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Marek Kaluba 2024-02-18 17:56:25 +01:00 committed by GitHub
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4 changed files with 289 additions and 164 deletions

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@ -112,7 +112,7 @@ function certify_solution(
!augmented && StarAlgebras.aug(elt) == StarAlgebras.aug(orderunit) == 0 !augmented && StarAlgebras.aug(elt) == StarAlgebras.aug(orderunit) == 0
Q = should_we_augment ? augment_columns!(Q) : Q Q = should_we_augment ? augment_columns!(Q) : Q
@time sos = compute_sos(parent(elt), Q; augmented = augmented) sos = compute_sos(parent(elt), Q; augmented = augmented)
@info "Checking in $(eltype(sos)) arithmetic with" λ @info "Checking in $(eltype(sos)) arithmetic with" λ
@ -123,11 +123,9 @@ function certify_solution(
end end
λ_int = IntervalArithmetic.@interval(λ) λ_int = IntervalArithmetic.@interval(λ)
Q_int = IntervalMatrices.IntervalMatrix([ Q_int = IntervalMatrices.IntervalMatrix(IntervalArithmetic.Interval.(Q))
IntervalArithmetic.@interval(q) for q in Q
])
check, sos_int = @time if should_we_augment check, sos_int = if should_we_augment
@info("Projecting columns of Q to the augmentation ideal...") @info("Projecting columns of Q to the augmentation ideal...")
Q_int = augment_columns!(Q_int) Q_int = augment_columns!(Q_int)
@info "Checking that sum of every column contains 0.0..." @info "Checking that sum of every column contains 0.0..."

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@ -156,12 +156,8 @@ function constraints(
mstr::StarAlgebras.MultiplicativeStructure; mstr::StarAlgebras.MultiplicativeStructure;
augmented = false, augmented = false,
) )
cnstrs = _constraints( id = basis[one(first(basis))]
mstr; cnstrs = _constraints(mstr; augmented = augmented, id = mstr[id, id])
augmented = augmented,
num_constraints = length(basis),
id = basis[one(first(basis))],
)
return Dict( return Dict(
basis[i] => ConstraintMatrix(c, size(mstr)..., 1) for basis[i] => ConstraintMatrix(c, size(mstr)..., 1) for
@ -172,11 +168,11 @@ end
function _constraints( function _constraints(
mstr::StarAlgebras.MultiplicativeStructure; mstr::StarAlgebras.MultiplicativeStructure;
augmented::Bool = false, augmented::Bool = false,
num_constraints = maximum(mstr),
id, id,
) )
cnstrs = [signed(eltype(mstr))[] for _ in 1:num_constraints] cnstrs = [signed(eltype(mstr))[] for _ in 1:maximum(mstr)]
LI = LinearIndices(size(mstr)) LI = LinearIndices(size(mstr))
id_ = mstr[id, id]
for ci in CartesianIndices(size(mstr)) for ci in CartesianIndices(size(mstr))
k = LI[ci] k = LI[ci]
@ -185,8 +181,8 @@ function _constraints(
push!(cnstrs[a_star_b], k) push!(cnstrs[a_star_b], k)
if augmented if augmented
# (1-a)'(1-b) = 1 - a' - b + a'b # (1-a)'(1-b) = 1 - a' - b + a'b
push!(cnstrs[id], k) push!(cnstrs[id_], k)
a_star, b = mstr[-i, id], j a_star, b = mstr[-i, id], mstr[j, id]
push!(cnstrs[a_star], -k) push!(cnstrs[a_star], -k)
push!(cnstrs[b], -k) push!(cnstrs[b], -k)
end end

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@ -67,13 +67,3 @@ function Sq(rootsystem::AbstractDict)
init = zero(first(values(rootsystem))), init = zero(first(values(rootsystem))),
) )
end end
function level(rootsystem, level::Integer)
1 level 4 || throw("level is implemented only for i ∈{1,2,3,4}")
level == 1 && return Adj(rootsystem, :C₁) # always positive
level == 2 && return Adj(rootsystem, :A₁) +
Adj(rootsystem, Symbol("C₁×C₁")) +
Adj(rootsystem, :C₂) # C₂ is not positive
level == 3 && return Adj(rootsystem, :A₂) + Adj(rootsystem, Symbol("A₁×C₁"))
level == 4 && return Adj(rootsystem, Symbol("A₁×A₁")) # positive
end

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@ -8,160 +8,301 @@ function countmap(f, v)
return counts return counts
end end
@testset "classify_root_system" begin @testset "Chevalley" begin
α = PropertyT.Roots.Root([1, -1, 0]) @testset "classify_root_system" begin
β = PropertyT.Roots.Root([0, 1, -1]) α = PropertyT.Roots.Root([1, -1, 0])
γ = PropertyT.Roots.Root([2, 0, 0]) β = PropertyT.Roots.Root([0, 1, -1])
γ = PropertyT.Roots.Root([2, 0, 0])
@test PropertyT.Roots.classify_root_system(α, β, (false, false)) == :A₂ @test PropertyT.Roots.classify_root_system(α, β, (false, false)) == :A₂
@test PropertyT.Roots.classify_root_system(α, γ, (false, true)) == :C₂ @test PropertyT.Roots.classify_root_system(α, γ, (false, true)) == :C₂
@test PropertyT.Roots.classify_root_system(β, γ, (false, true)) == @test PropertyT.Roots.classify_root_system(β, γ, (false, true)) ==
Symbol("A₁×C₁") Symbol("A₁×C₁")
end
@testset "Exceptional root systems" begin
@testset "F4" begin
F4 = let Σ = PermutationGroups.PermGroup(perm"(1,2,3,4)", perm"(1,2)")
long = let x = (1, 1, 0, 0) .// 1
PropertyT.Roots.Root.(
union(
(x^g for g in Σ),
((x .* (-1, 1, 1, 1))^g for g in Σ),
((-1 .* x)^g for g in Σ),
),
)
end
short = let x = (1, 0, 0, 0) .// 1
PropertyT.Roots.Root.(
union((x^g for g in Σ), ((-1 .* x)^g for g in Σ))
)
end
signs = collect(Iterators.product(fill([-1, +1], 4)...))
