345 lines
10 KiB
Julia
345 lines
10 KiB
Julia
using Test
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using AbstractAlgebra
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using GroupRings
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using SparseArrays
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@testset "GroupRings" begin
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@testset "Constructors: PermutationGroup" begin
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G = PermutationGroup(3)
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@test isa(GroupRing(G), AbstractAlgebra.NCRing)
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@test isa(GroupRing(G), GroupRing)
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RG = GroupRing(G)
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@test isdefined(RG, :basis) == true
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@test length(RG.basis) == 6
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@test isdefined(RG, :basis_dict) == true
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@test isdefined(RG, :pm) == false
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@test isa(GroupRing(PermutationGroup(6), rand(1:6, 6,6)), GroupRing)
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RG = GroupRing(G, cachedmul=true)
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@test isdefined(RG, :pm) == true
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@test RG.pm == zeros(Int, (6,6))
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@test isa(complete!(RG), GroupRing)
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@test all(RG.pm .> 0)
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@test RG.pm == GroupRings.initializepm!(GroupRing(G, cachedmul=false), fill=true).pm
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@test RG.basis_dict == GroupRings.reverse_dict(collect(G))
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@test isa(GroupRing(G, collect(G)), GroupRing)
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S = collect(G)
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pm = create_pm(S)
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@test isa(GroupRing(G, S), GroupRing)
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@test isa(GroupRing(G, S, pm), GroupRing)
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A = GroupRing(G, S)
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B = GroupRing(G, S, pm)
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@test RG == A
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@test RG == B
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end
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@testset "GroupRing constructors FreeGroup" begin
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using Groups
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F = FreeGroup(3)
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S = gens(F)
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append!(S, [inv(s) for s in S])
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basis, sizes = Groups.generate_balls(S, F(), radius=4)
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d = GroupRings.reverse_dict(basis)
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@test_throws KeyError create_pm(basis)
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pm = create_pm(basis, d, sizes[2])
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@test isa(GroupRing(F, basis, pm), GroupRing)
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@test isa(GroupRing(F, basis, d, pm), GroupRing)
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A = GroupRing(F, basis, pm)
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B = GroupRing(F, basis, d, pm)
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@test A == B
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RF = GroupRing(F, basis, d, create_pm(basis, d, check=false))
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nz1 = count(!iszero, RF.pm)
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@test nz1 > 1000
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GroupRings.complete!(RF)
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nz2 = count(!iszero, RF.pm)
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@test nz2 > nz1
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@test nz2 == 45469
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g = B()
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s = S[2]
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g[s] = 1
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@test g == B(s)
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@test g[s^2] == 0
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@test_throws KeyError g[s^10]
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end
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@testset "GroupRingElems constructors/basic manipulation" begin
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G = PermutationGroup(3)
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RG = GroupRing(G, cachedmul=true)
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a = rand(6)
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@test isa(GroupRingElem(a, RG), GroupRingElem)
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@test isa(RG(a), GroupRingElem)
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@test all(isa(RG(g), GroupRingElem) for g in G)
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@test_throws String GroupRingElem([1,2,3], RG)
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@test isa(RG(G([2,3,1])), GroupRingElem)
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p = G([2,3,1])
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a = RG(p)
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@test length(a) == 1
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@test isa(a.coeffs, SparseVector)
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@test a.coeffs[5] == 1
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@test a[5] == 1
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@test a[p] == 1
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@test string(a) == "(1,2,3)"
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@test string(-a) == " - 1(1,2,3)"
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@test RG([0,0,0,0,1,0]) == a
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s = G([1,2,3])
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@test a[s] == 0
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a[s] = 2
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@test a.coeffs[1] == 2
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@test a[1] == 2
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@test a[s] == 2
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@test string(a) == "2() + (1,2,3)"
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@test string(-a) == " - 2() - 1(1,2,3)"
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@test length(a) == 2
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@testset "RSL(3,Z)" begin
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N = 3
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halfradius = 2
