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update AutSigma_41 tests

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Marek Kaluba 2021-07-07 10:41:34 +02:00
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3 changed files with 161 additions and 74 deletions
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4
Project.toml
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@ -15,13 +15,15 @@ AbstractAlgebra = "0.15, 0.16"
GroupsCore = "^0.3" GroupsCore = "^0.3"
KnuthBendix = "^0.2.1" KnuthBendix = "^0.2.1"
OrderedCollections = "1" OrderedCollections = "1"
PermutationGroups = "^0.3"
ThreadsX = "^0.1.0" ThreadsX = "^0.1.0"
julia = "1.3, 1.4, 1.5, 1.6" julia = "1.3, 1.4, 1.5, 1.6"
[extras] [extras]
AbstractAlgebra = "c3fe647b-3220-5bb0-a1ea-a7954cac585d" AbstractAlgebra = "c3fe647b-3220-5bb0-a1ea-a7954cac585d"
BenchmarkTools = "6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf" BenchmarkTools = "6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf"
PermutationGroups = "8bc5a954-2dfc-11e9-10e6-cd969bffa420"
Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40" Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
[targets] [targets]
test = ["Test", "BenchmarkTools", "AbstractAlgebra"] test = ["Test", "BenchmarkTools", "AbstractAlgebra", "PermutationGroups"]

230
test/AutSigma_41.jl
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@ -1,12 +1,21 @@
using PermutationGroups using PermutationGroups
using Groups.KnuthBendix
@testset "Wajnryb presentation for Σ₄₁" begin @testset "Wajnryb presentation for Σ₄₁" begin
genus = 4 genus = 4
G = New.SpecialAutomorphismGroup(New.FreeGroup(2genus)) G = SpecialAutomorphismGroup(FreeGroup(2genus))
T = let G = G; (Tas, Tαs, Tes) = New.mcg_twists(genus) # symplectic pairing goes like this:
# in the free Group:
# f1 ↔ f5
# f2 ↔ f6
# f3 ↔ f7
# f4 ↔ f8
T = let G = G
(Tas, Tαs, Tes) = Groups.mcg_twists(G)
Ta = G.(Tas) Ta = G.(Tas)
Tα = G.(Tαs) Tα = G.(Tαs)
Tes = G.(Tes) Tes = G.(Tes)
@ -24,24 +33,58 @@ using PermutationGroups
a8 = T[8]^-1 # Tα₄ a8 = T[8]^-1 # Tα₄
b0 = T[2]^-1 # Ta₂ b0 = T[2]^-1 # Ta₂
a0 = (a1*a2*a3)^4*b0^-1 # from the 3-chain relation a0 = (a1 * a2 * a3)^4 * b0^-1 # from the 3-chain relation
X = a4*a5*a3*a4 # auxillary, not present in the Primer X = a4 * a5 * a3 * a4 # auxillary, not present in the Primer
b1 = X^-1*a0*X b1 = X^-1 * a0 * X
b2 = T[10]^-1 # Te₁₃ b2 = T[10]^-1 # Te₁₃
As = T[[1,5,9,6,12,7,14,8]] # the inverses of the elements a As = T[[1, 5, 9, 6, 12, 7, 14, 8]] # the inverses of the elements a
@testset "preserving relator" begin
F = Groups.object(G)
R = prod(commutator(gens(F, i), gens(F, i+genus)) for i in 1:genus)
## TODO: how to evaluate automorphisms properly??!!!
for g in T
w = one(word(g))
dg = Groups.domain(g)
gens_idcs = first.(word.(dg))
img = evaluate(g)
A = alphabet(first(dg))
ltrs_map = Vector{eltype(dg)}(undef, length(KnuthBendix.letters(A)))
for i in 1:length(KnuthBendix.letters(A))
if i in gens_idcs
ltrs_map[i] = img[findfirst(==(i), gens_idcs)]
else
ltrs_map[i] = inv(img[findfirst(==(inv(A, i)), gens_idcs)])
end
end
for l in word(R)
append!(w, word(ltrs_map[l]))
end
@test F(w) == R
end
end
@testset "commutation relations" begin @testset "commutation relations" begin
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. 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.
