mirror of
https://github.com/kalmarek/PropertyT.jl.git
synced 2024-12-26 10:35:29 +01:00
add G₂ script
This commit is contained in:
parent
92d4ba0c20
commit
a14c6d2669
179
scripts/G₂_Adj.jl
Normal file
179
scripts/G₂_Adj.jl
Normal file
@ -0,0 +1,179 @@
|
||||
using LinearAlgebra
|
||||
BLAS.set_num_threads(1)
|
||||
ENV["OMP_NUM_THREADS"] = 4
|
||||
|
||||
using MKL_jll
|
||||
include(joinpath(@__DIR__, "../test/optimizers.jl"))
|
||||
|
||||
using Groups
|
||||
import Groups.MatrixGroups
|
||||
using PropertyT
|
||||
|
||||
using SymbolicWedderburn
|
||||
using SymbolicWedderburn.StarAlgebras
|
||||
using PermutationGroups
|
||||
|
||||
include(joinpath(@__DIR__, "G₂_gens.jl"))
|
||||
|
||||
G, roots, Weyl = G₂_roots_weyl()
|
||||
|
||||
const HALFRADIUS = 2
|
||||
const UPPER_BOUND = Inf
|
||||
|
||||
RG, S, sizes = @time PropertyT.group_algebra(G, halfradius = HALFRADIUS)
|
||||
|
||||
Δ = RG(length(S)) - sum(RG(s) for s in S)
|
||||
|
||||
wd = let Σ = Weyl, RG = RG
|
||||
act = PropertyT.AlphabetPermutation{eltype(Σ),Int64}(
|
||||
Dict(g => PermutationGroups.perm(g) for g in Σ),
|
||||
)
|
||||
|
||||
@time SymbolicWedderburn.WedderburnDecomposition(
|
||||
Float64,
|
||||
Σ,
|
||||
act,
|
||||
basis(RG),
|
||||
StarAlgebras.Basis{UInt16}(@view basis(RG)[1:sizes[HALFRADIUS]]),
|
||||
semisimple = false,
|
||||
)
|
||||
end
|
||||
|
||||
elt = Δ^2
|
||||
unit = Δ
|
||||
|
||||
@time model, varP = PropertyT.sos_problem_primal(
|
||||
elt,
|
||||
unit,
|
||||
wd;
|
||||
upper_bound = UPPER_BOUND,
|
||||
augmented = true,
|
||||
show_progress = true,
|
||||
)
|
||||
warm = nothing
|
||||
|
||||
begin
|
||||
@time status, warm = PropertyT.solve(
|
||||
model,
|
||||
scs_optimizer(;
|
||||
linear_solver = SCS.MKLDirectSolver,
|
||||
eps = 1e-10,
|
||||
max_iters = 20_000,
|
||||
accel = 50,
|
||||
alpha = 1.95,
|
||||
),
|
||||
warm,
|
||||
)
|
||||
|
||||
@info "reconstructing the solution"
|
||||
Q = @time begin
|
||||
wd = wd
|
||||
Ps = [JuMP.value.(P) for P in varP]
|
||||
if any(any(isnan, P) for P in Ps)
|
||||
throw("solver was probably interrupted, no valid solution available")
|
||||
end
|
||||
Qs = real.(sqrt.(Ps))
|
||||
PropertyT.reconstruct(Qs, wd)
|
||||
end
|
||||
P = Q' * Q
|
||||
|
||||
@info "certifying the solution"
|
||||
@time certified, λ = PropertyT.certify_solution(
|
||||
elt,
|
||||
unit,
|
||||
JuMP.objective_value(model),
|
||||
Q;
|
||||
halfradius = HALFRADIUS,
|
||||
augmented = true,
|
||||
)
|
||||
end
|
||||
|
||||
### grading below
|
||||
|
||||
function desubscriptify(symbol::Symbol)
|
||||
digits = [
|
||||
Int(l) - 0x2080 for
|
||||
l in reverse(string(symbol)) if 0 ≤ Int(l) - 0x2080 ≤ 9
|
||||
]
|
||||
res = 0
|
||||
for (i, d) in enumerate(digits)
|
||||
res += 10^(i - 1) * d
|
||||
end
|
||||
return res
|
||||
end
|
||||
|
||||
function PropertyT.grading(g::MatrixGroups.MatrixElt, roots = roots)
|
||||
id = desubscriptify(g.id)
|
||||
return roots[id]
|
||||
end
|
||||
|
||||
Δs = PropertyT.laplacians(
|
||||
RG,
|
||||
S,
|
||||
x -> (gx = PropertyT.grading(x); Set([gx, -gx])),
|
||||
)
|
||||
|
||||
elt = PropertyT.Adj(Δs)
|
||||
elt == Δ^2 - PropertyT.Sq(Δs)
|
||||
unit = Δ
|
||||
|
||||
@time model, varP = PropertyT.sos_problem_primal(
|
||||
elt,
|
||||
unit,
|
||||
wd;
|
||||
upper_bound = UPPER_BOUND,
|
||||
augmented = true,
|
||||
)
|
||||
|
||||
warm = nothing
|
||||
|
||||
begin
|
||||
@time status, warm = PropertyT.solve(
|
||||
model,
|
||||
scs_optimizer(;
|
||||
linear_solver = SCS.MKLDirectSolver,
|
||||
eps = 1e-10,
|
||||
max_iters = 50_000,
|
||||
accel = 50,
|
||||
alpha = 1.95,
|
||||
),
|
||||
warm,
|
||||
)
|
||||
|
||||
@info "reconstructing the solution"
|
||||
Q = @time begin
|
||||
wd = wd
|
||||
Ps = [JuMP.value.(P) for P in varP]
|
||||
if any(any(isnan, P) for P in Ps)
|
||||
throw("solver was probably interrupted, no valid solution available")
|
||||
end
|
||||
Qs = real.(sqrt.(Ps))
