Move characters and Projections to a separate file
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@ -97,120 +97,14 @@ function matrix_repr(g::WreathProductElem, E, E_dict)
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return rep_matrix
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end
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function chars(G::PermutationGroup)
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permtype_unsorted(σ::Nemo.perm) = [length(c) for c in cycles(σ)]
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permtype(σ::Nemo.perm) = sort(permtype_unsorted(σ))
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χ_id(σ::Nemo.perm) = 1
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χ_sgn(σ::Nemo.perm) = (-1)^parity(σ)
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function χ_reg(σ::Nemo.perm)
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fixed_points = countnz([(x == y? 1 : 0) for (x,y) in enumerate(σ.d)])
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return fixed_points - 1
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end
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χ_regsgn(σ::Nemo.perm) = (-1)^parity(σ)*χ_reg(σ)
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function χ_regviaS3(σ::Nemo.perm)
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@assert parent(σ).n == 4
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t = permtype(σ)
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if t == [1,1,1,1]
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result = 2
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elseif t == [2,2]
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result = 2
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elseif t == [1,3]
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result = -1
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else
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result = 0
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end
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return result
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end
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chars = [χ_id, χ_sgn, χ_regviaS3, χ_reg, χ_regsgn]
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if G.n == 1
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return chars[1:1]
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elseif G.n == 2
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return chars[1:2]
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elseif G.n == 3
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return [chars[1:2]..., chars[4]]
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elseif G.n == 4
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return chars[1:5]
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else
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throw("Characters for $G unknown!")
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end
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end
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function epsilon(i, g::DirectProducts.DirectProductGroupElem)
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return reduce(*, 1, ((-1)^isone(g.elts[j]) for j in 1:i))
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end
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function central_projection(RG::GroupRing, char::Function, T::Type=Rational{Int})
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result = RG(T)
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for g in RG.basis
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result[g] = char(g)
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end
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return convert(T, char(RG.group())//Int(order(RG.group))*result)
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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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if G.n == 1 || G.n == 2
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return projections
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elseif G.n == 3
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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) - RG(RG.group([2,1,3])))*projections[3]
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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) - RG(RG.group([2,1,3,4])))*projections[3],
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1//2*(one(RG) - RG(RG.group([2,1,3,4])))*projections[4],
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1//2*(one(RG) + RG(RG.group([2,1,3,4])))*projections[5]]
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return rankone_projs
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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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end
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function rankOne_projections(BN::WreathProducts.WreathProduct, T::Type=Rational{Int})
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N = BN.P.n
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# projections as elements of the group rings RSₙ
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SNprojs_nc = [rankOne_projections(PermutationGroup(i), T) for i in 1:N]
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# embedding into group ring of BN
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RBN = GroupRing(BN)
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RFFFF_projs = [central_projection(GroupRing(BN.N), g->epsilon(i,g), T)
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for i in 0:BN.P.n]
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Qs = [RBN(q, g -> BN(g)) for q in RFFFF_projs]
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function incl(k::Int, g::perm, WP::WreathProduct=BN)
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@assert length(g.d) + k <= WP.P.n
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arr = [1:k; g.d .+ k; (length(g.d)+k+1):WP.P.n]
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return WP(WP.P(arr))
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end
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end
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all_projs=[Qs[1]*RBN(p, g-> incl(0,g)) for p in SNprojs_nc[N]]
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for i in 1:N-1
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Sk_first = [RBN(p, g->incl(0,g)) for p in SNprojs_nc[i]]
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Sk_last = [RBN(p, g->incl(i,g)) for p in SNprojs_nc[N-i]]
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append!(all_projs, [Qs[i+1]*p1*p2
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for (p1,p2) in Base.product(Sk_first,Sk_last)])
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end
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append!(all_projs, [Qs[N+1]*RBN(p, g-> incl(0,g)) for p in SNprojs_nc[N]])
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return all_projs
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end
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function Cstar_repr(x::GroupRingElem, matrix_reps)
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132
Projections.jl
Normal file
132
Projections.jl
Normal file
@ -0,0 +1,132 @@
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###############################################################################
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#
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# Characters of PermutationGroup
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#
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###############################################################################