for (j, aj) in enumerate(As) for (j, aj) in enumerate(As)
if abs(i-j) > 1 if abs(i - j) > 1
@test ai*aj == aj*ai @test ai * aj == aj * ai
elseif i ≠ j elseif i ≠ j
@test ai*aj != aj*ai @test ai * aj != aj * ai
end end
end end
if i != 4 if i != 4
@test a0*ai == ai*a0 @test a0 * ai == ai * a0
end end
end end
end end
@ -49,145 +92,186 @@ using PermutationGroups
@testset "braid relations" begin @testset "braid relations" begin
for (i, ai) in enumerate(As) #the element ai here is actually the inverse of ai before. It does not matter for braid relations. for (i, ai) in enumerate(As) #the element ai here is actually the inverse of ai before. It does not matter for braid relations.
for (j, aj) in enumerate(As) for (j, aj) in enumerate(As)
if abs(i-j) == 1 if abs(i - j) == 1
@test ai*aj*ai == aj*ai*aj @test ai * aj * ai == aj * ai * aj
end end
end end
end end
@test a0*a4*a0 == a4*a0*a4 # here, a0 and a4 are as before @test a0 * a4 * a0 == a4 * a0 * a4 # here, a0 and a4 are as before
end end
@testset "Lantern relation" begin @testset "Lantern relation" begin
@testset "b2 definition" begin @testset "b2 definition" begin
@test b2 == (a2*a3*a1*a2)^-1*b1*(a2*a3*a1*a2) @test b2 == (a2 * a3 * a1 * a2)^-1 * b1 * (a2 * a3 * a1 * a2)
# some additional tests, checking what explicitly happens to the generators of the π₁ under b2 # some additional tests, checking what explicitly happens to the generators of the π₁ under b2
d = New.domain(b2) d = Groups.domain(b2)
im = New.evaluate(b2) im = evaluate(b2)
z = im[7]*d[7]^-1 z = im[7] * d[7]^-1
@test im[1] == d[1] @test im[1] == d[1]
@test im[2] == z*d[2]*z^-1 @test im[2] == z * d[2] * z^-1
@test im[3] == z*d[3]*z^-1 @test im[3] == z * d[3] * z^-1
@test im[4] == d[4] @test im[4] == d[4]
@test im[5] == d[5]*z^-1 @test im[5] == d[5] * z^-1
@test im[6] == z*d[6]*z^-1 @test im[6] == z * d[6] * z^-1
@test im[7] == z*d[7] @test im[7] == z * d[7]
@test im[8] == d[8] @test im[8] == d[8]
end end
@testset "b2: commutation relations" begin @testset "b2: commutation relations" begin
@test b2*a1 == a1*b2 @test b2 * a1 == a1 * b2
@test b2*a2 != a2*b2 @test b2 * a2 != a2 * b2
@test b2*a3 == a3*b2 @test b2 * a3 == a3 * b2
@test b2*a4 == a4*b2 @test b2 * a4 == a4 * b2
@test b2*a5 == a5*b2 @test b2 * a5 == a5 * b2
@test b2*a6 != a6*b2 @test b2 * a6 != a6 * b2
end end
@testset "b2: braid relations" begin @testset "b2: braid relations" begin
@test a2*b2*a2 == b2*a2*b2 @test a2 * b2 * a2 == b2 * a2 * b2
@test a6*b2*a6 == b2*a6*b2 @test a6 * b2 * a6 == b2 * a6 * b2
end end
@testset "lantern" begin @testset "lantern" begin
u = (a6*a5)^-1*b1*(a6*a5) u = (a6 * a5)^-1 * b1 * (a6 * a5)
