Merge a05a0798ec into f4936dd50a
This commit is contained in:
Marek Kaluba
GitHub
commit
cb6c97a28d
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@ -112,7 +112,7 @@ function certify_solution(
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!augmented && StarAlgebras.aug(elt) == StarAlgebras.aug(orderunit) == 0
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Q = should_we_augment ? augment_columns!(Q) : Q
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@time sos = compute_sos(parent(elt), Q; augmented = augmented)
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sos = compute_sos(parent(elt), Q; augmented = augmented)
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@info "Checking in $(eltype(sos)) arithmetic with" λ
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@ -123,11 +123,9 @@ function certify_solution(
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end
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λ_int = IntervalArithmetic.@interval(λ)
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Q_int = IntervalMatrices.IntervalMatrix([
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IntervalArithmetic.@interval(q) for q in Q
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])
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Q_int = IntervalMatrices.IntervalMatrix(IntervalArithmetic.Interval.(Q))
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check, sos_int = @time if should_we_augment
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check, sos_int = if should_we_augment
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@info("Projecting columns of Q to the augmentation ideal...")
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Q_int = augment_columns!(Q_int)
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@info "Checking that sum of every column contains 0.0..."
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@ -156,12 +156,8 @@ function constraints(
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mstr::StarAlgebras.MultiplicativeStructure;
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augmented = false,
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)
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cnstrs = _constraints(
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mstr;
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augmented = augmented,
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num_constraints = length(basis),
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id = basis[one(first(basis))],
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)
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id = basis[one(first(basis))]
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cnstrs = _constraints(mstr; augmented = augmented, id = mstr[id, id])
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return Dict(
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basis[i] => ConstraintMatrix(c, size(mstr)..., 1) for
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@ -172,11 +168,11 @@ end
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function _constraints(
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mstr::StarAlgebras.MultiplicativeStructure;
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augmented::Bool = false,
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num_constraints = maximum(mstr),
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id,
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)
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cnstrs = [signed(eltype(mstr))[] for _ in 1:num_constraints]
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cnstrs = [signed(eltype(mstr))[] for _ in 1:maximum(mstr)]
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LI = LinearIndices(size(mstr))
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id_ = mstr[id, id]
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for ci in CartesianIndices(size(mstr))
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k = LI[ci]
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@ -185,8 +181,8 @@ function _constraints(
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push!(cnstrs[a_star_b], k)
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if augmented
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# (1-a)'(1-b) = 1 - a' - b + a'b
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push!(cnstrs[id], k)
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a_star, b = mstr[-i, id], j
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push!(cnstrs[id_], k)
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a_star, b = mstr[-i, id], mstr[j, id]
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push!(cnstrs[a_star], -k)
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push!(cnstrs[b], -k)
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end
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@ -67,13 +67,3 @@ function Sq(rootsystem::AbstractDict)
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init = zero(first(values(rootsystem))),
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)
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end
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function level(rootsystem, level::Integer)
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1 ≤ level ≤ 4 || throw("level is implemented only for i ∈{1,2,3,4}")
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level == 1 && return Adj(rootsystem, :C₁) # always positive
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level == 2 && return Adj(rootsystem, :A₁) +
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Adj(rootsystem, Symbol("C₁×C₁")) +
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Adj(rootsystem, :C₂) # C₂ is not positive
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level == 3 && return Adj(rootsystem, :A₂) + Adj(rootsystem, Symbol("A₁×C₁"))
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level == 4 && return Adj(rootsystem, Symbol("A₁×A₁")) # positive
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end
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@ -8,160 +8,301 @@ function countmap(f, v)
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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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@testset "Chevalley" begin
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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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@testset "F4" begin
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F4 = let Σ = PermutationGroups.PermGroup(perm"(1,2,3,4)", 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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@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 "E6-7-8 exceptional root systems" begin
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E8 =
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let Σ = PermutationGroups.PermGroup(
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perm"(1,2,3,4,5,6,7,8)",
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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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@testset "Exceptional root systems" begin
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@testset "F4" begin
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F4 =
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let Σ = PermutationGroups.PermGroup(
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perm"(1,2,3,4)",
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perm"(1,2)",
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)
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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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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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@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)) ==
