263 lines
7.7 KiB
Julia
263 lines
7.7 KiB
Julia
using SparseArrays
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@testset "Sq, Adj, Op" begin
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function isconstant_on_orbit(v, orb)
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isempty(orb) && return true
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k = v[first(orb)]
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return all(v[o] == k for o in orb)
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end
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@testset "unit tests" begin
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@test PropertyT.isopposite(PG.perm"(1,2,3)(4)", PG.perm"(1,4,2)")
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@test PropertyT.isadjacent(PG.perm"(1,2,3)", PG.perm"(1,2)(3)")
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@test !PropertyT.isopposite(PG.perm"(1,2,3)", PG.perm"(1,2)(3)")
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@test !PropertyT.isadjacent(PG.perm"(1,4)", PG.perm"(2,3)(4)")
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@test isconstant_on_orbit([1, 1, 1, 2, 2], [2, 3])
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@test !isconstant_on_orbit([1, 1, 1, 2, 2], [2, 3, 4])
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end
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@testset "Sq, Adj, Op in SL(4,Z)" begin
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N = 4
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G = MatrixGroups.SpecialLinearGroup{N}(Int8)
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@info "running tests for" G
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RG, S, sizes = PropertyT.group_algebra(G; halfradius = 2)
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Δ = let RG = RG, S = S
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RG(length(S)) - sum(RG(s) for s in S)
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end
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P = PG.PermGroup(PG.perm"(1,2)", PG.Perm(circshift(1:N, -1)))
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Σ = Groups.Constructions.WreathProduct(PG.PermGroup(PG.perm"(1,2)"), P)
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act = PropertyT.action_by_conjugation(G, Σ)
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wd = SW.WedderburnDecomposition(
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Float64,
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Σ,
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act,
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SA.basis(RG),
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SA.Basis{UInt16}(@view SA.basis(RG)[1:sizes[2]]),
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)
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@info wd
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ivs = SW.invariant_vectors(wd)
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sq, adj, op = PropertyT.SqAdjOp(RG, N)
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@test all(
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isconstant_on_orbit(sq, SparseArrays.nonzeroinds(iv)) for iv in ivs
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)
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@test all(
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isconstant_on_orbit(adj, SparseArrays.nonzeroinds(iv)) for iv in ivs
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)
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@test all(
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isconstant_on_orbit(op, SparseArrays.nonzeroinds(iv)) for iv in ivs
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)
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e = one(G)
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g = G([alphabet(G)[MatrixGroups.ElementaryMatrix{N}(1, 2, Int8(1))]])
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h = G([alphabet(G)[MatrixGroups.ElementaryMatrix{N}(1, 3, Int8(1))]])
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k = G([alphabet(G)[MatrixGroups.ElementaryMatrix{N}(3, 4, Int8(1))]])
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@test sq[e] == 120
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@test sq[g] == sq[h] == -8
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@test sq[g^2] == sq[h^2] == 1
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@test sq[g*h] == sq[h*g] == 0
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@test adj[e] == 384
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@test adj[g] == adj[h] == -32
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@test adj[g^2] == adj[h^2] == 0
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@test adj[g*h] == adj[h*g] == 2
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@test adj[k*h] == adj[h*k] == 1
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@test op[e] == 96
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@test op[g] == op[h] == -8
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@test op[g^2] == op[h^2] == 0
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@test op[g*h] == op[h*g] == 0
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@test op[g*k] == op[k*g] == 2
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@test op[h*k] == op[k*h] == 0
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end
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@testset "SAut(F₃)" begin
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n = 3
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G = SpecialAutomorphismGroup(FreeGroup(n))
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RG, S, sizes = PropertyT.group_algebra(G; halfradius = 2)
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sq, adj, op = PropertyT.SqAdjOp(RG, n)
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@test sq(one(G)) == 216
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@test all(sq(g) == -16 for g in gens(G))
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@test adj(one(G)) == 384
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@test all(adj(g) == -32 for g in gens(G))
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@test iszero(op)
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end
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end
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@testset "1812.03456 examples" begin
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@testset "SL(3,Z)" begin
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n = 3
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G = MatrixGroups.SpecialLinearGroup{n}(Int8)
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@info "running tests for" G
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RG, S, sizes = PropertyT.group_algebra(G; halfradius = 2)
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Δ = RG(length(S)) - sum(RG(s) for s in S)
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P = PG.PermGroup(PG.perm"(1,2)", PG.Perm(circshift(1:n, -1)))
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Σ = Groups.Constructions.WreathProduct(PG.PermGroup(PG.perm"(1,2)"), P)
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act = PropertyT.action_by_conjugation(G, Σ)
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wd = SW.WedderburnDecomposition(
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Float64,
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Σ,
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act,
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SA.basis(RG),
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SA.Basis{UInt16}(@view SA.basis(RG)[1:sizes[2]]),
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)
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@info wd
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sq, adj, op = PropertyT.SqAdjOp(RG, n)
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@testset "Sq₃ is SOS" begin
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elt = sq
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UB = Inf # λ ≈ 0.1040844
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status, certified, λ_cert = check_positivity(
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elt,
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Δ,
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wd;
