kalmar/Groups.jl
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create a general/saner homomorphism evaluation architecture

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kalmarek 2020-03-25 15:43:38 +01:00
parent ebefc7e399
commit 6d22c82ab3
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3 changed files with 49 additions and 31 deletions
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24
src/AutGroup.jl
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@ -297,24 +297,28 @@ function reduce!(w::Automorphism)
return W return W
end end
function linear_repr(A::Automorphism{N}, hom=matrix_repr) where N ###############################################################################
return reduce(*, linear_repr.(A.symbols, N, hom), init=hom(Identity(),N,1)) #
end # Abelianization (natural Representation to GL(N,Z))
#
linear_repr(a::AutSymbol, n::Int, hom) = hom(a.fn, n, a.pow) abelianize(A::Automorphism{N}) where N = image(A, abelianize; n=N)
function matrix_repr(a::Union{RTransvect, LTransvect}, n::Int, pow) # homomorphism definition
abelianize(; n::Integer=1) = Matrix{Int}(I, n, n)
abelianize(a::AutSymbol; n::Int=1) = abelianize(a.fn, n, a.pow)
function abelianize(a::Union{RTransvect, LTransvect}, n::Int, pow)
x = Matrix{Int}(I, n, n) x = Matrix{Int}(I, n, n)
x[a.i,a.j] = pow x[a.i,a.j] = pow
return x return x
end end
function matrix_repr(a::FlipAut, n::Int, pow) function abelianize(a::FlipAut, n::Int, pow)
x = Matrix{Int}(I, n, n) x = Matrix{Int}(I, n, n)
x[a.i,a.i] = -1^pow x[a.i,a.i] = -1
return x return x
end end
matrix_repr(a::PermAut, n::Int, pow) = Matrix{Int}(I, n, n)[(a.perm^pow).d, :] abelianize(a::PermAut, n::Integer, pow) = Matrix{Int}(I, n, n)[(a.perm^pow).d, :]
abelianize(a::Identity, n::Integer, pow) = abelianize(;n=n)
matrix_repr(a::Identity, n::Int, pow) = Matrix{Int}(I, n, n)

14
src/Groups.jl
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@ -13,6 +13,7 @@ import Base: deepcopy_internal
using LinearAlgebra using LinearAlgebra
using Markdown using Markdown
export gens, FreeGroup, Aut, SAut
include("types.jl") include("types.jl")
@ -89,4 +90,17 @@ function generate_balls(S::AbstractVector{T}, center::T=one(first(S));
return c.*B, sizes return c.*B, sizes
end end
@doc doc"""
image(A::GWord, homomorphism; kwargs...)
Evaluate homomorphism `homomorphism` on a GWord `A`.
`homomorphism` needs implement
> `hom(s; kwargs...)`,
where `hom(;kwargs...)` evaluates the value at the identity element.
"""
function image(w::GWord, hom; kwargs...)
return reduce(*,
(hom(s; kwargs...) for s in syllables(w)),
init = hom(;kwargs...))
end
end # of module Groups end # of module Groups

