add tests for Aut(Σ₄.₁)

Project.toml

@@ -22,7 +22,8 @@ julia = "1.3, 1.4, 1.5"
 
 [extras]
 BenchmarkTools = "6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf"
+PermutationGroups = "8bc5a954-2dfc-11e9-10e6-cd969bffa420"
 Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
 
 [targets]
-test = ["Test", "BenchmarkTools"]
+test = ["Test", "BenchmarkTools", "PermutationGroups"]

src/groups/mcg.jl

@@ -1,5 +1,3 @@
-include("symplectic_twists.jl")
-
 struct SurfaceGroup{T, S, R} <: AbstractFPGroup
     genus::Int
     boundaries::Int
@@ -8,6 +6,8 @@ struct SurfaceGroup{T, S, R} <: AbstractFPGroup
     rws::R
 end
 
+include("symplectic_twists.jl")
+
 genus(S::SurfaceGroup) = S.genus
 
 function Base.show(io::IO, S::SurfaceGroup)

test/AutSigma_41.jl

@@ -0,0 +1,193 @@
+using PermutationGroups
+
+@testset "Wajnryb presentation for Σ₄₁" begin
+
+    genus = 4
+
+    G = New.SpecialAutomorphismGroup(New.FreeGroup(2genus))
+
+    T = let G = G; (Tas, Tαs, Tes) = New.mcg_twists(genus)
+        Ta = G.(Tas)
+        Tα = G.(Tαs)
+        Tes = G.(Tes)
+
+        [Ta; Tα; Tes]
+    end
+
+    a1 = T[1]^-1 # Ta₁
+    a2 = T[5]^-1 # Tα₁
+    a3 = T[9]^-1 # Te₁₂
+    a4 = T[6]^-1 # Tα₂
+    a5 = T[12]^-1 # Te₂₃
+    a6 = T[7]^-1 # Tα₃
+    a7 = T[14]^-1 # Te₃₄
+    a8 = T[8]^-1 # Tα₄
+
+    b0 = T[2]^-1 # Ta₂
+    a0 = (a1*a2*a3)^4*b0^-1 # from the 3-chain relation
+    X = a4*a5*a3*a4 # auxillary, not present in the Primer
+    b1 = X^-1*a0*X
+    b2 = T[10]^-1 # Te₁₃
+
+    As = T[[1,5,9,6,12,7,14,8]] # the inverses of the elements a
+
+    @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 (j, aj) in enumerate(As)
+                if abs(i-j) > 1
+                    @test ai*aj == aj*ai
+                elseif i ≠ j
+                    @test ai*aj != aj*ai
+                end
+            end
+            if i != 4
+                @test a0*ai == ai*a0
+            end
+        end
+    end
+
+    @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 (j, aj) in enumerate(As)
+                if abs(i-j) == 1
+                    @test ai*aj*ai == aj*ai*aj
+                end
+            end
+        end
+        @test a0*a4*a0 == a4*a0*a4 # here, a0 and a4 are as before
+    end
+
+    @testset "Lantern relation" begin
+
+        @testset "b2 definition" begin
+                @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
+            d = New.domain(b2)
+            im = New.evaluate(b2)
+            z = im[7]*d[7]^-1
+
+            @test im[1] == d[1]
+            @test im[2] == z*d[2]*z^-1
+            @test im[3] == z*d[3]*z^-1
+            @test im[4] == d[4]
+
+            @test im[5] == d[5]*z^-1
+            @test im[6] == z*d[6]*z^-1
+            @test im[7] == z*d[7]
+            @test im[8] == d[8]
+
+        end
+
+        @testset "b2: commutation relations" begin
+            @test b2*a1 == a1*b2
+            @test b2*a2 != a2*b2
+            @test b2*a3 == a3*b2
+            @test b2*a4 == a4*b2
+            @test b2*a5 == a5*b2
+            @test b2*a6 != a6*b2
