mirror of
https://github.com/kalmarek/PropertyT.jl.git
synced 2024-09-07 07:11:42 +02:00
This includes: * PermutationGroups as PG * SymbolicWedderburn as SW * StarAlgebras as SA
171 lines
5.7 KiB
Julia
171 lines
5.7 KiB
Julia
countmap(v) = countmap(identity, v)
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function countmap(f, v)
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counts = Dict{eltype(f(first(v))),Int}()
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for x in v
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fx = f(x)
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counts[fx] = get!(counts, fx, 0) + 1
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end
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return counts
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end
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@testset "classify_root_system" begin
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α = PropertyT.Roots.Root([1, -1, 0])
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β = PropertyT.Roots.Root([0, 1, -1])
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γ = PropertyT.Roots.Root([2, 0, 0])
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@test PropertyT.Roots.classify_root_system(α, β, (false, false)) == :A₂
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@test PropertyT.Roots.classify_root_system(α, γ, (false, true)) == :C₂
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@test PropertyT.Roots.classify_root_system(β, γ, (false, true)) ==
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Symbol("A₁×C₁")
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end
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@testset "Exceptional root systems" begin
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Base.:^(t::NTuple{N}, p::PG.AbstractPermutation) where {N} =
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ntuple(i -> t[i^p], N)
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@testset "F4" begin
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F4 = let Σ = PG.PermGroup(PG.perm"(1,2,3,4)", PG.perm"(1,2)")
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long = let x = (1, 1, 0, 0) .// 1
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PropertyT.Roots.Root.(
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union(
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(x^g for g in Σ),
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((x .* (-1, 1, 1, 1))^g for g in Σ),
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((-1 .* x)^g for g in Σ),
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),
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)
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end
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short = let x = (1, 0, 0, 0) .// 1
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PropertyT.Roots.Root.(
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union((x^g for g in Σ), ((-1 .* x)^g for g in Σ))
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)
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end
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signs = collect(Iterators.product(fill([-1, +1], 4)...))
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halfs = let x = (1, 1, 1, 1) .// 2
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PropertyT.Roots.Root.(union(x .* sgn for sgn in signs))
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end
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union(long, short, halfs)
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end
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@test length(F4) == 48
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a = F4[1]
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@test isapprox(PropertyT.Roots.ℓ₂length(a), sqrt(2))
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b = F4[6]
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@test isapprox(PropertyT.Roots.ℓ₂length(b), sqrt(2))
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c = a + b
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@test isapprox(PropertyT.Roots.ℓ₂length(c), 2.0)
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@test PropertyT.Roots.classify_root_system(b, c, (false, true)) == :C₂
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long = F4[findfirst(r -> PropertyT.Roots.ℓ₂length(r) == sqrt(2), F4)]
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short = F4[findfirst(r -> PropertyT.Roots.ℓ₂length(r) == 1.0, F4)]
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subtypes = Set([:C₂, :A₂, Symbol("A₁×C₁")])
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let Ω = F4, α = long
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counts = countmap([
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PropertyT.Roots.classify_sub_root_system(Ω, α, γ) for
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γ in Ω if !PropertyT.Roots.isproportional(α, γ)
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])
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@test Set(keys(counts)) == subtypes
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d, r = divrem(counts[:C₂], 6)
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@test r == 0 && d == 3
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d, r = divrem(counts[:A₂], 4)
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@test r == 0 && d == 4
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end
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let Ω = F4, α = short
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counts = countmap([
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PropertyT.Roots.classify_sub_root_system(Ω, α, γ) for
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γ in Ω if !PropertyT.Roots.isproportional(α, γ)
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])
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@test Set(keys(counts)) == subtypes
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d, r = divrem(counts[:C₂], 6)
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@test r == 0 && d == 3
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d, r = divrem(counts[:A₂], 4)
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@test r == 0 && d == 4
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end
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end
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@testset "E6-7-8 exceptional root systems" begin
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E8 =
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let Σ = PG.PermGroup(
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PG.perm"(1,2,3,4,5,6,7,8)",
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PG.perm"(1,2)",
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)
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long = let x = (1, 1, 0, 0, 0, 0, 0, 0) .// 1
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PropertyT.Roots.Root.(
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union(
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(x^g for g in Σ),
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((x .* (-1, 1, 1, 1, 1, 1, 1, 1))^g for g in Σ),
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((-1 .* x)^g for g in Σ),
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),
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)
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end
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signs = collect(
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p for p in Iterators.product(fill([-1, +1], 8)...) if
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iseven(count(==(-1), p))
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)
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halfs = let x = (1, 1, 1, 1, 1, 1, 1, 1) .// 2
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rts = unique(PropertyT.Roots.Root(x .* sgn) for sgn in signs)
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end
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union(long, halfs)
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end
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subtypes = Set([:A₂, Symbol("A₁×A₁")])
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@testset "E8" begin
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@test length(E8) == 240
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@test all(r -> PropertyT.Roots.ℓ₂length(r) ≈ sqrt(2), E8)
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let Ω = E8, α = first(Ω)
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counts = countmap([
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PropertyT.Roots.classify_sub_root_system(Ω, α, γ) for
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γ in Ω if !PropertyT.Roots.isproportional(α, γ)
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])
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@test Set(keys(counts)) == subtypes
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d, r = divrem(counts[:A₂], 4)
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@test r == 0 && d == 28
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end
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end
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@testset "E7" begin
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E7 = filter(r -> iszero(sum(r.coord)), E8)
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@test length(E7) == 126
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let Ω = E7, α = first(Ω)
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counts = countmap([
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PropertyT.Roots.classify_sub_root_system(Ω, α, γ) for
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γ in Ω if !PropertyT.Roots.isproportional(α, γ)
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])
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@test Set(keys(counts)) == subtypes
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d, r = divrem(counts[:A₂], 4)
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@test r == 0 && d == 16
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end
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end
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@testset "E6" begin
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E6 = filter(
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r -> r.coord[end] == r.coord[end-1] == r.coord[end-2],
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E8,
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)
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@test length(E6) == 72
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let Ω = E6, α = first(Ω)
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counts = countmap([
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PropertyT.Roots.classify_sub_root_system(Ω, α, γ) for
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γ in Ω if !PropertyT.Roots.isproportional(α, γ)
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])
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@test Set(keys(counts)) == subtypes
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d, r = divrem(counts[:A₂], 4)
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@info d, r
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@test r == 0 && d == 10
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end
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end
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end
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end
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