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account for changes in order of characters
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@ -29,26 +29,27 @@ end
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function rankOne_projections(G::PermutationGroup, T::Type=Rational{Int})
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RG = GroupRing(G)
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projections = [central_projection(RG, χ, T) for χ in chars(G)]
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cprojs = [central_projection(RG, χ, T) for χ in (character(λ) for λ in IntPartitions(G.n))]
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if G.n == 1 || G.n == 2
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return projections
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return cprojs
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elseif G.n == 3
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p = 1//2*(one(RG, T) - RG(G([2,1,3]), T))
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rankone_projs = [
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projections[1],
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projections[2],
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1//2*(one(RG, T) - RG(G([2,1,3]), T))*projections[3]
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cprojs[1], # alternating
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p*cprojs[2], # regular
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cprojs[3] # trivial
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]
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return rankone_projs
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elseif G.n == 4
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rankone_projs = [
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projections[1],
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projections[2],
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1//2*(one(RG, T) - RG(G([2,1,3,4]), T))*projections[3],
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1//2*(one(RG, T) - RG(G([2,1,3,4]), T))*projections[4],
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1//2*(one(RG, T) + RG(G([2,1,3,4]), T))*projections[5]
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p⁺ = 1//2*(one(RG, T) + RG(G([2,1,3,4]), T))
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p⁻ = 1//2*(one(RG, T) - RG(G([2,1,3,4]), T))
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rankone_projs = [
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cprojs[1], # alternating
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p⁺*cprojs[2], # alt_regular
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p⁻*cprojs[3], # regular
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p⁻*cprojs[4], # via projection to S₃
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cprojs[5] # trivial
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]
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return rankone_projs
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elseif G.n == 5
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p⁺ = 1//2*(one(RG, T) + RG(G([2,1,3,4,5]), T))
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p⁻ = 1//2*(one(RG, T) - RG(G([2,1,3,4,5]), T))
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@ -57,17 +58,18 @@ function rankOne_projections(G::PermutationGroup, T::Type=Rational{Int})
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q⁻ = 1//2*(one(RG, T) - RG(G([1,2,4,3,5]), T))
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rankone_projs = [
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projections[1],
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projections[2],
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p⁻*projections[3],
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p⁺*projections[4],
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p⁺*q⁺*projections[5],
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p⁻*q⁻*projections[6],
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p⁺*q⁺*projections[7]
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cprojs[1], # alternating
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p⁺*cprojs[2], # alt_regular
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p⁺*q⁺*cprojs[3], # ψ
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p⁻*q⁻*cprojs[4], # alt_ϱ
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p⁻*cprojs[5], # regular
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p⁺*q⁺*cprojs[6], # ϱ
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cprojs[7] # trivial
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]
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else
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throw("Rank-one projections for $G unknown!")
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end
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return rankone_projs
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end
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function rankOne_projections(BN::WreathProduct, T::Type=Rational{Int})
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