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:C₂
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long =
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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 = PermutationGroups.PermGroup(
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perm"(1,2,3,4,5,6,7,8)",
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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
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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(
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PropertyT.Roots.Root(x .* sgn) for sgn in signs
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)
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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)
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for 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)
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for 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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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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@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)
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for 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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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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end
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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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@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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@testset "Levels in Sp2n" begin
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||||
function level(rootsystem, level::Integer)
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1 ≤ level ≤ 4 || throw("level is implemented only for i ∈{1,2,3,4}")
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level == 1 && return PropertyT.Adj(rootsystem, :C₁) # always positive
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level == 2 && return PropertyT.Adj(rootsystem, :A₁) +
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PropertyT.Adj(rootsystem, Symbol("C₁×C₁")) +
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PropertyT.Adj(rootsystem, :C₂) # C₂ is not positive
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level == 3 && return PropertyT.Adj(rootsystem, :A₂) +
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PropertyT.Adj(rootsystem, Symbol("A₁×C₁"))
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level == 4 && return PropertyT.Adj(rootsystem, Symbol("A₁×A₁")) # positive
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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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n = 5
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G = MatrixGroups.SymplecticGroup{2n}(Int8)
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RG, S, sizes = PropertyT.group_algebra(G; halfradius = 1)
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Weyl = let N = n
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P = PermGroup(perm"(1,2)", Perm(circshift(1:N, -1)))
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Groups.Constructions.WreathProduct(PermGroup(perm"(1,2)"), P)
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end
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act = PropertyT.action_by_conjugation(G, Weyl)
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||||
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||||
function ^ᵃ(x, w::Groups.Constructions.WreathProductElement)
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return SymbolicWedderburn.action(act, w, x)
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||||
end
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||||
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||||
Sₙ = S
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||||
Δsₙ = PropertyT.laplacians(
|
||||
RG,
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||||
Sₙ,
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x -> (gx = PropertyT.grading(x); Set([gx, -gx])),
|
||||
)
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||||
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function natural_embedding(i, Sp2m, Sp2n)
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_dim(::MatrixGroups.ElementarySymplectic{N}) where {N} = N
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n = _dim(first(alphabet(Sp2n))) ÷ 2
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m = _dim(first(alphabet(Sp2m))) ÷ 2
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l = alphabet(Sp2m)[i]
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||||
i, j = if l.symbol === :A
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||||
l.i, l.j
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||||
elseif l.symbol === :B
|
||||
ifelse(l.i ≤ m, (l.i, l.j - m + n), (l.i - m + n, l.j))
|
||||
else
|
||||
throw("unknown type: $(l.symbol)")
|
||||
end
|
||||
image_of_l =
|
||||
MatrixGroups.ElementarySymplectic{2n}(l.symbol, i, j, l.val)
|
||||
return Groups.word_type(Sp2n)([alphabet(Sp2n)[image_of_l]])
|
||||
end
|
||||
|
||||
@testset "Sp4 ↪ Sp12" begin
|
||||
m = 2
|
||||
Sₘ = let m = m, Sp2n = G
|
||||
Sp2m = MatrixGroups.SymplecticGroup{2m}(Int8)
|
||||
h = Groups.Homomorphism(
|
||||
natural_embedding,
|
||||
Sp2m,
|
||||
Sp2n;
|
||||
check = false,
|
||||
)
|
||||
S = h.(gens(Sp2m))
|
||||
S = union!(S, inv.(S))
|
||||
end
|
||||
|
||||
Δsₘ = PropertyT.laplacians(
|
||||
RG,
|
||||
Sₘ,
|
||||
x -> (gx = PropertyT.grading(x); Set([gx, -gx])),
|
||||
)
|
||||
@test length(E6) == 72
|
||||
|
||||
let Ω = E6, α = first(Ω)
|
||||
counts = countmap([
|
||||
PropertyT.Roots.classify_sub_root_system(Ω, α, γ) for
|
||||
γ in Ω if !PropertyT.Roots.isproportional(α, γ)
|
||||
])
|
||||
@test Set(keys(counts)) == subtypes
|
||||
d, r = divrem(counts[:A₂], 4)
|
||||
@info d, r
|
||||
@test r == 0 && d == 10
|
||||
function k(n, m, i)
|
||||
return 2^n * factorial(m) * factorial(n - i) ÷ factorial(m - i)
|
||||
end
|
||||
|
||||
@testset "Level $i" for i in 1:4
|
||||
Levᵢᵐ = level(Δsₘ, i)
|
||||
Levᵢⁿ = level(Δsₙ, i)
|
||||
|
||||
if 1 ≤ i ≤ 2
|
||||
@test !iszero(Levᵢᵐ)
|
||||
@time Σ_W_Levᵢᵐ = sum(Levᵢᵐ^ᵃw for w in Weyl)
|
||||
|
||||
@test isinteger(Σ_W_Levᵢᵐ[one(G)] / Levᵢⁿ[one(G)])
|
||||
@test Σ_W_Levᵢᵐ[one(G)] / Levᵢⁿ[one(G)] == k(n, m, i)
|
||||
@test Σ_W_Levᵢᵐ == k(n, m, i) * Levᵢⁿ
|
||||
else
|
||||
@test iszero(Levᵢᵐ)
|
||||
@test !iszero(Levᵢⁿ)
|
||||
end
|
||||
end
|
||||
end
|
||||
|
||||
@testset "Sp8 ↪ Sp12" begin
|
||||
m = 4
|
||||
Sₘ = let m = m, Sp2n = G
|
||||
Sp2m = MatrixGroups.SymplecticGroup{2m}(Int8)
|
||||
h = Groups.Homomorphism(
|
||||
natural_embedding,
|
||||
Sp2m,
|
||||
Sp2n;
|
||||
check = false,
|
||||
)
|
||||
S = h.(gens(Sp2m))
|
||||
S = union!(S, inv.(S))
|
||||
end
|
||||
|
||||
Δsₘ = PropertyT.laplacians(
|
||||
RG,
|
||||
Sₘ,
|
||||
x -> (gx = PropertyT.grading(x); Set([gx, -gx])),
|
||||
)
|
||||
|
||||
function k(n, m, i)
|
||||
return 2^n * factorial(m) * factorial(n - i) ÷ factorial(m - i)
|
||||
end
|
||||
|
||||
@testset "Level $i" for i in 1:4
|
||||
Levᵢᵐ = level(Δsₘ, i)
|
||||
Levᵢⁿ = level(Δsₙ, i)
|
||||
|
||||
@test !iszero(Levᵢᵐ)
|
||||
@time Σ_W_Levᵢᵐ = sum(Levᵢᵐ^ᵃw for w in Weyl)
|
||||
|
||||
@test isinteger(Σ_W_Levᵢᵐ[one(G)] / Levᵢⁿ[one(G)])
|
||||
@test Σ_W_Levᵢᵐ[one(G)] / Levᵢⁿ[one(G)] == k(n, m, i)
|
||||
@test Σ_W_Levᵢᵐ == k(n, m, i) * Levᵢⁿ
|
||||
end
|
||||
end
|
||||
end
|
||||
|
|
|
|||
Loading…
Reference in New Issue