x = (a6*a5*a4*a3*a2*u*a1^-1*a2^-1*a3^-1*a4^-1) # yet another auxillary x = (a6 * a5 * a4 * a3 * a2 * u * a1^-1 * a2^-1 * a3^-1 * a4^-1) # yet another auxillary
# x = (a4^-1*a3^-1*a2^-1*a1^-1*u*a2*a3*a4*a5*a6) # x = (a4^-1*a3^-1*a2^-1*a1^-1*u*a2*a3*a4*a5*a6)
@time New.evaluate(x) @time evaluate(x)
b3 = x*a0*x^-1 b3 = x * a0 * x^-1
@time New.evaluate(b3) @time evaluate(b3)
@test a0*b2*b1 == a1*a3*a5*b3 @test a0 * b2 * b1 == a1 * a3 * a5 * b3
end end
end end
@testset "Te₁₂ definition" begin @testset "Te₁₂ definition" begin
G = parent(first(T)) G = parent(first(T))
F₈ = New.object(G) F₈ = Groups.object(G)
(a, b, c, d, α, β, γ, δ) = gens(F₈) (a, b, c, d, α, β, γ, δ) = Groups.gens(F₈)
A = KnuthBendix.alphabet(G) A = alphabet(G)
λ = [i == j ? one(G) : G([A[New.λ(i,j)]]) for i in 1:8, j in 1:8] λ = [i == j ? one(G) : G([A[Groups.λ(i, j)]]) for i in 1:8, j in 1:8]
ϱ = [i == j ? one(G) : G([A[New.ϱ(i,j)]]) for i in 1:8, j in 1:8] ϱ = [i == j ? one(G) : G([A[Groups.ϱ(i, j)]]) for i in 1:8, j in 1:8]
g = one(G) g = one(G)
# @show g
# @show g(Groups.domain(G))
# β ↦ α*β # β ↦ α*β
g *= λ[6,5] g *= λ[6, 5]
@test New.evaluate(g)[6] == α*β @test evaluate(g)[6] == α * β
# α ↦ a*α*b^-1 # α ↦ a*α*b^-1
g *= λ[5,1]*inv(ϱ[5,2]) g *= λ[5, 1] * inv(ϱ[5, 2])
@test New.evaluate(g)[5] == a*α*b^-1 @test evaluate(g)[5] == a * α * b^-1
# β ↦ b*α^-1*a^-1*α*β # β ↦ b*α^-1*a^-1*α*β
g *= inv(λ[6,5]) g *= inv(λ[6, 5])
@test New.evaluate(g)[6] == b*α^-1*a^-1*α*β @test evaluate(g)[6] == b * α^-1 * a^-1 * α * β
# b ↦ α # b ↦ α
g *= λ[2,5]*inv(λ[2,1]); g *= λ[2, 5] * inv(λ[2, 1])
@test New.evaluate(g)[2] == α @test evaluate(g)[2] == α
# b ↦ b*α^-1*a^-1*α # b ↦ b*α^-1*a^-1*α
g *= inv(λ[2,5]); g *= inv(λ[2, 5])
@test New.evaluate(g)[2] == b*α^-1*a^-1*α @test evaluate(g)[2] == b * α^-1 * a^-1 * α
# b ↦ b*α^-1*a^-1*α*b*α^-1 # b ↦ b*α^-1*a^-1*α*b*α^-1
g *= inv(ϱ[2,5])*ϱ[2,1]; g *= inv(ϱ[2, 5]) * ϱ[2, 1]
@test New.evaluate(g)[2] == b*α^-1*a^-1*α*b*α^-1 @test evaluate(g)[2] == b * α^-1 * a^-1 * α * b * α^-1
# b ↦ b*α^-1*a^-1*α*b*α^-1*a*α*b^-1 # b ↦ b*α^-1*a^-1*α*b*α^-1*a*α*b^-1
g *= ϱ[2,5]; g *= ϱ[2, 5]
@test New.evaluate(g)[2] == b*α^-1*a^-1*α*b*α^-1*a*α*b^-1 @test evaluate(g)[2] == b * α^-1 * a^-1 * α * b * α^-1 * a * α * b^-1
x = b*α^-1*a^-1*α x = b * α^-1 * a^-1 * α
@test New.evaluate(g) == # (a, b, c, d, α, β, γ, δ) @test evaluate(g) ==
(a, x*b*x^-1, c, d, α*x^-1, x*β, γ, δ) (a, x * b * x^-1, c, d, α * x^-1, x * β, γ, δ)
# (a, b, c, d, α, β, γ, δ)
@test g == T[9] @test g == T[9]
end end
Base.conj(t::New.Transvection, p::Perm) = Base.conj(t::Groups.Transvection, p::Perm) =
New.Transvection(t.id, t.i^p, t.j^p, t.inv) Groups.Transvection(t.id, t.i^p, t.j^p, t.inv)
