2017-06-22 14:12:35 +02:00
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###############################################################################
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#
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2017-07-27 22:05:15 +02:00
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# Characters of Symmetric Group and DirectProduct
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2017-06-22 14:12:35 +02:00
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#
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###############################################################################
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2017-07-27 22:05:15 +02:00
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abstract AbstractCharacter <: Function
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immutable PermCharacter <: AbstractCharacter
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p::Partition
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end
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immutable DirectProdCharacter <: AbstractCharacter
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i::Int
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end
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function (chi::PermCharacter)(g::Nemo.perm)
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R = Nemo.partitionseq(chi.p)
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p = Partition(Nemo.permtype(g))
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return Int(Nemo.MN1inner(R, p, 1, Nemo._charvalsTable))
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end
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## NOTE: this works only for Z/2!!!!
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function (chi::DirectProdCharacter)(g::DirectProductGroupElem)
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return reduce(*, 1, ((-1)^isone(g.elts[j]) for j in 1:chi.i))
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2017-06-22 14:12:35 +02:00
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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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2017-07-17 10:18:26 +02:00
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function central_projection(RG::GroupRing, chi::Function, T::Type=Rational{Int})
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2017-06-22 14:12:35 +02:00
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result = RG(T)
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2017-07-12 20:18:42 +02:00
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result.coeffs = full(result.coeffs)
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2017-07-17 10:18:26 +02:00
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dim = chi(RG.group())
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ord = Int(order(RG.group))
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2017-06-22 14:12:35 +02:00
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for g in RG.basis
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2017-07-27 20:36:22 +02:00
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result[g] = convert(T, (Int(dim)//ord)*Int(chi(g)))
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2017-06-22 14:12:35 +02:00
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end
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2017-07-17 10:18:26 +02:00
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2017-07-17 10:28:18 +02:00
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return result
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2017-06-22 14:12:35 +02:00
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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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2017-07-27 20:36:22 +02:00
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cprojs = [central_projection(RG, χ, T) for χ in (character(λ) for λ in Partitions(G.n))]
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2017-06-22 14:12:35 +02:00
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if G.n == 1 || G.n == 2
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2017-07-17 12:27:28 +02:00
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return cprojs
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2017-06-22 14:12:35 +02:00
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elseif G.n == 3
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2017-07-17 12:27:28 +02:00
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p = 1//2*(one(RG, T) - RG(G([2,1,3]), T))
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2017-06-22 14:12:35 +02:00
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rankone_projs = [
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2017-07-17 12:27:28 +02:00
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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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2017-07-12 20:23:00 +02:00
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]
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2017-06-22 14:12:35 +02:00
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elseif G.n == 4
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2017-07-17 12:27:28 +02:00
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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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2017-07-12 20:23:00 +02:00
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]
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2017-07-12 20:23:27 +02:00
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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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q⁺ = 1//2*(one(RG, T) + RG(G([1,2,4,3,5]), T))
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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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2017-07-17 12:27:28 +02:00
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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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2017-07-27 20:35:39 +02:00
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p⁺*q⁺*cprojs[4], # alt_ϱ
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2017-07-17 12:27:28 +02:00
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p⁻*cprojs[5], # regular
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2017-07-27 20:35:39 +02:00
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p⁻*q⁻*cprojs[6], # ϱ
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2017-07-17 12:27:28 +02:00
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cprojs[7] # trivial
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2017-07-12 20:23:27 +02:00
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]
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2017-06-22 14:12:35 +02:00
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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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2017-07-17 12:27:28 +02:00
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return rankone_projs
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2017-06-22 14:12:35 +02:00
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end
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2017-06-22 15:14:15 +02:00
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function rankOne_projections(BN::WreathProduct, T::Type=Rational{Int})
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2017-06-22 14:12:35 +02:00
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N = BN.P.n
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2017-07-17 12:28:17 +02:00
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# projections as elements of the group rings RSₙ
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2017-06-22 14:12:35 +02:00
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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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2017-07-17 15:40:54 +02:00
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RFFFF_projs = [
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central_projection(GroupRing(BN.N), g->epsilon(i,g), T) for i in 1:BN.P.n
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]
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2017-06-22 14:12:35 +02:00
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2017-07-17 15:40:54 +02:00
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e0 = central_projection(GroupRing(BN.N), g->epsilon(0,g), T)
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Q0 = RBN(e0, g -> BN(g))
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Qs = [RBN(q, g -> BN(g)) for q in RFFFF_projs]
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2017-06-22 14:12:35 +02:00
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2017-07-17 15:40:54 +02:00
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all_projs = [Q0*RBN(p, g->BN(g)) for p in SNprojs_nc[N]]
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2017-06-22 14:12:35 +02:00
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2017-07-17 15:40:54 +02:00
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range = collect(1:N)
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for i in 1:N-1
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2017-07-17 12:28:17 +02:00
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2017-07-17 15:40:54 +02:00
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Sk_first = [RBN(p, g->BN(Nemo.emb!(BN.P(), g, range[1:i]))) for p in SNprojs_nc[i]]
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Sk_last = [RBN(p, g->BN(Nemo.emb!(BN.P(), g, range[i+1:end]))) for p in SNprojs_nc[N-i]]
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2017-06-22 14:12:35 +02:00
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2017-07-17 15:40:54 +02:00
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append!(all_projs,
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[Qs[i]*p1*p2 for (p1,p2) in Base.product(Sk_first,Sk_last)])
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end
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2017-06-22 14:12:35 +02:00
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2017-07-17 15:40:54 +02:00
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append!(all_projs, [Qs[N]*RBN(p, g->BN(g)) for p in SNprojs_nc[N]])
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2017-07-17 12:28:17 +02:00
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2017-07-17 15:40:54 +02:00
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return all_projs
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2017-06-22 14:12:35 +02:00
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end
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