adjust sq, adj, op to the new world
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@ -17,6 +17,8 @@ include("constraint_matrix.jl")
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include("sos_sdps.jl")
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include("certify.jl")
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include("sqadjop.jl")
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include("1712.07167.jl")
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include("1812.03456.jl")
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134
src/sqadjop.jl
134
src/sqadjop.jl
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@ -1,98 +1,80 @@
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isopposite(σ::Generic.Perm, τ::Generic.Perm, i=1, j=2) =
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σ[i] ≠ τ[i] && σ[i] ≠ τ[j] &&
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σ[j] ≠ τ[i] && σ[j] ≠ τ[j]
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import PermutationGroups.AbstractPerm
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isadjacent(σ::Generic.Perm, τ::Generic.Perm, i=1, j=2) =
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(σ[i] == τ[i] && σ[j] ≠ τ[j]) || # first equal, second differ
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(σ[j] == τ[j] && σ[i] ≠ τ[i]) || # second equal, first differ
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(σ[i] == τ[j] && σ[j] ≠ τ[i]) || # first σ equal to second τ
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(σ[j] == τ[i] && σ[i] ≠ τ[j]) # second σ equal to first τ
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# move to Groups
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Base.keys(a::Alphabet) = keys(1:length(a))
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Base.div(X::GroupRingElem, x::Number) = parent(X)(X.coeffs.÷x)
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## the old 1812.03456 definitions
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function Sq(RG::GroupRing, N::Integer)
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S₂ = generating_set(RG.group, 2)
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ℤ = Int64
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Δ₂ = length(S₂)*one(RG, ℤ) - RG(S₂, ℤ);
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isopposite(σ::AbstractPerm, τ::AbstractPerm, i=1, j=2) =
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i^σ ≠ i^τ && i^σ ≠ j^τ && j^σ ≠ i^τ && j^σ ≠ j^τ
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Alt_N = [g for g in SymmetricGroup(N) if parity(g) == 0]
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isadjacent(σ::AbstractPerm, τ::AbstractPerm, i=1, j=2) =
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(i^σ == i^τ && j^σ ≠ j^τ) || # first equal, second differ
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(j^σ == j^τ && i^σ ≠ i^τ) || # second equal, first differ
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(i^σ == j^τ && j^σ ≠ i^τ) || # first σ equal to second τ
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(j^σ == i^τ && i^σ ≠ j^τ) # second σ equal to first τ
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sq = RG()
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for σ in Alt_N
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GroupRings.addeq!(sq, *(Δ₂^σ, Δ₂^σ, false))
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function _ncycle(start, length, n=start + length - 1)
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p = Perm(Int8(n))
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@assert n ≥ start + length - 1
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for k in start:start+length-2
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p[k] = k + 1
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end
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return sq÷factorial(N-2)
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p[start+length-1] = start
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return p
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end
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function Adj(RG::GroupRing, N::Integer)
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S₂ = generating_set(RG.group, 2)
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ℤ = Int64
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Δ₂ = length(S₂)*one(RG, ℤ) - RG(S₂, ℤ);
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alternating_group(n::Integer) = PermGroup([_ncycle(i, 3) for i in 1:n-2])
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Alt_N = [g for g in SymmetricGroup(N) if parity(g) == 0]
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Δ₂s = Dict(σ=>Δ₂^σ for σ in Alt_N)
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adj = RG()
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for σ in Alt_N
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for τ in Alt_N
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if isadjacent(σ, τ)
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GroupRings.addeq!(adj, *(Δ₂s[σ], Δ₂s[τ], false))
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end
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end
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function small_gens(G::MatrixGroups.SpecialLinearGroup)
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A = alphabet(G)
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S = map([(1, 2), (2, 1)]) do (i, j)
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k = findfirst(l -> (l.i == i && l.j == j), A)
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@assert !isnothing(k)
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G(Groups.word_type(G)([k]))
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end
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return adj÷factorial(N-2)^2
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return union!(S, inv.(S))
