mirror of
https://github.com/kalmarek/Groups.jl.git
synced 2024-11-19 14:35:28 +01:00
260 lines
8.8 KiB
Julia
260 lines
8.8 KiB
Julia
@testset "Automorphisms" begin
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G = PermutationGroup(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 isa(f, Groups.GSymbol)
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@test isa(f, Groups.AutSymbol)
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@test isa(Groups.perm_autsymbol(Int8.([1,2,3,4])), Groups.AutSymbol)
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@test isa(Groups.rmul_autsymbol(1,2), Groups.AutSymbol)
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@test isa(Groups.lmul_autsymbol(3,4), Groups.AutSymbol)
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@test isa(Groups.flip_autsymbol(3), Groups.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_autsymbol correctness" begin
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@test Groups.flip_autsymbol(1)(deepcopy(D)) == (a^-1, b,c,d)
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@test Groups.flip_autsymbol(2)(deepcopy(D)) == (a, b^-1,c,d)
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@test Groups.flip_autsymbol(3)(deepcopy(D)) == (a, b,c^-1,d)
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@test Groups.flip_autsymbol(4)(deepcopy(D)) == (a, b,c,d^-1)
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@test inv(Groups.flip_autsymbol(1))(deepcopy(D)) == (a^-1, b,c,d)
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@test inv(Groups.flip_autsymbol(2))(deepcopy(D)) == (a, b^-1,c,d)
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@test inv(Groups.flip_autsymbol(3))(deepcopy(D)) == (a, b,c^-1,d)
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@test inv(Groups.flip_autsymbol(4))(deepcopy(D)) == (a, b,c,d^-1)
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end
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@testset "perm_autsymbol correctness" begin
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σ = Groups.perm_autsymbol([1,2,3,4])
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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.perm_autsymbol([2,3,4,1])
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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.perm_autsymbol([2,1,4,3])
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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.perm_autsymbol([2,3,1,4])
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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/lmul_autsymbol correctness" begin
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i,j = 1,2
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r = Groups.rmul_autsymbol(i,j)
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l = Groups.lmul_autsymbol(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.rmul_autsymbol(i,j)
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l = Groups.lmul_autsymbol(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.rmul_autsymbol(i,j)
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l = Groups.lmul_autsymbol(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.rmul_autsymbol(i,j)
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l = Groups.lmul_autsymbol(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 isa(Groups.gens(A), Vector{Automorphism{1}})
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@test length(Groups.gens(A)) == 1
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A = AutGroup(FreeGroup(1), special=true)
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@test length(Groups.gens(A)) == 0
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A = AutGroup(FreeGroup(2))
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@test length(Groups.gens(A)) == 7
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gens = Groups.gens(A)
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@test isa(A(Groups.rmul_autsymbol(1,2)), Automorphism)
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@test A(Groups.rmul_autsymbol(1,2)) in gens
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@test isa(A(Groups.rmul_autsymbol(2,1)), Automorphism)
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@test A(Groups.rmul_autsymbol(2,1)) in gens
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@test isa(A(Groups.lmul_autsymbol(1,2)), Automorphism)
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@test A(Groups.lmul_autsymbol(1,2)) in gens
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@test isa(A(Groups.lmul_autsymbol(2,1)), Automorphism)
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@test A(Groups.lmul_autsymbol(2,1)) in gens
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@test isa(A(Groups.flip_autsymbol(1)), Automorphism)
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@test A(Groups.flip_autsymbol(1)) in gens
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@test isa(A(Groups.flip_autsymbol(2)), Automorphism)
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@test A(Groups.flip_autsymbol(2)) in gens
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@test isa(A(Groups.perm_autsymbol([2,1])), Automorphism)
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@test A(Groups.perm_autsymbol([2,1])) in gens
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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.perm_autsymbol([2,3,4,1])
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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.perm_autsymbol([2,1,4,3])
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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.perm_autsymbol([2,3,4,1])
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f² = Groups.r_multiply(A(f), [f], reduced=false)
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@test Groups.simplifyperms!(f²) == false
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@test f²^2 == one(A)
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a = A(Groups.rmul_autsymbol(1,2))*Groups.flip_autsymbol(2)
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b = Groups.flip_autsymbol(2)*A(inv(Groups.rmul_autsymbol(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.rmul_autsymbol(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.rmul_autsymbol(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.lmul_autsymbol(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_autsymbol(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.perm_autsymbol([2,1,3,4]))(Groups.domain(A))
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@test g == A(Groups.perm_autsymbol([2,1,3,4]))
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g_im = g(Groups.domain(A))
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@test length(g_im[1]) == 5
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@test length(g_im[2]) == 3
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@test length(g_im[3]) == 1
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@test length(g_im[4]) == 1
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@test length.(Groups.reduce!.(g_im)) == (1,1,1,1)
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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 "linear_repr tests" begin
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N = 3
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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.linear_repr(one(G)) isa Matrix{Int}
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@test Groups.linear_repr(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.rmul_autsymbol(1,2))
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λ₁₂ = G(Groups.rmul_autsymbol(1,2))
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@test Groups.linear_repr(ϱ₁₂) == M
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@test Groups.linear_repr(λ₁₂) == M
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M[1,2] = -1
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@test Groups.linear_repr(ϱ₁₂^-1) == M
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@test Groups.linear_repr(λ₁₂^-1) == M
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@test Groups.linear_repr(ϱ₁₂*λ₁₂^-1) == Matrix{Int}(I, N, N)
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@test Groups.linear_repr(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_autsymbol(2))
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@test Groups.linear_repr(ε₂) == M
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@test Groups.linear_repr(ε₂^2) == Matrix{Int}(I, N, N)
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M = [0 1 0; 0 0 1; 1 0 0]
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σ = G(Groups.perm_autsymbol([2,3,1]))
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@test Groups.linear_repr(σ) == M
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@test Groups.linear_repr(σ^3) == Matrix{Int}(I, 3, 3)
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@test Groups.linear_repr(σ)^3 == Matrix{Int}(I, 3, 3)
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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.linear_repr.(elts)) == Groups.linear_repr(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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