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update AutSigma_41 tests
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@ -15,13 +15,15 @@ AbstractAlgebra = "0.15, 0.16"
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GroupsCore = "^0.3"
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KnuthBendix = "^0.2.1"
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OrderedCollections = "1"
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PermutationGroups = "^0.3"
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ThreadsX = "^0.1.0"
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julia = "1.3, 1.4, 1.5, 1.6"
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[extras]
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AbstractAlgebra = "c3fe647b-3220-5bb0-a1ea-a7954cac585d"
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BenchmarkTools = "6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf"
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PermutationGroups = "8bc5a954-2dfc-11e9-10e6-cd969bffa420"
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Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
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[targets]
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test = ["Test", "BenchmarkTools", "AbstractAlgebra"]
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test = ["Test", "BenchmarkTools", "AbstractAlgebra", "PermutationGroups"]
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@ -1,12 +1,21 @@
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using PermutationGroups
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using Groups.KnuthBendix
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@testset "Wajnryb presentation for Σ₄₁" begin
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genus = 4
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G = New.SpecialAutomorphismGroup(New.FreeGroup(2genus))
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G = SpecialAutomorphismGroup(FreeGroup(2genus))
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T = let G = G; (Tas, Tαs, Tes) = New.mcg_twists(genus)
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# symplectic pairing goes like this:
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# in the free Group:
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# f1 ↔ f5
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# f2 ↔ f6
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# f3 ↔ f7
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# f4 ↔ f8
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T = let G = G
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(Tas, Tαs, Tes) = Groups.mcg_twists(G)
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Ta = G.(Tas)
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Tα = G.(Tαs)
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Tes = G.(Tes)
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@ -24,24 +33,58 @@ using PermutationGroups
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a8 = T[8]^-1 # Tα₄
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b0 = T[2]^-1 # Ta₂
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a0 = (a1*a2*a3)^4*b0^-1 # from the 3-chain relation
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X = a4*a5*a3*a4 # auxillary, not present in the Primer
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b1 = X^-1*a0*X
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a0 = (a1 * a2 * a3)^4 * b0^-1 # from the 3-chain relation
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X = a4 * a5 * a3 * a4 # auxillary, not present in the Primer
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b1 = X^-1 * a0 * X
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b2 = T[10]^-1 # Te₁₃
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As = T[[1,5,9,6,12,7,14,8]] # the inverses of the elements a
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As = T[[1, 5, 9, 6, 12, 7, 14, 8]] # the inverses of the elements a
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@testset "preserving relator" begin
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F = Groups.object(G)
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R = prod(commutator(gens(F, i), gens(F, i+genus)) for i in 1:genus)
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## TODO: how to evaluate automorphisms properly??!!!
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for g in T
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w = one(word(g))
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dg = Groups.domain(g)
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gens_idcs = first.(word.(dg))
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img = evaluate(g)
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A = alphabet(first(dg))
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ltrs_map = Vector{eltype(dg)}(undef, length(KnuthBendix.letters(A)))
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for i in 1:length(KnuthBendix.letters(A))
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if i in gens_idcs
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ltrs_map[i] = img[findfirst(==(i), gens_idcs)]
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else
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ltrs_map[i] = inv(img[findfirst(==(inv(A, i)), gens_idcs)])
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end
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end
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for l in word(R)
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append!(w, word(ltrs_map[l]))
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end
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@test F(w) == R
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end
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end
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@testset "commutation relations" begin
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for (i, ai) in enumerate(As) #the element ai here is actually the inverse of ai before. It does not matter for commutativity. Also, a0 is as defined before.
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for (j, aj) in enumerate(As)
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if abs(i-j) > 1
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@test ai*aj == aj*ai
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if abs(i - j) > 1
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@test ai * aj == aj * ai
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elseif i ≠ j
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@test ai*aj != aj*ai
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@test ai * aj != aj * ai
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end
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end
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if i != 4
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@test a0*ai == ai*a0
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@test a0 * ai == ai * a0
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end
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end
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end
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@ -49,145 +92,186 @@ using PermutationGroups
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@testset "braid relations" begin
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for (i, ai) in enumerate(As) #the element ai here is actually the inverse of ai before. It does not matter for braid relations.
