorphism{3}(f), Automorphism)
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@test isa(AutGroup(FreeGroup(3)), AbstractAlgebra.Group)
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@test isa(AutGroup(FreeGroup(1)), Groups.AbstractFPGroup)
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A = AutGroup(FreeGroup(1))
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@test Groups.gens(A) isa Vector{Automorphism{1}}
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@test length(Groups.gens(A)) == 1
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@test length(Groups.gens(Aut(FreeGroup(1)))) == 1
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@test Groups.gens(A) == [A(Groups.flip(1))]
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A = AutGroup(FreeGroup(1), special=true)
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@test isempty(Groups.gens(A))
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@test Groups.gens(SAut(FreeGroup(1))) == Automorphism{1}[]
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A = AutGroup(FreeGroup(2))
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@test length(Groups.gens(A)) == 7
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Agens = Groups.gens(A)
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@test A(first(Agens)) isa Automorphism
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@test A(Groups.transvection_R(1,2)) isa Automorphism
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@test A(Groups.transvection_R(1,2)) in Agens
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@test A(Groups.transvection_R(2,1)) isa Automorphism
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@test A(Groups.transvection_R(2,1)) in Agens
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@test A(Groups.transvection_R(1,2)) isa Automorphism
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@test A(Groups.transvection_R(1,2)) in Agens
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@test A(Groups.transvection_R(2,1)) isa Automorphism
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@test A(Groups.transvection_R(2,1)) in Agens
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@test A(Groups.flip(1)) isa Automorphism
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@test A(Groups.flip(1)) in Agens
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@test A(Groups.flip(2)) isa Automorphism
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@test A(Groups.flip(2)) in Agens
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@test A(Groups.AutSymbol(perm"(1,2)")) isa Automorphism
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@test A(Groups.AutSymbol(perm"(1,2)")) in Agens
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@test A(Groups.id_autsymbol()) isa Automorphism
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end
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A = AutGroup(FreeGroup(4))
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@testset "eltary functions" begin
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f = Groups.AutSymbol(perm"(1,2,3,4)")
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@test (Groups.change_pow(f, 2)).pow == 1
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@test (Groups.change_pow(f, -2)).pow == 1
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@test (inv(f)).pow == 1
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f = Groups.AutSymbol(perm"(1,2)(3,4)")
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@test isa(inv(f), Groups.AutSymbol)
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@test_throws MethodError f*f
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@test A(f)^-1 == A(inv(f))
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end
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@testset "reductions/arithmetic" begin
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f = Groups.AutSymbol(perm"(1,2,3,4)")
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f² = push!(A(f), f)
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@test Groups.simplifyperms!(Bool, f²) == false
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@test f²^2 == one(A)
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@test !isone(f²)
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a = A(Groups.λ(1,2))*Groups.ε(2)
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b = Groups.ε(2)*A(inv(Groups.λ(1,2)))
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@test a*b == b*a
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@test a^3 * b^3 == one(A)
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g,h = Groups.gens(A)[[1,8]] # (g, h) = (ϱ₁₂, ϱ₃₂)
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@test Groups.domain(A) == NTuple{4, FreeGroupElem}(gens(A.objectGroup))
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@test (g*h)(Groups.domain(A)) == (h*g)(Groups.domain(A))
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@test (g*h).savedhash == zero(UInt)
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@test (h*g).savedhash == zero(UInt)
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a = g*h
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b = h*g
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@test hash(a) != zero(UInt)
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@test hash(b) == hash(a)
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@test a.savedhash == b.savedhash
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@test length(unique([a,b])) == 1
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@test length(unique([g*h, h*g])) == 1
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# Not so simple arithmetic: applying starting on the left:
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# ϱ₁₂*ϱ₂₁⁻¹*λ₁₂*ε₂ == σ₂₁₃₄
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g = A(Groups.transvection_R(1,2))
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x1, x2, x3, x4 = Groups.domain(A)
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@test g(Groups.domain(A)) == (x1*x2, x2, x3, x4)
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g = g*inv(A(Groups.transvection_R(2,1)))
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@test g(Groups.domain(A)) == (x1*x2, x1^-1, x3, x4)