42
test/AutGroup-tests.jl
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@ -9,7 +9,7 @@
@test isa(f, Groups.GSymbol) @test isa(f, Groups.GSymbol)
@test isa(f, Groups.AutSymbol) @test isa(f, Groups.AutSymbol)
@test isa(Groups.AutSymbol(perm"(4)"), Groups.AutSymbol) @test isa(Groups.AutSymbol(perm"(4)"), Groups.AutSymbol)
@test isa(Groups.AutSymbol([2,3,4,1]), Groups.AutSymbol) @test isa(Groups.AutSymbol(perm"(1,2,3,4)"), Groups.AutSymbol)
@test isa(Groups.transvection_R(1,2), Groups.AutSymbol) @test isa(Groups.transvection_R(1,2), Groups.AutSymbol)
@test isa(Groups.transvection_R(3,4), Groups.AutSymbol) @test isa(Groups.transvection_R(3,4), Groups.AutSymbol)
@test isa(Groups.flip(3), Groups.AutSymbol) @test isa(Groups.flip(3), Groups.AutSymbol)
@ -205,48 +205,48 @@
@test length(unique(B_2)) == 1777 @test length(unique(B_2)) == 1777
end end
@testset "linear_repr tests" begin @testset "abelianization homomorphism" begin
N = 3 N = 4
G = AutGroup(FreeGroup(N)) G = AutGroup(FreeGroup(N))
S = unique([gens(G); inv.(gens(G))]) S = unique([gens(G); inv.(gens(G))])
R = 3 R = 3
@test Groups.linear_repr(one(G)) isa Matrix{Int} @test Groups.abelianize(one(G)) isa Matrix{Int}
@test Groups.linear_repr(one(G)) == Matrix{Int}(I, N, N) @test Groups.abelianize(one(G)) == Matrix{Int}(I, N, N)
M = Matrix{Int}(I, N, N) M = Matrix{Int}(I, N, N)
M[1,2] = 1 M[1,2] = 1
ϱ₁₂ = G(Groups.transvection_R(1,2)) ϱ₁₂ = G(Groups.ϱ(1,2))
λ₁₂ = G(Groups.transvection_R(1,2)) λ₁₂ = G(Groups.λ(1,2))
@test Groups.linear_repr(ϱ₁₂) == M @test Groups.abelianize(ϱ₁₂) == M
@test Groups.linear_repr(λ₁₂) == M @test Groups.abelianize(λ₁₂) == M
M[1,2] = -1 M[1,2] = -1
@test Groups.linear_repr(ϱ₁₂^-1) == M @test Groups.abelianize(ϱ₁₂^-1) == M
@test Groups.linear_repr(λ₁₂^-1) == M @test Groups.abelianize(λ₁₂^-1) == M
@test Groups.linear_repr(ϱ₁₂*λ₁₂^-1) == Matrix{Int}(I, N, N) @test Groups.abelianize(ϱ₁₂*λ₁₂^-1) == Matrix{Int}(I, N, N)
@test Groups.linear_repr(λ₁₂^-1*ϱ₁₂) == Matrix{Int}(I, N, N) @test Groups.abelianize(λ₁₂^-1*ϱ₁₂) == Matrix{Int}(I, N, N)
M = Matrix{Int}(I, N, N) M = Matrix{Int}(I, N, N)
M[2,2] = -1 M[2,2] = -1
ε₂ = G(Groups.flip(2)) ε₂ = G(Groups.flip(2))
@test Groups.linear_repr(ε₂) == M @test Groups.abelianize(ε₂) == M
@test Groups.linear_repr(ε₂^2) == Matrix{Int}(I, N, N) @test Groups.abelianize(ε₂^2) == Matrix{Int}(I, N, N)
M = [0 1 0; 0 0 1; 1 0 0] M = [0 1 0 0; 0 0 0 1; 0 0 1 0; 1 0 0 0]
σ = G(Groups.AutSymbol(perm"(1,2,3)")) σ = G(Groups.AutSymbol(perm"(1,2,4)"))
@test Groups.linear_repr(σ) == M @test Groups.abelianize(σ) == M
@test Groups.linear_repr(σ^3) == Matrix{Int}(I, 3, 3) @test Groups.abelianize(σ^3) == Matrix{Int}(I, N, N)
@test Groups.linear_repr(σ)^3 == Matrix{Int}(I, 3, 3) @test Groups.abelianize(σ)^3 == Matrix{Int}(I, N, N)
function test_homomorphism(S, r) function test_homomorphism(S, r)
for elts in Iterators.product([[g for g in S] for _ in 1:r]...) for elts in Iterators.product([[g for g in S] for _ in 1:r]...)
prod(Groups.linear_repr.(elts)) == Groups.linear_repr(prod(elts)) || error("linear representaton test failed at $elts") prod(Groups.abelianize.(elts)) == Groups.abelianize(prod(elts)) || error("linear representaton test failed at $elts")
end end
return 0 return 0
end end