111 lines
2.5 KiB
Julia
111 lines
2.5 KiB
Julia
struct SpecialLinearGroup <: SymmetrizedGroup
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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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R = Nemo.NmodRing(UInt(p))
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G = MatrixSpace(R, 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(n::DirectProductGroupElem, p::perm)
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Id = parent(n.elts[1])()
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elt = diagm([(-1)^(el == Id ? 0 : 1) for el in n.elts])
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return elt[:, p.d]
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end
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function (g::WreathProductElem)(A::MatElem)
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g_inv = inv(g)
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G = matrix_emb(g.n, g_inv.p)
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G_inv = matrix_emb(g_inv.n, g.p)
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M = parent(A)
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return M(G)*A*M(G_inv)
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end
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import Base.*
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doc"""
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*(x::AbstractAlgebra.MatElem, P::Generic.perm)
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> Apply the pemutation $P$ to the rows of the matrix $x$ and return the result.
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"""
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function *(x::AbstractAlgebra.MatElem, P::Generic.perm)
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z = similar(x)
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m = rows(x)
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n = cols(x)
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for i = 1:m
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for j = 1:n
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z[i, j] = x[i,P[j]]
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end
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end
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return z
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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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return p*A*inv(p)
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end
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