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function chars(G::PermutationGroup)
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permtype_unsorted(σ::Nemo.perm) = [length(c) for c in cycles(σ)]
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permtype(σ::Nemo.perm) = sort(permtype_unsorted(σ))
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χ_id(σ::Nemo.perm) = 1
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χ_sgn(σ::Nemo.perm) = (-1)^parity(σ)
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function χ_reg(σ::Nemo.perm)
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fixed_points = countnz([(x == y? 1 : 0) for (x,y) in enumerate(σ.d)])
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return fixed_points - 1
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end
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χ_regsgn(σ::Nemo.perm) = (-1)^parity(σ)*χ_reg(σ)
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function χ_regviaS3(σ::Nemo.perm)
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@assert parent(σ).n == 4
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t = permtype(σ)
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if t == [1,1,1,1]
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result = 2
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elseif t == [2,2]
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result = 2
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elseif t == [1,3]
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result = -1
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else
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result = 0
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end
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return result
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end
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chars = [χ_id, χ_sgn, χ_regviaS3, χ_reg, χ_regsgn]
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if G.n == 1
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return chars[1:1]
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elseif G.n == 2
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return chars[1:2]
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elseif G.n == 3
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return [chars[1:2]..., chars[4]]
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elseif G.n == 4
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return chars[1:5]
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else
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throw("Characters for $G unknown!")
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end
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end
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###############################################################################
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#
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# Character of DirectProducts
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#
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###############################################################################
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function epsilon(i, g::DirectProducts.DirectProductGroupElem)
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return reduce(*, 1, ((-1)^isone(g.elts[j]) for j in 1:i))
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end
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###############################################################################
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#
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# Projections
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#
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###############################################################################
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function central_projection(RG::GroupRing, char::Function, T::Type=Rational{Int})
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result = RG(T)
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for g in RG.basis
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result[g] = char(g)
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end
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return convert(T, char(RG.group())//Int(order(RG.group))*result)
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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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if G.n == 1 || G.n == 2
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return projections
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elseif G.n == 3
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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) - RG(RG.group([2,1,3])))*projections[3]
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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) - RG(RG.group([2,1,3,4])))*projections[3],
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1//2*(one(RG) - RG(RG.group([2,1,3,4])))*projections[4],
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1//2*(one(RG) + RG(RG.group([2,1,3,4])))*projections[5]]
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return rankone_projs
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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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end
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function rankOne_projections(BN::WreathProducts.WreathProduct, T::Type=Rational{Int})
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N = BN.P.n
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# projections as elements of the group rings RSₙ
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SNprojs_nc = [rankOne_projections(PermutationGroup(i), T) for i in 1:N]
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# embedding into group ring of BN
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RBN = GroupRing(BN)
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RFFFF_projs = [central_projection(GroupRing(BN.N), g->epsilon(i,g), T)
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for i in 0:BN.P.n]
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Qs = [RBN(q, g -> BN(g)) for q in RFFFF_projs]
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function incl(k::Int, g::perm, WP::WreathProduct=BN)
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@assert length(g.d) + k <= WP.P.n
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arr = [1:k; g.d .+ k; (length(g.d)+k+1):WP.P.n]
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return WP(WP.P(arr))
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end
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all_projs=[Qs[1]*RBN(p, g-> incl(0,g)) for p in SNprojs_nc[N]]
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for i in 1:N-1
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Sk_first = [RBN(p, g->incl(0,g)) for p in SNprojs_nc[i]]
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Sk_last = [RBN(p, g->incl(i,g)) for p in SNprojs_nc[N-i]]
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append!(all_projs, [Qs[i+1]*p1*p2
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for (p1,p2) in Base.product(Sk_first,Sk_last)])
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end
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append!(all_projs, [Qs[N+1]*RBN(p, g-> incl(0,g)) for p in SNprojs_nc[N]])
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return all_projs
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end
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