function Base.conj(elt::New.FPGroupElement, p::Perm) function Base.conj(elt::FPGroupElement, p::Perm)
G = parent(elt) G = parent(elt)
A = New.alphabet(elt) A = alphabet(elt)
return G([A[conj(A[idx], p)] for idx in New.word(elt)]) return G([A[conj(A[idx], p)] for idx in word(elt)])
end end
@testset "Te₂₃ definition" begin @testset "Te₂₃ definition" begin
Te₁₂, Te₂₃ = T[9], T[12] Te₁₂, Te₂₃ = T[9], T[12]
G = parent(Te₁₂) G = parent(Te₁₂)
F₈ = New.object(G) F₈ = Groups.object(G)
(a, b, c, d, α, β, γ, δ) = gens(F₈) (a, b, c, d, α, β, γ, δ) = Groups.gens(F₈)
img_Te₂₃ = New.evaluate(Te₂₃) img_Te₂₃ = evaluate(Te₂₃)
y = c*β^-1*b^-1*β y = c * β^-1 * b^-1 * β
@test img_Te₂₃ == (a, b, y*c*y^-1, d, α, β*y^-1, y*γ, δ) @test img_Te₂₃ == (a, b, y * c * y^-1, d, α, β * y^-1, y * γ, δ)
σ = perm"(1,2,3)(5,6,7)(8)" σ = perm"(1,2,3)(5,6,7)(8)"
Te₂₃_σ = conj(Te₁₂, σ) Te₂₃_σ = conj(Te₁₂, σ)
# @test New.word(Te₂₃_σ) == New.word(Te₂₃) # @test word(Te₂₃_σ) == word(Te₂₃)
@test New.evaluate(Te₂₃_σ) == New.evaluate(Te₂₃) @test evaluate(Te₂₃_σ) == evaluate(Te₂₃)
@test Te₂₃ == Te₂₃_σ @test Te₂₃ == Te₂₃_σ
end end
@testset "Te₃₄ definition" begin @testset "Te₃₄ definition" begin
Te₁₂, Te₃₄ = T[9], T[14] Te₁₂, Te₃₄ = T[9], T[14]
G = parent(Te₁₂) G = parent(Te₁₂)
F₈ = New.object(G) F₈ = Groups.object(G)
(a, b, c, d, α, β, γ, δ) = Groups.gens(F₈) (a, b, c, d, α, β, γ, δ) = Groups.gens(F₈)
σ = perm"(1,3)(2,4)(5,7)(6,8)" σ = perm"(1,3)(2,4)(5,7)(6,8)"
Te₃₄_σ = conj(Te₁₂, σ) Te₃₄_σ = conj(Te₁₂, σ)
@test Te₃₄ == Te₃₄_σ @test Te₃₄ == Te₃₄_σ
end end
@testset "hyperelliptic τ is central" begin
A = alphabet(G)
λ = Groups.ΡΛ(:λ, A, 2genus)
ϱ = Groups.ΡΛ(:ϱ, A, 2genus)
import Groups: Ta, Tα, Te
halftwists = map(1:genus-1) do i
j = i + 1
x = Ta(λ, j) * inv(A, Ta(λ, i)) * Tα(λ, j) * Te(λ, ϱ, i, j)
δ = x * Tα(λ, i) * inv(A, x)
c =
inv(A, Ta(λ, j)) *
Te(λ, ϱ, i, j) *
Tα(λ, i)^2 *
inv(A, δ) *
inv(A, Ta(λ, j)) *
Ta(λ, i) *
δ
z =
Te(λ, ϱ, j, i) *
inv(A, Ta(λ, i)) *
Tα(λ, i) *
Ta(λ, i) *
inv(A, Te(λ, ϱ, j, i))
G(Ta(λ, i) * inv(A, Ta(λ, j) * Tα(λ, j))^6 * (Ta(λ, j) * Tα(λ, j) * z)^4 * c)
end
τ = (G(Ta(λ, 1) * Tα(λ, 1))^6) * prod(halftwists, init = one(G))
# τ^genus is trivial but only in autπ₁Σ₄
# here we check its centrality
τᵍ = τ^genus
@test_broken all(a * τᵍ == τᵍ * a for a in Groups.gens(G))
end
end end

1
test/runtests.jl
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@ -26,6 +26,7 @@ include(joinpath(pathof(GroupsCore), "..", "..", "test", "conformance_test.jl"))
include("fp_groups.jl") include("fp_groups.jl")
include("AutFn.jl") include("AutFn.jl")
include("AutSigma_41.jl")
# if !haskey(ENV, "CI") # if !haskey(ENV, "CI")
# include("benchmarks.jl") # include("benchmarks.jl")