72 lines
2.0 KiB
Julia
72 lines
2.0 KiB
Julia
function _abelianize(
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i::Integer,
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source::AutomorphismGroup{<:FreeGroup},
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target::MatrixGroups.SpecialLinearGroup{N,T},
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) where {N,T}
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n = ngens(object(source))
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@assert n == N
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aut = alphabet(source)[i]
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if aut isa Transvection
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# we change (i,j) to (j, i) to be consistent with the action:
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# Automorphisms act on the right which corresponds to action on
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# the columns in the matrix case
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eij = MatrixGroups.ElementaryMatrix{N}(
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aut.j,
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aut.i,
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ifelse(aut.inv, -one(T), one(T)),
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)
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k = alphabet(target)[eij]
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return word_type(target)([k])
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else
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throw("unexpected automorphism symbol: $(typeof(aut))")
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end
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end
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function _abelianize(
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i::Integer,
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source::AutomorphismGroup{<:Groups.SurfaceGroup},
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target::MatrixGroups.SpecialLinearGroup{N,T},
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) where {N,T}
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n = ngens(Groups.object(source))
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@assert n == N
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g = alphabet(source)[i].autFn_word
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result = one(target)
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for l in word(g)
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append!(word(result), _abelianize(l, parent(g), target))
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end
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return word(result)
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end
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function Groups._abelianize(
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i::Integer,
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source::AutomorphismGroup{<:Groups.SurfaceGroup},
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target::MatrixGroups.SymplecticGroup{N,T},
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) where {N,T}
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@assert iseven(N)
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As = alphabet(source)
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At = alphabet(target)
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SlN = let genus = Groups.genus(Groups.object(source))
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@assert 2genus == N
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MatrixGroups.SpecialLinearGroup{2genus}(T)
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end
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ab = Groups.Homomorphism(Groups._abelianize, source, SlN; check = false)
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matrix_spn_map = let S = gens(target)
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Dict(MatrixGroups.matrix(g) => word(g) for g in union(S, inv.(S)))
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end
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# renumeration:
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# (f1, f2, f3, f4, f5, f6) = (a₁, a₂, a₃, b₁, b₂, b₃) →
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# → (b₃, a₃, b₂, a₂, b₁, a₁)
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# hence p = [6, 4, 2, 5, 3, 1]
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p = [reverse(2:2:N); reverse(1:2:N)]
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g = source([i])
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Mg = MatrixGroups.matrix(ab(g))[p, p]
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return matrix_spn_map[Mg]
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end
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