GroupsWithPropertyT/groups/speciallinear.jl

module SpecialLinear

using Nemo
using Groups

import Nemo.Generic.perm

###############################################################################
#
#  Generating set
#
###############################################################################

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 SLsize(n,p)
    p == 0 && return Inf
    result = BigInt(1)
    for k in 0:n-1
        result *= p^n - p^k
    end
    return div(result, p-1)
end

function generatingset(n::Int, X::Bool=false)
    indexing = [(i,j) for i in 1:n for j in 1:n if i≠j]
    G = MatrixSpace(ZZ, n, n)
    if X
        S = [E(i,j,G,v) for (i,j) in indexing for v in [1, 100]]
    else
        S = [E(i,j,G,v) for (i,j) in indexing for v in [1]]
    end
    S = vcat(S, [inv(x) for x in S])
    return G, unique(S)
end

function generatingset(n::Int, p::Int, X::Bool=false)
    (p > 1 && n > 1) || throw("Both n and p should be positive integers!")
    info("Size(SL($n,$p)) = $(SLsize(n,p))")
    F = ResidueRing(ZZ, p)
    G = MatrixSpace(F, n, n)
    indexing = [(i,j) for i in 1:n for j in 1:n if i≠j]
    S = [E(i, j, G) for (i,j) in indexing]
    S = vcat(S, [inv(x) for x in S])
    return G, unique(S)
end

function generatingset(parsed_args)
    N = parsed_args["N"]
    p = parsed_args["p"]
    X = parsed_args["X"]

    if p == 0
        return generatingset(N, X)
    else
        return generatingset(N, p, X)
    end
end

###############################################################################
#
#  Action of WreathProductElems on Nemo.MatElem
#
###############################################################################

function matrix_emb(MM::MatSpace, g::WreathProductElem)
    parent(g).P.n == MM.cols == MM.rows || throw("No natural embedding of $(parent(g)) in ")
    powers = [(elt == parent(elt)() ? 0: 1) for elt in g.n.elts]
    elt = diagm([(-1)^(elt == parent(elt)() ? 0: 1) for elt in g.n.elts])
    return MM(elt)*MM(Nemo.matrix_repr(g.p)')
end

function (g::WreathProductElem)(A::MatElem)
    G = matrix_emb(parent(A), g)
    inv_G = matrix_emb(parent(A), inv(g))
    return G*A*inv_G
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))")
    R = parent(A)
    return p*A*inv(p)
end

autS(N::Int) = WreathProduct(PermutationGroup(2), PermutationGroup(N))

function autS(parsed_args)
   return autS(parsed_args["N"])
end

###############################################################################
#
#  Misc
#
###############################################################################

function groupname(parsed_args)
    N = parsed_args["N"]
    p = parsed_args["p"]
    X = parsed_args["X"]

    return groupname(N, p, X), N
end

function groupname(N::Int, p::Int=0, X::Bool=false)
    if p == 0
       if X
          name = "SL$(N)Z⟨X⟩"
       else
          name = "SL$(N)Z"
       end
    else
       name = "SL$(N)_$p"
    end
    return name
end

end # of module SpecialLinear