halfs = let x = (1, 1, 1, 1) .// 2
PropertyT.Roots.Root.(union(x .* sgn for sgn in signs))
end
union(long, short, halfs)
end
@test length(F4) == 48
a = F4[1]
@test isapprox(PropertyT.Roots.₂length(a), sqrt(2))
b = F4[6]
@test isapprox(PropertyT.Roots.₂length(b), sqrt(2))
c = a + b
@test isapprox(PropertyT.Roots.₂length(c), 2.0)
@test PropertyT.Roots.classify_root_system(b, c, (false, true)) == :C₂
long = F4[findfirst(r -> PropertyT.Roots.₂length(r) == sqrt(2), F4)]
short = F4[findfirst(r -> PropertyT.Roots.₂length(r) == 1.0, F4)]
subtypes = Set([:C₂, :A₂, Symbol("A₁×C₁")])
let Ω = F4, α = long
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[:C₂], 6)
@test r == 0 && d == 3
d, r = divrem(counts[:A₂], 4)
@test r == 0 && d == 4
end
let Ω = F4, α = short
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[:C₂], 6)
@test r == 0 && d == 3
d, r = divrem(counts[:A₂], 4)
@test r == 0 && d == 4
end
end end
@testset "E6-7-8 exceptional root systems" begin @testset "Exceptional root systems" begin
E8 = @testset "F4" begin
let Σ = PermutationGroups.PermGroup( F4 =
perm"(1,2,3,4,5,6,7,8)", let Σ = PermutationGroups.PermGroup(
perm"(1,2)", perm"(1,2,3,4)",
) perm"(1,2)",
long = let x = (1, 1, 0, 0, 0, 0, 0, 0) .// 1
PropertyT.Roots.Root.(
union(
(x^g for g in Σ),
((x .* (-1, 1, 1, 1, 1, 1, 1, 1))^g for g in Σ),
((-1 .* x)^g for g in Σ),
),
) )
long = let x = (1, 1, 0, 0) .// 1
PropertyT.Roots.Root.(
union(
(x^g for g in Σ),
((x .* (-1, 1, 1, 1))^g for g in Σ),
((-1 .* x)^g for g in Σ),
),
)
end
short = let x = (1, 0, 0, 0) .// 1
PropertyT.Roots.Root.(
union((x^g for g in Σ), ((-1 .* x)^g for g in Σ))
)
end
signs = collect(Iterators.product(fill([-1, +1], 4)...))
