63 lines
1.4 KiB
Julia
63 lines
1.4 KiB
Julia
struct SpecialLinearGroup{N} <: SymmetrizedGroup
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group::AbstractAlgebra.Group
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p::Int
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X::Bool
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end
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function SpecialLinearGroup(args::Dict)
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N = args["SL"]
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p = args["p"]
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X = args["X"]
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if p == 0
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G = MatrixSpace(Nemo.ZZ, N, N)
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else
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R = Nemo.NmodRing(UInt(p))
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G = MatrixSpace(R, N, N)
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end
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return SpecialLinearGroup{N}(G, p, X)
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end
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function name(G::SpecialLinearGroup{N}) where N
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if G.p == 0
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R = (G.X ? "Z[x]" : "Z")
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else
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R = "F$(G.p)"
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end
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return SL($(G.N),$R)
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end
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group(G::SpecialLinearGroup) = G.group
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function generatingset(G::SpecialLinearGroup{N}) where N
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G.p > 0 && G.X && throw("SL(n, F_p[x]) not implemented")
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SL = group(G)
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return generatingset(SL, G.X)
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end
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# r is the injectivity radius of
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# SL(n, Z[X]) -> SL(n, Z) induced by X -> 100
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function generatingset(SL::MatSpace, X::Bool=false, r=5)
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n = SL.cols
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indexing = [(i,j) for i in 1:n for j in 1:n if i≠j]
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if !X
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S = [E(idx[1],idx[2],SL) for idx in indexing]
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else
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S = [E(i,j,SL,v) for (i,j) in indexing for v in [1, 100*r]]
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end
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return unique([S; inv.(S)])
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end
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function E(i::Int, j::Int, M::MatSpace, val=one(M.base_ring))
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@assert i≠j
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m = one(M)
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m[i,j] = val
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return m
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end
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function autS(G::SpecialLinearGroup{N}) where N
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return WreathProduct(PermutationGroup(2), PermutationGroup(N))
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end
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