|
||||
PropertyT.reconstruct(Qs, wd)
|
||||
end
|
||||
P = Q' * Q
|
||||
|
||||
@info "certifying the solution"
|
||||
@time certified, λ = PropertyT.certify_solution(
|
||||
elt,
|
||||
unit,
|
||||
JuMP.objective_value(model),
|
||||
Q;
|
||||
halfradius = HALFRADIUS,
|
||||
augmented = true,
|
||||
)
|
||||
end
|
||||
|
||||
# Δ² - 1 / 1 · Sq → -0.8818044647162608
|
||||
# Δ² - 2 / 3 · Sq → -0.1031738
|
||||
# Δ² - 1 / 2 · Sq → 0.228296213895906
|
||||
# Δ² - 1 / 3 · Sq → 0.520
|
||||
# Δ² - 0 / 1 · Sq → 0.9676851592000731
|
||||
# Sq → 0.333423
|
||||
|
||||
# vals = [
|
||||
# 1.0 -0.8818
|
||||
# 2/3 -0.1032
|
||||
# 1/2 0.2282
|
||||
# 1/3 0.520
|
||||
# 0 0.9677
|
||||
# ]
|
||||
308
scripts/G₂_gens.jl
Normal file
308
scripts/G₂_gens.jl
Normal file
@ -0,0 +1,308 @@
|
||||
#= GAP code to generate matrices
|
||||
alg := SimpleLieAlgebra("G", 2, Rationals);
|
||||
root_sys := RootSystem(alg);
|
||||
pos_gens := PositiveRootVectors(root_sys);
|
||||
pos_rts := PositiveRoots(root_sys);
|
||||
|
||||
neg_gens := NegativeRootVectors(root_sys);
|
||||
neg_rts := NegativeRoots(root_sys);
|
||||
|
||||
alg_gens := ShallowCopy(pos_gens);;
|
||||
Append(alg_gens, neg_gens);
|
||||
grading := ShallowCopy(pos_rts);
|
||||
Append(grading, neg_rts);
|
||||
|
||||
mats := List(alg_gens, x->AdjointMatrix(Basis(alg), x));
|
||||
|
||||
W := WeylGroup(root_sys);
|
||||
PW := Action(W, grading, OnRight);
|
||||
=#
|
||||
|
||||
using LinearAlgebra
|
||||
|
||||
function matrix_exp(M::AbstractMatrix{<:Integer})
|
||||
res = zeros(Rational{eltype(M)}, size(M))
|
||||
res += I
|
||||
k = 0
|
||||
expM = one(M)
|
||||
while !iszero(expM)
|
||||
k += 1
|
||||
expM *= M
|
||||
@. res += 1 // factorial(k) * expM
|
||||
if k == 20
|
||||
@warn "matrix exponential did not converge" norm(expM - exp(M))
|
||||
break
|
||||
end
|
||||
end
|
||||
@debug "matrix_exp converged after $k iterations"
|
||||
return res
|
||||
end
|
||||
|
||||
const gap_adj_mats = [
|
||||
[
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 1],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
],
|
||||
[
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, -2],
|
||||
[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0],
|
||||
],
|
||||
[
|
||||
[0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1],
|
||||
[2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0],
|
||||
],
|
||||
[
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0],
|
||||
[3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0],
|
||||
],
|
||||
[
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 1],
|
||||
[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0],
|
||||
],
|
||||
[
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0],
|
||||
],
|
||||
[
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -1],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
],
|
||||
[
|
||||
[0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 2],
|
||||
[0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
],
|
||||
[
|
||||
[0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[-3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1],
|
||||
[0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0],
|
||||
[0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
],
|
||||
[
|
||||
[0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[-2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0],
|
||||
[0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
],
|
||||
[
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, -1],
|
||||
[0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
],
|
||||
[
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1],
|
||||
[0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
[0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0],
|
||||
],
|
||||
]
|
||||
|
||||
function G₂_matrices_roots()
|
||||
adj_mats = map(gap_adj_mats) do m
|
||||
return hcat(m...)