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upper_bound = UB,
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halfradius = 2,
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optimizer = scs_optimizer(; accel = -50, alpha = 1.9),
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)
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@test status == JuMP.OPTIMAL
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@test certified
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@test λ_cert > 104 // 1000
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end
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@testset "Adj₃ is SOS" begin
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elt = adj
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UB = Inf # λ ≈ 0.15858018
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status, certified, λ_cert = check_positivity(
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elt,
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Δ,
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wd;
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upper_bound = UB,
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halfradius = 2,
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optimizer = cosmo_optimizer(; accel = 50, alpha = 1.9),
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)
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@test status == JuMP.OPTIMAL
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@test certified
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@test λ_cert > 1585 // 10000
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m, _ = PropertyT.sos_problem_primal(elt, wd)
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PropertyT.solve(
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m,
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scs_optimizer(; max_iters = 1000, accel = 50, alpha = 1.9),
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)
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@test JuMP.termination_status(m) in
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(JuMP.ALMOST_OPTIMAL, JuMP.OPTIMAL, JuMP.ITERATION_LIMIT)
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@test abs(JuMP.objective_value(m)) < 1e-3
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end
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@testset "Op₃ is empty, so can not be certified" begin
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elt = op
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@test iszero(op)
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UB = Inf
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status, certified, λ_cert = check_positivity(
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elt,
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Δ,
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wd;
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upper_bound = UB,
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halfradius = 2,
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optimizer = scs_optimizer(; accel = 50, alpha = 1.9),
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)
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@test status == JuMP.OPTIMAL
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@test !certified
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@test λ_cert < 0
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end
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end
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@testset "SL(4,Z)" begin
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n = 4
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G = MatrixGroups.SpecialLinearGroup{n}(Int8)
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@info "running tests for" G
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RG, S, sizes = PropertyT.group_algebra(G; halfradius = 2)
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Δ = RG(length(S)) - sum(RG(s) for s in S)
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P = PG.PermGroup(PG.perm"(1,2)", PG.Perm(circshift(1:n, -1)))
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Σ = Groups.Constructions.WreathProduct(PG.PermGroup(PG.perm"(1,2)"), P)
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act = PropertyT.action_by_conjugation(G, Σ)
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wd = SW.WedderburnDecomposition(
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Float64,
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Σ,
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act,
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SA.basis(RG),
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SA.Basis{UInt16}(@view SA.basis(RG)[1:sizes[2]]),
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)
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@info wd
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sq, adj, op = PropertyT.SqAdjOp(RG, n)
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@testset "Sq is SOS" begin
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elt = sq
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UB = Inf # λ ≈ 0.31670
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status, certified, λ_cert = check_positivity(
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elt,
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Δ,
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wd;
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upper_bound = UB,
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halfradius = 2,
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optimizer = cosmo_optimizer(; accel = 50, alpha = 1.9),
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)
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@test status == JuMP.OPTIMAL
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@test certified
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@test λ_cert > 316 // 1000
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end
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@testset "Adj is SOS" begin
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elt = adj
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UB = 0.541 # λ ≈ 0.545710
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status, certified, λ_cert = check_positivity(
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elt,
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Δ,
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wd;
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upper_bound = UB,
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halfradius = 2,
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optimizer = cosmo_optimizer(; accel = 50, alpha = 1.9),
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)
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@test status == JuMP.OPTIMAL
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@test certified
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@test λ_cert > 54 // 100
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end
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@testset "Op is a sum of squares, but not an order unit" begin
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elt = op
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UB = Inf
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status, certified, λ_cert = check_positivity(
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elt,
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Δ,
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wd;
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upper_bound = UB,
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halfradius = 2,
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optimizer = cosmo_optimizer(; accel = 50, alpha = 1.9),
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)
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@test status == JuMP.OPTIMAL
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@test !certified
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@test -1e-2 < λ_cert < 0
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end
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end
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end
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