halfs = let x = (1, 1, 1, 1) .// 2
PropertyT.Roots.Root.(union(x .* sgn for sgn in signs))
end
union(long, short, halfs)
end end
signs = collect( @test length(F4) == 48
p for p in Iterators.product(fill([-1, +1], 8)...) if
iseven(count(==(-1), p)) a = F4[1]
@test isapprox(PropertyT.Roots.₂length(a), sqrt(2))
b = F4[6]
@test isapprox(PropertyT.Roots.₂length(b), sqrt(2))
c = a + b
@test isapprox(PropertyT.Roots.₂length(c), 2.0)
@test PropertyT.Roots.classify_root_system(b, c, (false, true)) ==
:C₂
long =
F4[findfirst(r -> PropertyT.Roots.₂length(r) == sqrt(2), F4)]
short = F4[findfirst(r -> PropertyT.Roots.₂length(r) == 1.0, F4)]
subtypes = Set([:C₂, :A₂, Symbol("A₁×C₁")])
let Ω = F4, α = long
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[:C₂], 6)
@test r == 0 && d == 3
d, r = divrem(counts[:A₂], 4)
@test r == 0 && d == 4
end
let Ω = F4, α = short
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[:C₂], 6)
@test r == 0 && d == 3
d, r = divrem(counts[:A₂], 4)
@test r == 0 && d == 4
end
end
@testset "E6-7-8 exceptional root systems" begin
E8 =
let Σ = PermutationGroups.PermGroup(
perm"(1,2,3,4,5,6,7,8)",
perm"(1,2)",
)
long = let x = (1, 1, 0, 0, 0, 0, 0, 0) .// 1
PropertyT.Roots.Root.(
union(
(x^g for g in Σ),
((x .* (-1, 1, 1, 1, 1, 1, 1, 1))^g for g in Σ),
((-1 .* x)^g for g in Σ),
),
)
end
signs = collect(
p for
p in Iterators.product(fill([-1, +1], 8)...) if
iseven(count(==(-1), p))
)
halfs = let x = (1, 1, 1, 1, 1, 1, 1, 1) .// 2
rts = unique(
PropertyT.Roots.Root(x .* sgn) for sgn in signs
)
end
union(long, halfs)
end
subtypes = Set([:A₂, Symbol("A₁×A₁")])
@testset "E8" begin
@test length(E8) == 240
@test all(r -> PropertyT.Roots.₂length(r) sqrt(2), E8)
let Ω = E8, 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)
@test r == 0 && d == 28
end
end
@testset "E7" begin
E7 = filter(r -> iszero(sum(r.coord)), E8)
@test length(E7) == 126
let Ω = E7, 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)
@test r == 0 && d == 16
end
end
@testset "E6" begin
E6 = filter(
r -> r.coord[end] == r.coord[end-1] == r.coord[end-2],
E8,
) )
halfs = let x = (1, 1, 1, 1, 1, 1, 1, 1) .// 2 @test length(E6) == 72
rts = unique(PropertyT.Roots.Root(x .* sgn) for sgn in signs)
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
end end
union(long, halfs)
end
subtypes = Set([:A₂, Symbol("A₁×A₁")])
@testset "E8" begin
@test length(E8) == 240
@test all(r -> PropertyT.Roots.₂length(r) sqrt(2), E8)
let Ω = E8, 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)
@test r == 0 && d == 28
end end
end end
@testset "E7" begin end
E7 = filter(r -> iszero(sum(r.coord)), E8)
@test length(E7) == 126
let Ω = E7, α = first(Ω) @testset "Levels in Sp2n" begin
counts = countmap([ function level(rootsystem, level::Integer)
PropertyT.Roots.classify_sub_root_system(Ω, α, γ) for 1 level 4 || throw("level is implemented only for i ∈{1,2,3,4}")
γ in Ω if !PropertyT.Roots.isproportional(α, γ) level == 1 && return PropertyT.Adj(rootsystem, :C₁) # always positive
]) level == 2 && return PropertyT.Adj(rootsystem, :A₁) +
@test Set(keys(counts)) == subtypes PropertyT.Adj(rootsystem, Symbol("C₁×C₁")) +
d, r = divrem(counts[:A₂], 4) PropertyT.Adj(rootsystem, :C₂) # C₂ is not positive
@test r == 0 && d == 16 level == 3 && return PropertyT.Adj(rootsystem, :A₂) +
end PropertyT.Adj(rootsystem, Symbol("A₁×C₁"))
level == 4 && return PropertyT.Adj(rootsystem, Symbol("A₁×A₁")) # positive
end end
@testset "E6" begin n = 5
E6 = filter( G = MatrixGroups.SymplecticGroup{2n}(Int8)
r -> r.coord[end] == r.coord[end-1] == r.coord[end-2], RG, S, sizes = PropertyT.group_algebra(G; halfradius = 1)
E8,
Weyl = let N = n
P = PermGroup(perm"(1,2)", Perm(circshift(1:N, -1)))
Groups.Constructions.WreathProduct(PermGroup(perm"(1,2)"), P)
end
act = PropertyT.action_by_conjugation(G, Weyl)
function ^ᵃ(x, w::Groups.Constructions.WreathProductElement)
return SymbolicWedderburn.action(act, w, x)
end
Sₙ = S
Δsₙ = PropertyT.laplacians(
RG,
Sₙ,
x -> (gx = PropertyT.grading(x); Set([gx, -gx])),
)
function natural_embedding(i, Sp2m, Sp2n)
_dim(::MatrixGroups.ElementarySymplectic{N}) where {N} = N
n = _dim(first(alphabet(Sp2n))) ÷ 2
m = _dim(first(alphabet(Sp2m))) ÷ 2
l = alphabet(Sp2m)[i]
i, j = if l.symbol === :A
l.i, l.j
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) function k(n, m, i)
counts = countmap([ return 2^n * factorial(m) * factorial(n - i) ÷ factorial(m - i)
PropertyT.Roots.classify_sub_root_system(Ω, α, γ) for end
γ in Ω if !PropertyT.Roots.isproportional(α, γ)
]) @testset "Level $i" for i in 1:4
@test Set(keys(counts)) == subtypes Levᵢᵐ = level(Δsₘ, i)
d, r = divrem(counts[:A₂], 4) Levᵢⁿ = level(Δsₙ, i)
@info d, r
@test r == 0 && d == 10 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 end
end end