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for (j, aj) in enumerate(As)
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if abs(i-j) == 1
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@test ai*aj*ai == aj*ai*aj
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if abs(i - j) == 1
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@test ai * aj * ai == aj * ai * aj
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end
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end
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end
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@test a0*a4*a0 == a4*a0*a4 # here, a0 and a4 are as before
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@test a0 * a4 * a0 == a4 * a0 * a4 # here, a0 and a4 are as before
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end
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@testset "Lantern relation" begin
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@testset "b2 definition" begin
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@test b2 == (a2*a3*a1*a2)^-1*b1*(a2*a3*a1*a2)
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@test b2 == (a2 * a3 * a1 * a2)^-1 * b1 * (a2 * a3 * a1 * a2)
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# some additional tests, checking what explicitly happens to the generators of the π₁ under b2
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d = New.domain(b2)
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im = New.evaluate(b2)
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z = im[7]*d[7]^-1
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d = Groups.domain(b2)
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im = evaluate(b2)
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z = im[7] * d[7]^-1
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@test im[1] == d[1]
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@test im[2] == z*d[2]*z^-1
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@test im[3] == z*d[3]*z^-1
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@test im[2] == z * d[2] * z^-1
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@test im[3] == z * d[3] * z^-1
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@test im[4] == d[4]
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@test im[5] == d[5]*z^-1
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@test im[6] == z*d[6]*z^-1
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@test im[7] == z*d[7]
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@test im[5] == d[5] * z^-1
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@test im[6] == z * d[6] * z^-1
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@test im[7] == z * d[7]
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@test im[8] == d[8]
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end
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@testset "b2: commutation relations" begin
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@test b2*a1 == a1*b2
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@test b2*a2 != a2*b2
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@test b2*a3 == a3*b2
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@test b2*a4 == a4*b2
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@test b2*a5 == a5*b2
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@test b2*a6 != a6*b2
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@test b2 * a1 == a1 * b2
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@test b2 * a2 != a2 * b2
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@test b2 * a3 == a3 * b2
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@test b2 * a4 == a4 * b2
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@test b2 * a5 == a5 * b2
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@test b2 * a6 != a6 * b2
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end
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@testset "b2: braid relations" begin
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@test a2*b2*a2 == b2*a2*b2
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@test a6*b2*a6 == b2*a6*b2
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@test a2 * b2 * a2 == b2 * a2 * b2
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@test a6 * b2 * a6 == b2 * a6 * b2
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end
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@testset "lantern" begin
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u = (a6*a5)^-1*b1*(a6*a5)
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x = (a6*a5*a4*a3*a2*u*a1^-1*a2^-1*a3^-1*a4^-1) # yet another auxillary
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u = (a6 * a5)^-1 * b1 * (a6 * a5)
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x = (a6 * a5 * a4 * a3 * a2 * u * a1^-1 * a2^-1 * a3^-1 * a4^-1) # yet another auxillary
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# x = (a4^-1*a3^-1*a2^-1*a1^-1*u*a2*a3*a4*a5*a6)
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@time New.evaluate(x)
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b3 = x*a0*x^-1
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@time New.evaluate(b3)
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@test a0*b2*b1 == a1*a3*a5*b3
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@time evaluate(x)
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b3 = x * a0 * x^-1
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@time evaluate(b3)
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@test a0 * b2 * b1 == a1 * a3 * a5 * b3
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end
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end
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@testset "Te₁₂ definition" begin
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G = parent(first(T))
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F₈ = New.object(G)
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(a, b, c, d, α, β, γ, δ) = gens(F₈)
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F₈ = Groups.object(G)
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(a, b, c, d, α, β, γ, δ) = Groups.gens(F₈)
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A = KnuthBendix.alphabet(G)
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A = alphabet(G)
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λ = [i == j ? one(G) : G([A[New.λ(i,j)]]) for i in 1:8, j in 1:8]
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ϱ = [i == j ? one(G) : G([A[New.ϱ(i,j)]]) for i in 1:8, j in 1:8]
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λ = [i == j ? one(G) : G([A[Groups.λ(i, j)]]) for i in 1:8, j in 1:8]
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ϱ = [i == j ? one(G) : G([A[Groups.ϱ(i, j)]]) for i in 1:8, j in 1:8]
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g = one(G)
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# @show g
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# @show g(Groups.domain(G))
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# β ↦ α*β
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g *= λ[6,5]
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@test New.evaluate(g)[6] == α*β
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g *= λ[6, 5]
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@test evaluate(g)[6] == α * β
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# α ↦ a*α*b^-1
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g *= λ[5,1]*inv(ϱ[5,2])
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@test New.evaluate(g)[5] == a*α*b^-1
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g *= λ[5, 1] * inv(ϱ[5, 2])
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@test evaluate(g)[5] == a * α * b^-1
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# β ↦ b*α^-1*a^-1*α*β
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g *= inv(λ[6,5])
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@test New.evaluate(g)[6] == b*α^-1*a^-1*α*β
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g *= inv(λ[6, 5])
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@test evaluate(g)[6] == b * α^-1 * a^-1 * α * β
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# b ↦ α
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g *= λ[2,5]*inv(λ[2,1]);
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@test New.evaluate(g)[2] == α
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g *= λ[2, 5] * inv(λ[2, 1])
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@test evaluate(g)[2] == α
