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102 lines
3.4 KiB
Julia
102 lines
3.4 KiB
Julia
@testset "actions on Group[Rings]" begin
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Eij = PropertyT.EltaryMat
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ssgs(M::MatAlgebra, i, j) = (S = [Eij(M, i, j), Eij(M, j, i)];
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S = unique([S; inv.(S)]); S)
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function ssgs(A::AutGroup, i, j)
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rmuls = [Groups.transvection_R(i,j), Groups.transvection_R(j,i)]
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lmuls = [Groups.transvection_L(i,j), Groups.transvection_L(j,i)]
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gen_set = A.([rmuls; lmuls])
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return unique([gen_set; inv.(gen_set)])
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end
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@testset "actions on SL(3,Z) and its group ring" 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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S = PropertyT.generating_set(M)
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E_R, sizes = Groups.wlmetric_ball(S, one(M), radius=2halfradius);
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rdict = GroupRings.reverse_dict(E_R)
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pm = GroupRings.create_pm(E_R, rdict, sizes[halfradius]; twisted=false);
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RG = GroupRing(M, E_R, rdict, pm)
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@testset "correctness of actions" begin
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Δ = length(S)*RG(1) - sum(RG(s) for s in S)
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@test Δ == PropertyT.spLaplacian(RG, S)
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elt = S[5]
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x = RG(1) - RG(elt)
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elt2 = E_R[rand(sizes[1]:sizes[2])]
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y = 2RG(elt2) - RG(elt)
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for G in [SymmetricGroup(N), WreathProduct(SymmetricGroup(2), SymmetricGroup(N))]
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@test all(one(M)^g == one(M) for g in G)
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@test all(rdict[m^g] <= sizes[1] for g in G for m in S)
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@test all(m^g*n^g == (m*n)^g for g in G for m in S for n in S)
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@test all(Δ^g == Δ for g in G)
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@test all(x^g == RG(1) - RG(elt^g) for g in G)
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@test all(2RG(elt2^g) - RG(elt^g) == y^g for g in G)
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end
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end
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@testset "small Laplacians" begin
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for (i,j) in PropertyT.indexing(N)
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Sij = ssgs(M, i,j)
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Δij= PropertyT.spLaplacian(RG, Sij)
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@test all(Δij^p == PropertyT.spLaplacian(RG, ssgs(M, p[i], p[j])) for p in SymmetricGroup(N))
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@test all(Δij^g == PropertyT.spLaplacian(RG, ssgs(M, g.p[i], g.p[j])) for g in WreathProduct(SymmetricGroup(2), SymmetricGroup(N)))
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end
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end
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end
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@testset "actions on SAut(F_3) and its group ring" begin
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N = 3
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halfradius = 2
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M = SAut(FreeGroup(N))
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S = PropertyT.generating_set(M)
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E_R, sizes = Groups.wlmetric_ball(S, one(M), radius=2halfradius);
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rdict = GroupRings.reverse_dict(E_R)
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pm = GroupRings.create_pm(E_R, rdict, sizes[halfradius]; twisted=false);
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RG = GroupRing(M, E_R, rdict, pm)
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@testset "correctness of actions" begin
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Δ = length(S)*RG(1) - sum(RG(s) for s in S)
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@test Δ == PropertyT.spLaplacian(RG, S)
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elt = S[5]
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x = RG(1) - RG(elt)
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elt2 = E_R[rand(sizes[1]:sizes[2])]
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y = 2RG(elt2) - RG(elt)
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for G in [SymmetricGroup(N), WreathProduct(SymmetricGroup(2), SymmetricGroup(N))]
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@test all(one(M)^g == one(M) for g in G)
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@test all(rdict[m^g] <= sizes[1] for g in G for m in S)
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@test all(m^g*n^g == (m*n)^g for g in G for m in S for n in S)
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@test all(Δ^g == Δ for g in G)
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@test all(x^g == RG(1) - RG(elt^g) for g in G)
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@test all(2RG(elt2^g) - RG(elt^g) == y^g for g in G)
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end
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end
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for (i,j) in PropertyT.indexing(N)
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Sij = ssgs(M, i,j)
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Δij= PropertyT.spLaplacian(RG, Sij)
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@test all(Δij^p == PropertyT.spLaplacian(RG, ssgs(M, p[i], p[j])) for p in SymmetricGroup(N))
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@test all(Δij^g == PropertyT.spLaplacian(RG, ssgs(M, g.p[i], g.p[j])) for g in WreathProduct(SymmetricGroup(2), SymmetricGroup(N)))
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end
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end
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end
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