99 lines
2.8 KiB
Julia
99 lines
2.8 KiB
Julia
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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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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Base.div(X::GroupRingElem, x::Number) = parent(X)(X.coeffs.÷x)
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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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Alt_N = [g for g in SymmetricGroup(N) if parity(g) == 0]
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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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end
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return sq÷factorial(N-2)
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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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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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end
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return adj÷factorial(N-2)^2
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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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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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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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for σ in Alt_N
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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=>σ(Δ₂) 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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elseif 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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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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end
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