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# b ↦ b*α^-1*a^-1*α
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g *= inv(λ[2,5]);
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@test New.evaluate(g)[2] == b*α^-1*a^-1*α
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g *= inv(λ[2, 5])
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@test evaluate(g)[2] == b * α^-1 * a^-1 * α
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# b ↦ b*α^-1*a^-1*α*b*α^-1
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g *= inv(ϱ[2,5])*ϱ[2,1];
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@test New.evaluate(g)[2] == b*α^-1*a^-1*α*b*α^-1
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g *= inv(ϱ[2, 5]) * ϱ[2, 1]
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@test evaluate(g)[2] == b * α^-1 * a^-1 * α * b * α^-1
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# b ↦ b*α^-1*a^-1*α*b*α^-1*a*α*b^-1
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g *= ϱ[2,5];
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@test New.evaluate(g)[2] == b*α^-1*a^-1*α*b*α^-1*a*α*b^-1
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g *= ϱ[2, 5]
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@test evaluate(g)[2] == b * α^-1 * a^-1 * α * b * α^-1 * a * α * b^-1
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x = b*α^-1*a^-1*α
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@test New.evaluate(g) == # (a, b, c, d, α, β, γ, δ)
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(a, x*b*x^-1, c, d, α*x^-1, x*β, γ, δ)
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x = b * α^-1 * a^-1 * α
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@test evaluate(g) ==
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(a, x * b * x^-1, c, d, α * x^-1, x * β, γ, δ)
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# (a, b, c, d, α, β, γ, δ)
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@test g == T[9]
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end
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Base.conj(t::New.Transvection, p::Perm) =
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New.Transvection(t.id, t.i^p, t.j^p, t.inv)
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Base.conj(t::Groups.Transvection, p::Perm) =
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Groups.Transvection(t.id, t.i^p, t.j^p, t.inv)
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function Base.conj(elt::New.FPGroupElement, p::Perm)
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function Base.conj(elt::FPGroupElement, p::Perm)
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G = parent(elt)
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A = New.alphabet(elt)
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return G([A[conj(A[idx], p)] for idx in New.word(elt)])
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A = alphabet(elt)
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return G([A[conj(A[idx], p)] for idx in word(elt)])
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end
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@testset "Te₂₃ definition" begin
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Te₁₂, Te₂₃ = T[9], T[12]
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G = parent(Te₁₂)
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F₈ = New.object(G)
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(a, b, c, d, α, β, γ, δ) = gens(F₈)
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F₈ = Groups.object(G)
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(a, b, c, d, α, β, γ, δ) = Groups.gens(F₈)
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img_Te₂₃ = New.evaluate(Te₂₃)
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y = c*β^-1*b^-1*β
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@test img_Te₂₃ == (a, b, y*c*y^-1, d, α, β*y^-1, y*γ, δ)
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img_Te₂₃ = evaluate(Te₂₃)
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y = c * β^-1 * b^-1 * β
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@test img_Te₂₃ == (a, b, y * c * y^-1, d, α, β * y^-1, y * γ, δ)
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σ = perm"(1,2,3)(5,6,7)(8)"
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Te₂₃_σ = conj(Te₁₂, σ)
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# @test New.word(Te₂₃_σ) == New.word(Te₂₃)
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# @test word(Te₂₃_σ) == word(Te₂₃)
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@test New.evaluate(Te₂₃_σ) == New.evaluate(Te₂₃)
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@test evaluate(Te₂₃_σ) == evaluate(Te₂₃)
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@test Te₂₃ == Te₂₃_σ
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end
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@testset "Te₃₄ definition" begin
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Te₁₂, Te₃₄ = T[9], T[14]
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G = parent(Te₁₂)
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F₈ = New.object(G)
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F₈ = Groups.object(G)
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(a, b, c, d, α, β, γ, δ) = Groups.gens(F₈)
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σ = perm"(1,3)(2,4)(5,7)(6,8)"
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Te₃₄_σ = conj(Te₁₂, σ)
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@test Te₃₄ == Te₃₄_σ
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end
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@testset "hyperelliptic τ is central" begin
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A = alphabet(G)
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λ = Groups.ΡΛ(:λ, A, 2genus)
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ϱ = Groups.ΡΛ(:ϱ, A, 2genus)
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import Groups: Ta, Tα, Te
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halftwists = map(1:genus-1) do i
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j = i + 1
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x = Ta(λ, j) * inv(A, Ta(λ, i)) * Tα(λ, j) * Te(λ, ϱ, i, j)
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δ = x * Tα(λ, i) * inv(A, x)
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c =
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inv(A, Ta(λ, j)) *
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Te(λ, ϱ, i, j) *
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Tα(λ, i)^2 *
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inv(A, δ) *
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inv(A, Ta(λ, j)) *
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Ta(λ, i) *
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δ
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z =
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Te(λ, ϱ, j, i) *
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inv(A, Ta(λ, i)) *
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Tα(λ, i) *
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Ta(λ, i) *
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inv(A, Te(λ, ϱ, j, i))
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G(Ta(λ, i) * inv(A, Ta(λ, j) * Tα(λ, j))^6 * (Ta(λ, j) * Tα(λ, j) * z)^4 * c)
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end
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τ = (G(Ta(λ, 1) * Tα(λ, 1))^6) * prod(halftwists, init = one(G))
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# τ^genus is trivial but only in autπ₁Σ₄
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# here we check its centrality
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τᵍ = τ^genus
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@test_broken all(a * τᵍ == τᵍ * a for a in Groups.gens(G))
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end
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end
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@ -26,6 +26,7 @@ include(joinpath(pathof(GroupsCore), "..", "..", "test", "conformance_test.jl"))
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include("fp_groups.jl")
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include("AutFn.jl")
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include("AutSigma_41.jl")
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# if !haskey(ENV, "CI")
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# include("benchmarks.jl")
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