2017-06-22 14:12:35 +02:00
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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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2017-06-22 15:14:15 +02:00
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function epsilon(i, g::DirectProductGroupElem)
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2017-06-22 14:12:35 +02:00
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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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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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2017-07-17 10:18:26 +02:00
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result[g] = convert(T, (dim//ord)*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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2017-07-17 12:27:28 +02:00
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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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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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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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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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# 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,
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[Qs[i+1]*p1*p2 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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