GroupsWithPropertyT/SL_orbit.jl

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using ArgParse
using SCS
using Nemo
using PropertyT
using Groups
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###############################################################################
#
# Action of WreathProductElems on Nemo.MatElem
#
###############################################################################
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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 ")
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
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function (g::WreathProductElem)(A::MatElem)
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G = matrix_emb(parent(A), g)
inv_G = matrix_emb(parent(A), inv(g))
return G*A*inv_G
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), $(A.c))")
R = parent(A)
return p*A*inv(p)
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end
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###############################################################################
#
# Generating set
#
###############################################################################
function E(i::Int, j::Int, M::MatSpace, val=one(M.base_ring))
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@assert i≠j
m = one(M)
m[i,j] = val
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return m
end
function SLsize(n,p)
result = BigInt(1)
for k in 0:n-1
result *= p^n - p^k
end
return div(result, p-1)
end
function SL_generatingset(n::Int, X::Bool=false)
indexing = [(i,j) for i in 1:n for j in 1:n if i≠j]
G = MatrixSpace(ZZ, n, n)
if X
S = [E(i,j,G,v) for (i,j) in indexing for v in [1, 100]]
else
S = [E(i,j,G,v) for (i,j) in indexing for v in [1]]
end
S = vcat(S, [inv(x) for x in S])
return G, unique(S)
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end
function SL_generatingset(n::Int, p::Int, X::Bool=false)
p == 0 && return SL_generatingset(n, X)
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(p > 1 && n > 1) || throw("Both n and p should be positive integers!")
info("Size(SL($n,$p)) = $(SLsize(n,p))")
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F = ResidueRing(ZZ, p)
G = MatrixSpace(F, n, n)
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indexing = [(i,j) for i in 1:n for j in 1:n if i≠j]
S = [E(i, j, G, v) for (i,j) in indexing]
S = vcat(S, [inv(x) for x in S])
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return G, unique(S)
end
###############################################################################
#
# Parsing command line
#
###############################################################################
function parse_commandline()
settings = ArgParseSettings()
@add_arg_table settings begin
"--tol"
help = "set numerical tolerance for the SDP solver (default: 1e-5)"
arg_type = Float64
default = 1e-5
"--iterations"
help = "set maximal number of iterations for the SDP solver (default: 20000)"
arg_type = Int
default = 20000
"--upper-bound"
help = "Set an upper bound for the spectral gap (default: Inf)"
arg_type = Float64
default = Inf
"--cpus"
help = "Set number of cpus used by solver (default: auto)"
arg_type = Int
required = false
"-N"
help = "Consider automorphisms of free group on N generators (default: N=2)"
arg_type = Int
default = 2
"-p"
help = "Matrices over filed of p-elements (default: p=0 => over ZZ)"
arg_type = Int
default = 0
"--radius"
help = "Find the decomposition over B_r(e,S)"
arg_type = Int
default = 2
"-X"
help = "Matrices are over ZZ⟨X⟩"
action = :store_true
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end
return parse_args(settings)
end
###############################################################################
#
# main
#
###############################################################################
function main()
parsed_args = parse_commandline()
N = parsed_args["N"]
p = parsed_args["p"]
if p == 0
if parsed_args["X"]
dirname = "oSL$(N)Z⟨X⟩"
else
dirname = "oSL$(N)Z"
end
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else
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dirname = "oSL$(N)_$p"
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end
radius = parsed_args["radius"]
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tol = parsed_args["tol"]
iterations = parsed_args["iterations"]
upper_bound = parsed_args["upper-bound"]
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dirname = "$(dirname)_$(upper_bound)_r=$radius"
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isdir(dirname) || mkdir(dirname)
logger = PropertyT.setup_logging(dirname)
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info(logger, "Group: $dirname")
info(logger, "Iterations: $iterations")
info(logger, "Precision: $tol")
info(logger, "Upper bound: $upper_bound")
G, S = SL_generatingset(N, p, parsed_args["X"])
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info(logger, G)
info(logger, "Symmetric generating set of size $(length(S))")
info(logger, S)
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AutS = WreathProduct(FiniteField(2,1, "a")[1], PermutationGroup(N))
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# AutS = PermutationGroup(N)
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solver = SCS.SCSSolver(eps=tol, max_iters=iterations, verbose=true, linearsolver=SCS.Direct)
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sett = Settings(dirname, N, G, S, AutS, radius, solver, upper_bound, tol)
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PropertyT.check_property_T(sett)
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end
main()