struct SpecialLinearGroup <: SymmetrizedGroup args::Dict{String,Any} group::AbstractAlgebra.Group function SpecialLinearGroup(args::Dict) n = args["N"] p = args["p"] X = args["X"] if p == 0 G = MatrixSpace(Nemo.ZZ, n, n) else R = Nemo.NmodRing(UInt(p)) G = MatrixSpace(R, n, n) end return new(args, G) end end function name(G::SpecialLinearGroup) N = G.args["N"] p = G.args["p"] X = G.args["X"] if p == 0 R = (X ? "Z[x]" : "Z") else R = "F$p" end if G.args["nosymmetry"] return "SL($N,$R)" else return "oSL($N,$R)" end end group(G::SpecialLinearGroup) = G.group function E(i::Int, j::Int, M::MatSpace, val=one(M.base_ring)) @assert i≠j m = one(M) m[i,j] = val return m end function generatingset(G::SpecialLinearGroup) n = G.args["N"] p = G.args["p"] X = G.args["X"] p > 0 && X && throw("SL(n, F_p[x]) not implemented") SL = group(G) r = G.args["radius"] return generatingset(SL, r, X) end function generatingset(SL::MatSpace, radius::Integer, X::Bool=false) n = SL.cols indexing = [(i,j) for i in 1:n for j in 1:n if i≠j] if !X S = [E(idx[1],idx[2],SL) for idx in indexing] else S = [E(i,j,SL,v) for (i,j) in indexing for v in [1, 100*r]] end return unique([S; inv.(S)]) end function autS(G::SpecialLinearGroup) N = G.args["N"] return WreathProduct(PermutationGroup(2), PermutationGroup(N)) end ############################################################################### # # Action of WreathProductElems on Nemo.MatElem # ############################################################################### function matrix_emb(n::DirectProductGroupElem, p::perm) Id = parent(n.elts[1])() elt = diagm([(-1)^(el == Id ? 0 : 1) for el in n.elts]) return elt[:, p.d] end function (g::WreathProductElem)(A::MatElem) g_inv = inv(g) G = matrix_emb(g.n, g_inv.p) G_inv = matrix_emb(g_inv.n, g.p) M = parent(A) return M(G)*A*M(G_inv) end import Base.* doc""" *(x::AbstractAlgebra.MatElem, P::Generic.perm) > Apply the pemutation $P$ to the rows of the matrix $x$ and return the result. """ function *(x::AbstractAlgebra.MatElem, P::Generic.perm) z = similar(x) m = rows(x) n = cols(x) for i = 1:m for j = 1:n z[i, j] = x[i,P[j]] end end return z end function (p::perm)(A::MatElem) length(p.d) == A.r == A.c || throw("Can't act via $p on matrix of size ($(A.r), $(A.c))") return p*A*inv(p) end