93 lines
2.6 KiB
Julia
93 lines
2.6 KiB
Julia
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function (p::perm)(A::GroupRingElem)
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RG = parent(A)
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result = zero(RG, eltype(A.coeffs))
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for (idx, c) in enumerate(A.coeffs)
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if c!= zero(eltype(A.coeffs))
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result[p(RG.basis[idx])] = c
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end
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end
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return result
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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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###############################################################################
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#
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# Action of WreathProductElems on AutGroupElem
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#
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###############################################################################
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function AutFG_emb(A::AutGroup, g::WreathProductElem)
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isa(A.objectGroup, FreeGroup) || throw("Not an Aut(Fₙ)")
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parent(g).P.n == length(A.objectGroup.gens) || throw("No natural embedding of $(parent(g)) into $A")
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elt = A()
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Id = parent(g.n.elts[1])()
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flips = Groups.AutSymbol[Groups.flip_autsymbol(i) for i in 1:length(g.p.d) if g.n.elts[i] != Id]
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Groups.r_multiply!(elt, flips, reduced=false)
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Groups.r_multiply!(elt, [Groups.perm_autsymbol(g.p)])
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return elt
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end
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function AutFG_emb(A::AutGroup, p::perm)
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isa(A.objectGroup, FreeGroup) || throw("Not an Aut(Fₙ)")
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parent(p).n == length(A.objectGroup.gens) || throw("No natural embedding of $(parent(p)) into $A")
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return A(Groups.perm_autsymbol(p))
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end
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function (g::WreathProductElem)(a::Groups.Automorphism)
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A = parent(a)
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g = AutFG_emb(A,g)
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res = A()
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Groups.r_multiply!(res, g.symbols, reduced=false)
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Groups.r_multiply!(res, a.symbols, reduced=false)
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Groups.r_multiply!(res, [inv(s) for s in reverse!(g.symbols)])
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return res
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end
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function (p::perm)(a::Groups.Automorphism)
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g = AutFG_emb(parent(a),p)
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return g*a*inv(g)
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end
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