110 lines
2.8 KiB
Julia
110 lines
2.8 KiB
Julia
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module MappingClassGroups
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using Nemo
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using Groups
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###############################################################################
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#
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# Generating set
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#
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###############################################################################
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Comm(x,y) = x*y*x^-1*y^-1
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function generatingset(N::Int)
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if N == 2
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MCGroup = Groups.FPGroup(["a1","a2","a3","a4","a5"]);
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S = Nemo.gens(MCGroup)
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N = length(S)
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A = prod(reverse(S))*prod(S)
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relations = [
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[Comm(S[i], S[j]) for i in 1:N for j in 1:N if abs(i-j) > 1]...,
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[S[i]*S[i+1]*S[i]*inv(S[i+1]*S[i]*S[i+1]) for i in 1:N-1]...,
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(S[1]*S[2]*S[3])^4*inv(S[5])^2,
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Comm(A, S[1]),
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A^2
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]
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relations = [relations; [inv(rel) for rel in relations]]
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Groups.add_rels!(MCGroup, Dict(rel => MCGroup() for rel in relations))
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return MCGroup
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elseif N < 2
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throw("Genus must be at least 2!")
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end
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MCGroup = Groups.FPGroup(["a$i" for i in 0:2N])
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S = Nemo.gens(MCGroup)
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a0 = S[1]
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A = S[2:end]
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k = length(A)
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relations = [
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[Comm(A[i], A[j]) for i in 1:k for j in 1:k if abs(i-j) > 1]...,
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[Comm(a0, A[i]) for i in 1:k if i != 4]...,
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[A[i]*A[(i+1)]*A[i]*inv(A[i+1]*A[i]*A[i+1]) for i in 1:k-1]...,
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A[4]*a0*A[4]*inv(a0*A[4]*a0)
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]
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# 3-chain relation
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c = prod(reverse(A[1:4]))*prod(A[1:4])
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b0 = c*a0*inv(c)
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push!(relations, (A[1]*A[2]*A[3])^4*inv(a0*b0))
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# Lantern relation
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b1 = inv(A[4]*A[5]*A[3]*A[4])*a0*(A[4]*A[5]*A[3]*A[4])
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b2 = inv(A[2]*A[3]*A[1]*A[2])*b1*(A[2]*A[3]*A[1]*A[2])
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u = inv(A[6]*A[5])*b1*(A[6]*A[5])
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x = prod(reverse(A[2:6]))*u*prod(inv.(A[1:4]))
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b3 = x*a0*inv(x)
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push!(relations, a0*b2*b1*inv(A[1]*A[3]*A[5]*b3))
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# Hyperelliptic relation
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X = prod(reverse(A))*prod(A)
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function n(i::Int, b=b0)
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if i == 1
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return A[1]
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elseif i == 2
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return b
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else
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return w(i-2)*n(i-2)*w(i-2)
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end
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end
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function w(i::Int)
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(A[2i+4]*A[2i+3]*A[2i+2]* n(i+1))*(A[2i+1]*A[2i] *A[2i+2]*A[2i+1])*
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(A[2i+3]*A[2i+2]*A[2i+4]*A[2i+3])*( n(i+1)*A[2i+2]*A[2i+1]*A[2i] )
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end
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push!(relations, X*n(N)*inv(n(N)*X))
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relations = [relations; [inv(rel) for rel in relations]]
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Groups.add_rels!(MCGroup, Dict(rel => MCGroup() for rel in relations))
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return MCGroup, gens(MCGroup)
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end
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function generatingset(parsed_args)
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N = parsed_args["N"]
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return generatingset(N)
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end
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###############################################################################
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#
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# Misc
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#
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###############################################################################
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function groupname(parsed_args)
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N = parsed_args["N"]
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return groupname(N), N
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end
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groupname(N::Int) = "MCG$(N)"
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end #of module MappingClassGroups
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