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M = MatrixAlgebra(zz, N)
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E(M, i,j) = (e_ij = one(M); e_ij[i,j] = 1; e_ij)
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S = [E(M, i,j) for i in 1:N for j in 1:N if i≠j]
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S = unique([S; inv.(S)])
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E_R, sizes = Groups.generate_balls(S, radius=2*halfradius)
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E_rdict = GroupRings.reverse_dict(E_R)
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pm = GroupRings.create_pm(E_R, E_rdict, sizes[halfradius]; twisted=true);
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@test GroupRing(M, E_R, E_rdict, pm) isa GroupRing
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end
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end
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@testset "Arithmetic" begin
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G = PermutationGroup(3)
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RG = GroupRing(G, cachedmul=true)
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a = RG(ones(Int, order(G)))
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@testset "scalar operators" begin
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@test isa(-a, GroupRingElem)
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@test (-a).coeffs == -(a.coeffs)
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@test isa(2*a, GroupRingElem)
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@test eltype(2*a) == typeof(2)
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@test (2*a).coeffs == 2 .*(a.coeffs)
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ww = "Scalar and coeffs are in different rings! Promoting result to Float64"
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@test isa(2.0*a, GroupRingElem)
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@test_logs (:warn, ww) eltype(2.0*a) == typeof(2.0)
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@test_logs (:warn, ww) (2.0*a).coeffs == 2.0.*(a.coeffs)
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@test_logs (:warn, ww) (a/2).coeffs == a.coeffs./2
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b = a/2
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@test isa(b, GroupRingElem)
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@test eltype(b) == typeof(1/2)
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@test (b/2).coeffs == 0.25*(a.coeffs)
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@test isa(convert(Rational{Int}, a), GroupRingElem)
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@test eltype(convert(Rational{Int}, a)) == Rational{Int}
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@test convert(Rational{Int}, a).coeffs ==
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convert(Vector{Rational{Int}}, a.coeffs)
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b = convert(Rational{Int}, a)
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@test isa(b//4, GroupRingElem)
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@test eltype(b//4) == Rational{Int}
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@test isa(b//big(4), NCRingElem)
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@test eltype(b//(big(4)//1)) == Rational{BigInt}
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@test isa(a//1, GroupRingElem)
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@test eltype(a//1) == Rational{Int}
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@test (1.0a)//1 == (1.0a)
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end
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@testset "Additive structure" begin
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@test RG(ones(Int, order(G))) == sum(RG(g) for g in G)
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a = RG(ones(Int, order(G)))
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b = sum((-1)^parity(g)*RG(g) for g in G)
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@test 1/2*(a+b).coeffs == [1.0, 0.0, 1.0, 0.0, 1.0, 0.0]
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a = RG(1) + RG(perm"(2,3)") + RG(perm"(1,2,3)")
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b = RG(1) - RG(perm"(1,2)(3)") - RG(perm"(1,2,3)")
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@test a - b == RG(perm"(2,3)") + RG(perm"(1,2)(3)") + 2RG(perm"(1,2,3)")
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@test 1//2*2a == a
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@test a + 2a == (3//1)*a
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@test 2a - (1//1)*a == a
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end
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@testset "Multiplicative structure" begin
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for g in G, h in G
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a = RG(g)
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b = RG(h)
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@test a*b == RG(g*h)
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@test (a+b)*(a+b) == a*a + a*b + b*a + b*b
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end
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for g in G
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@test star(RG(g)) == RG(inv(g))
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@test (one(RG)-RG(g))*star(one(RG)-RG(g)) ==
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2*one(RG) - RG(g) - RG(inv(g))
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@test aug((one(RG)-RG(g))) == 0
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end
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a = RG(1) + RG(perm"(2,3)") + RG(perm"(1,2,3)")
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b = RG(1) - RG(perm"(1,2)(3)") - RG(perm"(1,2,3)")
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@test a*b == mul!(a,a,b)
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@test aug(a) == 3
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@test aug(b) == -1
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@test aug(a)*aug(b) == aug(a*b) == aug(b*a)
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z = sum((one(RG)-RG(g))*star(one(RG)-RG(g)) for g in G)
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@test aug(z) == 0
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@test supp(z) == parent(z).basis
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@test supp(RG(1) + RG(perm"(2,3)")) == [G(), perm"(2,3)"]
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@test supp(a) == [perm"(3)", perm"(2,3)", perm"(1,2,3)"]
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end
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@testset "HPC multiplicative operations" begin
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G = PermutationGroup(5)
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RG = GroupRing(G, cachedmul=true)
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RG2 = GroupRing(G, cachedmul=false)
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Z = RG()
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W = RG()
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for g in [rand(G) for _ in 1:30]