|
||||
end
|
||||
adj_mats = filter!(!isdiag, adj_mats) # remove the ones from center
|
||||
|
||||
gens_mats = [convert(Matrix{Int}, matrix_exp(m')) for m in adj_mats]
|
||||
|
||||
#=
|
||||
The roots from
|
||||
|
||||
G₂roots_gap = [
|
||||
[2, -1], # α = e₁ - e₂
|
||||
[-3, 2], # A = -α + β = -e₁ + 2e₂ - e₃
|
||||
[-1, 1], # β = e₂ - e₃
|
||||
[1, 0], # α + β = e₁ - e₃
|
||||
[3, -1], # B = 2α + β = 2e₁ - e₂ - e₃
|
||||
[0, 1], # A + B = α + 2β = e₁ + e₂ - 2e₃
|
||||
[-2, 1], # -α
|
||||
[3, -2], # -A
|
||||
[1, -1], # -β
|
||||
[-1, 0], # -α - β
|
||||
[-3, 1], # -B
|
||||
[0, -1], # -A - B
|
||||
]
|
||||
|
||||
G₂roots_gap are the ones from cartan matrix. To obtain the standard
|
||||
(hexagonal) picture map them by `T` defined as follows:
|
||||
```julia
|
||||
cartan = hcat(G₂roots_gap[1:2]...)
|
||||
rot(α) = [cos(α) -sin(α); sin(α) cos(α)]
|
||||
|
||||
c₁ = [√2, 0]
|
||||
c₂ = rot(5π / 6) * [√2, 0] * √3 # (= 1/2[√6, 1])
|
||||
|
||||
T = hcat(c₁, c₂) * inv(cartan)
|
||||
```
|
||||
By plotting one against the others (or by blind calculation) one can see
|
||||
the following assignment. Here `⟨α, β⟩_ℤ = A₂` and `⟨A, B⟩_ℤ ≅ √3/√2 A₂`.
|
||||
=#
|
||||
e₁ = PropertyT.Roots.𝕖(3, 1)
|
||||
e₂ = PropertyT.Roots.𝕖(3, 2)
|
||||
e₃ = PropertyT.Roots.𝕖(3, 3)
|
||||
|
||||
α = e₁ - e₂
|
||||
β = e₂ - e₃
|
||||
A = -α + β
|
||||
B = α + (α + β)
|
||||
|
||||
roots = [α, A, β, α + β, B, A + B, -α, -A, -β, -α - β, -B, -A - B]
|
||||
|
||||
return gens_mats, roots
|
||||
end
|
||||
|
||||
function G₂_roots_weyl()
|
||||
(mats, roots) = G₂_matrices_roots()
|
||||
d = size(first(mats), 1)
|
||||
G₂ = Groups.MatrixGroup{d}(mats)
|
||||
|
||||
m = Groups.gens(G₂)
|
||||
|
||||
σ = let w = m[1] * inv(m[7]) * m[1], m = union(m, inv.(m))
|
||||
PermutationGroups.Perm([findfirst(==(inv(w) * x * w), m) for x in m])
|
||||
end
|
||||
|
||||
τ = let w = m[2] * inv(m[8]) * m[2], m = union(m, inv.(m))
|
||||
PermutationGroups.Perm([findfirst(==(inv(w) * x * w), m) for x in m])
|
||||
end
|
||||
|
||||
W = PermGroup(σ, τ)
|
||||
|
||||
return G₂, roots, W
|
||||
end
|
||||
Loading…
Reference in New Issue
Block a user