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g = g*A(Groups.transvection_L(1,2))
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@test g(Groups.domain(A)) == (x2, x1^-1, x3, x4)
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g = g*A(Groups.flip(2))
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@test g(Groups.domain(A)) == (x2, x1, x3, x4)
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@test g(Groups.domain(A)) == A(Groups.AutSymbol(perm"(1,2)(4)"))(Groups.domain(A))
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@test g == A(Groups.AutSymbol(perm"(1,2)(4)"))
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g_im = g(Groups.domain(A))
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@test length.(g_im) == (1,1,1,1)
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g = A(Groups.σ(perm"(1,2)(4)"))
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h = A(Groups.σ(perm"(2,3,4)"))
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@test g*h isa Groups.Automorphism{4}
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f = g*h
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Groups
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@test Groups.syllablelength(f) == 2
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@test Groups.reduce!(f) isa Groups.Automorphism{4}
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@test Groups.syllablelength(f) == 1
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end
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@testset "specific Aut(F4) tests" begin
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N = 4
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G = AutGroup(FreeGroup(N))
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S = G.gens
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@test isa(S, Vector{Groups.AutSymbol})
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S = [G(s) for s in unique(S)]
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@test isa(S, Vector{Automorphism{N}})
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@test S == gens(G)
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@test length(S) == 51
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S_inv = [S..., [inv(s) for s in S]...]
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@test length(unique(S_inv)) == 75
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G = AutGroup(FreeGroup(N), special=true)
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S = gens(G)
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S_inv = [one(G), S..., [inv(s) for s in S]...]
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S_inv = unique(S_inv)
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B_2 = [i*j for (i,j) in Base.product(S_inv, S_inv)]
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@test length(B_2) == 2401
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@test length(unique(B_2)) == 1777
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end
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@testset "abelianization homomorphism" begin
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N = 4
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G = AutGroup(FreeGroup(N))
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S = unique([gens(G); inv.(gens(G))])
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R = 3
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@test Groups.abelianize(one(G)) isa Matrix{Int}
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@test Groups.abelianize(one(G)) == Matrix{Int}(I, N, N)
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M = Matrix{Int}(I, N, N)
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M[1,2] = 1
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ϱ₁₂ = G(Groups.ϱ(1,2))
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λ₁₂ = G(Groups.λ(1,2))
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@test Groups.abelianize(ϱ₁₂) == M
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@test Groups.abelianize(λ₁₂) == M
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M[1,2] = -1
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@test Groups.abelianize(ϱ₁₂^-1) == M
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@test Groups.abelianize(λ₁₂^-1) == M
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@test Groups.abelianize(ϱ₁₂*λ₁₂^-1) == Matrix{Int}(I, N, N)
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@test Groups.abelianize(λ₁₂^-1*ϱ₁₂) == Matrix{Int}(I, N, N)
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M = Matrix{Int}(I, N, N)
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M[2,2] = -1
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ε₂ = G(Groups.flip(2))
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@test Groups.abelianize(ε₂) == M
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@test Groups.abelianize(ε₂^2) == Matrix{Int}(I, N, N)
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M = [0 1 0 0; 0 0 0 1; 0 0 1 0; 1 0 0 0]
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σ = G(Groups.AutSymbol(perm"(1,2,4)"))
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@test Groups.abelianize(σ) == M
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@test Groups.abelianize(σ^3) == Matrix{Int}(I, N, N)
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@test Groups.abelianize(σ)^3 == Matrix{Int}(I, N, N)
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function test_homomorphism(S, r)
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for elts in Iterators.product([[g for g in S] for _ in 1:r]...)
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prod(Groups.abelianize.(elts)) == Groups.abelianize(prod(elts)) || error("linear representaton test failed at $elts")
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end
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return 0
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end
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@test test_homomorphism(S, R) == 0
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end
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end
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