+        end
+
+        @testset "b2: braid relations" begin
+            @test a2*b2*a2 == b2*a2*b2
+            @test a6*b2*a6 == b2*a6*b2
+        end
+
+        @testset "lantern" begin
+            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 = (a4^-1*a3^-1*a2^-1*a1^-1*u*a2*a3*a4*a5*a6)
+            @time New.evaluate(x)
+            b3 = x*a0*x^-1
+            @time New.evaluate(b3)
+            @test a0*b2*b1 == a1*a3*a5*b3
+        end
+    end
+
+
+    @testset "Te₁₂ definition" begin
+        G = parent(first(T))
+        F₈ = New.object(G)
+        (a, b, c, d, α, β, γ, δ) = gens(F₈)
+
+        A = KnuthBendix.alphabet(G)
+
+        λ = [i == j ? one(G) : G([A[New.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]
+
+        g = one(G)
+        # @show g
+        # @show g(Groups.domain(G))
+
+        # β ↦ α*β
+        g *= λ[6,5]
+        @test New.evaluate(g)[6] == α*β
+
+        # α ↦ a*α*b^-1
+        g *= λ[5,1]*inv(ϱ[5,2])
+        @test New.evaluate(g)[5] == a*α*b^-1
+
+        # β ↦ b*α^-1*a^-1*α*β
+        g *= inv(λ[6,5])
+        @test New.evaluate(g)[6] == b*α^-1*a^-1*α*β
+
+        # b ↦ α
+        g *= λ[2,5]*inv(λ[2,1]);
+        @test New.evaluate(g)[2] == α
+
+        # b ↦ b*α^-1*a^-1*α
+        g *= inv(λ[2,5]);
+        @test New.evaluate(g)[2] == b*α^-1*a^-1*α
+
+        # b ↦ b*α^-1*a^-1*α*b*α^-1
+        g *= inv(ϱ[2,5])*ϱ[2,1];
+        @test New.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
+        g *= ϱ[2,5];
+        @test New.evaluate(g)[2] == b*α^-1*a^-1*α*b*α^-1*a*α*b^-1
+
+        x = b*α^-1*a^-1*α
+        @test New.evaluate(g) == # (a, b,        c, d, α,      β,   γ, δ)
+                                   (a, x*b*x^-1, c, d, α*x^-1, x*β, γ, δ)
+        @test g == T[9]
+    end
+
+    Base.conj(t::New.Transvection, p::Perm) =
+        New.Transvection(t.id, t.i^p, t.j^p, t.inv)
+
+    function Base.conj(elt::New.FPGroupElement, p::Perm)
+        G = parent(elt)
+        A = New.alphabet(elt)
+        return G([A[conj(A[idx], p)] for idx in New.word(elt)])
+    end
+
+    @testset "Te₂₃ definition" begin
+        Te₁₂, Te₂₃ = T[9], T[12]
+        G = parent(Te₁₂)
+        F₈ = New.object(G)
+        (a, b, c, d, α, β, γ, δ) = gens(F₈)
+
+        img_Te₂₃ = New.evaluate(Te₂₃)
+        y = c*β^-1*b^-1*β
+        @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)"
+        Te₂₃_σ = conj(Te₁₂, σ)
+        # @test New.word(Te₂₃_σ) == New.word(Te₂₃)
+
+        @test New.evaluate(Te₂₃_σ) == New.evaluate(Te₂₃)
+        @test Te₂₃ == Te₂₃_σ
+    end
+
+    @testset "Te₃₄ definition" begin
+        Te₁₂, Te₃₄ = T[9], T[14]
+        G = parent(Te₁₂)
+        F₈ = New.object(G)
+        (a, b, c, d, α, β, γ, δ) = Groups.gens(F₈)
+
+        σ = perm"(1,3)(2,4)(5,7)(6,8)"
+        Te₃₄_σ = conj(Te₁₂, σ)
+        @test Te₃₄ == Te₃₄_σ
+    end
+end

test/runtests.jl

@@ -35,6 +35,8 @@ using LinearAlgebra
       include("fp_groups.jl")
 
       include("AutFn.jl")
+      include("AutSigma_41.jl")
+
       include("benchmarks.jl")
    end
 end