118 lines
2.9 KiB
Julia
118 lines
2.9 KiB
Julia
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module SLNs
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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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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 SLsize(n,p)
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result = BigInt(1)
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for k in 0:n-1
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result *= p^n - p^k
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end
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return div(result, p-1)
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end
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function generatingset(n::Int, X::Bool=false)
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indexing = [(i,j) for i in 1:n for j in 1:n if i≠j]
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G = MatrixSpace(ZZ, n, n)
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if X
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S = [E(i,j,G,v) for (i,j) in indexing for v in [1, 100]]
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else
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S = [E(i,j,G,v) for (i,j) in indexing for v in [1]]
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end
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S = vcat(S, [inv(x) for x in S])
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return G, unique(S)
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end
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function generatingset(n::Int, p::Int, X::Bool=false)
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(p > 1 && n > 1) || throw("Both n and p should be positive integers!")
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info("Size(SL($n,$p)) = $(SLsize(n,p))")
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F = ResidueRing(ZZ, p)
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G = MatrixSpace(F, n, n)
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indexing = [(i,j) for i in 1:n for j in 1:n if i≠j]
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S = [E(i, j, G) for (i,j) in indexing]
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S = vcat(S, [inv(x) for x in S])
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return G, unique(S)
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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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p = parsed_args["p"]
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X = parsed_args["X"]
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if p == 0
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return generatingset(N, X)
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else
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return generatingset(N, p, X)
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end
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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), $(A.c))")
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R = parent(A)
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return p*A*inv(p)
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end
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function autS(parsed_args)
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N = parsed_args["N"]
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return WreathProduct(PermutationGroup(2), PermutationGroup(N))
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# return WreathProduct(FiniteField(2,1, "a")[1], PermutationGroup(N))
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end
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###############################################################################
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#
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# Misc
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#
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###############################################################################
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function groupname(parsed_args)
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N = parsed_args["N"]
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p = parsed_args["p"]
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X = parsed_args["X"]
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if p == 0
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if X
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name = "oSL$(N)Z⟨X⟩"
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else
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name = "oSL$(N)Z"
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end
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else
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name = "oSL$(N)_$p"
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end
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return name, N
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end
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end # of module SLNs
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