92 lines
2.2 KiB
Julia
92 lines
2.2 KiB
Julia
struct SpecialLinearGroup <: SymmetricGroup
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args::Dict{String,Any}
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group::AbstractAlgebra.Group
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function SpecialLinearGroup(args::Dict)
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n = args["N"]
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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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G = MatrixSpace(FiniteField(p), n, n)
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end
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return new(args, G)
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end
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end
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function name(G::SpecialLinearGroup)
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N = G.args["N"]
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p = G.args["p"]
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X = G.args["X"]
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if p == 0
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R = (X ? "Z[x]" : "Z")
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else
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R = "F$p"
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end
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if G.args["nosymmetry"]
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return "SL($N,$R)"
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else
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return "oSL($N,$R)"
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end
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end
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group(G::SpecialLinearGroup) = G.group
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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 generatingset(G::SpecialLinearGroup)
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n = G.args["N"]
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p = G.args["p"]
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X = G.args["X"]
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SL = group(G)
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indexing = [(i,j) for i in 1:n for j in 1:n if i≠j]
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p > 0 && X && throw("SL(n, F_p[x]) not implemented")
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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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r = G.args["radius"]
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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 autS(G::SpecialLinearGroup)
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N = G.args["N"]
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return WreathProduct(PermutationGroup(2), PermutationGroup(N))
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end
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###############################################################################
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#
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# Action of WreathProductElems on Nemo.MatElem
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#
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###############################################################################
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function matrix_emb(MM::MatSpace, g::WreathProductElem)
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parent(g).P.n == MM.cols == MM.rows || throw("No natural embedding of $(parent(g)) in ")
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powers = [(elt == parent(elt)() ? 0: 1) for elt in g.n.elts]
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elt = diagm([(-1)^(elt == parent(elt)() ? 0: 1) for elt in g.n.elts])
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return MM(elt)*MM(Nemo.matrix_repr(g.p)')
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end
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function (g::WreathProductElem)(A::MatElem)
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G = matrix_emb(parent(A), g)
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inv_G = matrix_emb(parent(A), inv(g))
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return G*A*inv_G
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end
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function (p::perm)(A::MatElem)
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length(p.d) == A.r == A.c || throw("Can't act via $p on matrix of size ($(A.r), $(A.c))")
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R = parent(A)
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return p*A*inv(p)
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end
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