106 lines
2.4 KiB
Julia
106 lines
2.4 KiB
Julia
struct SpecialLinearGroup{N} <: SymmetrizedGroup
|
|
group::AbstractAlgebra.Group
|
|
p::Int
|
|
X::Bool
|
|
end
|
|
|
|
function SpecialLinearGroup(args::Dict)
|
|
N = args["SL"]
|
|
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 SpecialLinearGroup{N}(G, p, X)
|
|
end
|
|
|
|
function name(G::SpecialLinearGroup{N}) where N
|
|
if G.p == 0
|
|
R = (G.X ? "Z[x]" : "Z")
|
|
else
|
|
R = "F$(G.p)"
|
|
end
|
|
return SL($(G.N),$R)
|
|
end
|
|
|
|
group(G::SpecialLinearGroup) = G.group
|
|
|
|
function generatingset(G::SpecialLinearGroup{N}) where N
|
|
G.p > 0 && G.X && throw("SL(n, F_p[x]) not implemented")
|
|
SL = group(G)
|
|
return generatingset(SL, G.X)
|
|
end
|
|
|
|
# r is the injectivity radius of
|
|
# SL(n, Z[X]) -> SL(n, Z) induced by X -> 100
|
|
|
|
function generatingset(SL::MatSpace, X::Bool=false, r=5)
|
|
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 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 autS(G::SpecialLinearGroup{N}) where 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
|