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X = RG(g)
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Y = -RG(inv(g))
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for i in 1:10
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X[rand(G)] += rand(1:3)
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Y[rand(G)] -= rand(1:3)
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end
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@test X*Y ==
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RG2(X)*RG2(Y) ==
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GroupRings.mul!(Z, X, Y)
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@test Z.coeffs == GroupRings.GRmul!(W.coeffs, X.coeffs, Y.coeffs, RG.pm) == W.coeffs
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@test (2*X*Y).coeffs ==
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GroupRings.fmac!(W.coeffs, X.coeffs, Y.coeffs, RG.pm) ==
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GroupRings.mul!(2, X*Y).coeffs
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end
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end
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end
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@testset "SumOfSquares in group rings" begin
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∗ = star
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G = FreeGroup(["g", "h", "k", "l"])
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S = G.(G.gens)
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S = [S; inv.(S)]
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ID = G()
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RADIUS=3
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@time E_R, sizes = Groups.generate_balls(S, ID, radius=2*RADIUS);
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@test sizes == [9, 65, 457, 3201, 22409, 156865]
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E_rdict = GroupRings.reverse_dict(E_R)
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pm = GroupRings.create_pm(E_R, E_rdict, sizes[RADIUS]; twisted=true);
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RG = GroupRing(G, E_R, E_rdict, pm)
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g = RG.basis[2]
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h = RG.basis[3]
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k = RG.basis[4]
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l = RG.basis[5]
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G = (1-RG(g))
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@test G^2 == 2 - RG(g) - ∗(RG(g))
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G = (1-RG(g))
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H = (1-RG(h))
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K = (1-RG(k))
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L = (1-RG(l))
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GH = (1-RG(g*h))
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KL = (1-RG(k*l))
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X = (2 - ∗(RG(g)) - RG(h))
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Y = (2 - ∗(RG(g*h)) - RG(k))
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@test -(2 - RG(g*h) - ∗(RG(g*h))) + 2G^2 + 2H^2 == X^2
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@test (2 - RG(g*h) - ∗(RG(g*h))) == GH^2
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@test -(2 - RG(g*h*k) - ∗(RG(g*h*k))) + 2GH^2 + 2K^2 == Y^2
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@test -(2 - RG(g*h*k) - ∗(RG(g*h*k))) +
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2(GH^2 - 2G^2 - 2H^2) +
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4G^2 + 4H^2 + 2K^2 ==
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Y^2
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@test GH^2 - 2G^2 - 2H^2 == - X^2
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@test -(2 - RG(g*h*k) - ∗(RG(g*h*k))) + 4G^2 + 4H^2 + 2K^2 == 2X^2 + Y^2
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@test GH^2 == 2G^2 + 2H^2 - (2 - ∗(RG(g)) - RG(h))^2
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@test KL^2 == 2K^2 + 2L^2 - (2 - ∗(RG(k)) - RG(l))^2
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@test -(2 - ∗(RG(g*h*k*l)) - RG(g*h*k*l)) + 2*GH^2 + 2*KL^2 ==
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(2 - ∗(RG(g*h)) - RG(k*l))^2
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@test -(2 - ∗(RG(g*h*k*l)) - RG(g*h*k*l)) +
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2(2G^2 + 2H^2 - (2 - ∗(RG(g)) - RG(h))^2) +
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2(2K^2 + 2L^2 - (2 - ∗(RG(k)) - RG(l))^2) ==
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(2 - ∗(RG(g*h)) - RG(k*l))^2
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@test -(2 - ∗(RG(g*h*k*l)) - RG(g*h*k*l)) +
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2(2G^2 + 2H^2) +
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2(2K^2 + 2L^2) ==
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(2 - ∗(RG(g*h)) - RG(k*l))^2 +
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2(2 - ∗(RG(g)) - RG(h))^2 +
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2(2 - ∗(RG(k)) - RG(l))^2
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@test -(2 - ∗(RG(g*h*k*l)) - RG(g*h*k*l)) +
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2(2 - ∗(RG(g*h*k)) - RG(g*h*k)) + 2L^2 ==
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(2 - ∗(RG(g*h*k)) - RG(l))^2
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@test 2 - ∗(RG(g*h*k)) - RG(g*h*k) ==
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2GH^2 + 2K^2 - (2 - ∗(RG(g*h)) - RG(k))^2
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@test -(2 - ∗(RG(g*h*k*l)) - RG(g*h*k*l)) +
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2(2GH^2 + 2K^2 - (2 - ∗(RG(g*h)) - RG(k))^2) + 2L^2 ==
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(2 - ∗(RG(g*h*k)) - RG(l))^2
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@test -(2 - ∗(RG(g*h*k*l)) - RG(g*h*k*l)) +
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2(2GH^2 + 2K^2) + 2L^2 ==
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(2 - ∗(RG(g*h*k)) - RG(l))^2 +
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2(2 - ∗(RG(g*h)) - RG(k))^2
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@test -(2 - ∗(RG(g*h*k*l)) - RG(g*h*k*l)) +
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8G^2 + 8H^2 + 4K^2 + 2L^2 ==
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(2 - ∗(RG(g*h*k)) - RG(l))^2 + 2(2 - ∗(RG(g*h)) - RG(k))^2 + 4(2 - ∗(RG(g)) - RG(h))^2
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@test -(2 - ∗(RG(g*h*k*l)) - RG(g*h*k*l)) +
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2GH^2 + 2KL^2 == (2 - ∗(RG(g*h)) - RG(k*l))^2
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@test -(2 - ∗(RG(g*h*k*l)) - RG(g*h*k*l)) + 2(2G^2 + 2H^2) + 2(2K^2 + 2L^2) ==
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(2 - ∗(RG(g*h)) - RG(k*l))^2 + 2(2 - ∗(RG(k)) - RG(l))^2 + 2(2 - ∗(RG(g)) - RG(h))^2
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end
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end
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