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end
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function Op(RG::GroupRing, N::Integer)
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if N < 4
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return RG()
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function small_gens(G::Groups.AutomorphismGroup{<:FreeGroup})
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A = alphabet(G)
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S = map([(:ϱ, 1, 2), (:ϱ, 2, 1), (:λ, 1, 2), (:λ, 2, 1)]) do (id, i, j)
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k = findfirst(l -> (l.id == id && l.i == i && l.j == j), A)
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@assert !isnothing(k)
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G(Groups.word_type(G)([k]))
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end
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S₂ = generating_set(RG.group, 2)
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ℤ = Int64
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Δ₂ = length(S₂)*one(RG, ℤ) - RG(S₂, ℤ);
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return union!(S, inv.(S))
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end
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Alt_N = [g for g in SymmetricGroup(N) if parity(g) == 0]
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Δ₂s = Dict(σ=>Δ₂^σ for σ in Alt_N)
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op = RG()
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function small_laplacian(RG::StarAlgebra)
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G = StarAlgebras.object(RG)
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S₂ = small_gens(G)
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return length(S₂) * one(RG) - sum(RG(s) for s in S₂)
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end
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for σ in Alt_N
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for τ in Alt_N
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function SqAdjOp(A::StarAlgebra, n::Integer, Δ₂=small_laplacian(A))
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@assert parent(Δ₂) === A
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alt_n = alternating_group(n)
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G = StarAlgebras.object(A)
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act = action_by_conjugation(G, alt_n)
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Δ₂s = Dict(σ => SymbolicWedderburn.action(act, σ, Δ₂) for σ in alt_n)
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sq, adj, op = zero(A), zero(A), zero(A)
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tmp = zero(A)
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for σ in alt_n
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StarAlgebras.add!(sq, sq, StarAlgebras.mul!(tmp, Δ₂s[σ], Δ₂s[σ]))
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for τ in alt_n
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if isopposite(σ, τ)
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GroupRings.addeq!(op, *(Δ₂s[σ], Δ₂s[τ], false))
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end
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end
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end
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return op÷factorial(N-2)^2
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end
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for Elt in [:Sq, :Adj, :Op]
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@eval begin
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$Elt(RG::GroupRing{AutGroup{N}}) where N = $Elt(RG, N)
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$Elt(RG::GroupRing{<:MatAlgebra}) = $Elt(RG, nrows(RG.group))
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end
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end
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function SqAdjOp(RG::GroupRing, N::Integer)
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S₂ = generating_set(RG.group, 2)
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ℤ = Int64
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Δ₂ = length(S₂)*one(RG, ℤ) - RG(S₂, ℤ);
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Alt_N = [σ for σ in SymmetricGroup(N) if parity(σ) == 0]
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sq, adj, op = RG(), RG(), RG()
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Δ₂s = Dict(σ=>Δ₂^σ for σ in Alt_N)
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for σ in Alt_N
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GroupRings.addeq!(sq, *(Δ₂s[σ], Δ₂s[σ], false))
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for τ in Alt_N
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if isopposite(σ, τ)
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GroupRings.addeq!(op, *(Δ₂s[σ], Δ₂s[τ], false))
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StarAlgebras.add!(op, op, StarAlgebras.mul!(tmp, Δ₂s[σ], Δ₂s[τ]))
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elseif isadjacent(σ, τ)
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GroupRings.addeq!(adj, *(Δ₂s[σ], Δ₂s[τ], false))
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StarAlgebras.add!(adj, adj, StarAlgebras.mul!(tmp, Δ₂s[σ], Δ₂s[τ]))
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end
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end
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end
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k = factorial(N-2)
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return sq÷k, adj÷k^2, op÷k^2
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k = factorial(n - 2)
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return sq ÷ k, adj ÷ k^2, op ÷ k^2
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end
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