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109 lines
3.3 KiB
Julia
109 lines
3.3 KiB
Julia
@testset "Adj via grading" begin
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@testset "SL(n,Z) & Aut(F₄)" begin
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n = 4
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halfradius = 1
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SL = MatrixGroups.SpecialLinearGroup{n}(Int8)
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RSL, S, sizes = PropertyT.group_algebra(SL, halfradius=halfradius, twisted=true)
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Δ = RSL(length(S)) - sum(RSL(s) for s in S)
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Δs = let ψ = identity
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PropertyT.laplacians(
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RSL,
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S,
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x -> (gx = PropertyT.grading(ψ(x)); Set([gx, -gx])),
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)
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end
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sq, adj, op = PropertyT.SqAdjOp(RSL, n)
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@test PropertyT.Adj(Δs, :A₁) == sq
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@test PropertyT.Adj(Δs, :A₂) == adj
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@test PropertyT.Adj(Δs, Symbol("A₁×A₁")) == op
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halfradius = 1
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G = SpecialAutomorphismGroup(FreeGroup(n))
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RG, S, sizes = PropertyT.group_algebra(G, halfradius=halfradius, twisted=true)
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Δ = RG(length(S)) - sum(RG(s) for s in S)
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Δs = let ψ = Groups.Homomorphism(Groups._abelianize, G, SL)
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PropertyT.laplacians(
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RG,
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S,
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x -> (gx = PropertyT.grading(ψ(x)); Set([gx, -gx])),
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)
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end
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sq, adj, op = PropertyT.SqAdjOp(RG, n)
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@test PropertyT.Adj(Δs, :A₁) == sq
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@test PropertyT.Adj(Δs, :A₂) == adj
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@test PropertyT.Adj(Δs, Symbol("A₁×A₁")) == op
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end
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@testset "Symplectic group" begin
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@testset "Sp2(ℤ)" begin
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genus = 2
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halfradius = 1
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SpN = MatrixGroups.SymplecticGroup{2genus}(Int8)
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RSpN, S_sp, sizes_sp = PropertyT.group_algebra(SpN, halfradius=halfradius, twisted=true)
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Δ, Δs = let RG = RSpN, S = S_sp, ψ = identity
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Δ = RG(length(S)) - sum(RG(s) for s in S)
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Δs = PropertyT.laplacians(
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RG,
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S,
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x -> (gx = PropertyT.grading(ψ(x)); Set([gx, -gx])),
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)
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Δ, Δs
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end
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sq = sum(Δᵢ^2 for Δᵢ in values(Δs))
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@test PropertyT.Adj(Δs, :C₂) + sq == Δ^2
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end
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genus = 3
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halfradius = 1
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SpN = MatrixGroups.SymplecticGroup{2genus}(Int8)
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RSpN, S_sp, sizes_sp = PropertyT.group_algebra(SpN, halfradius=halfradius, twisted=true)
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Δ, Δs = let RG = RSpN, S = S_sp, ψ = identity
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Δ = RG(length(S)) - sum(RG(s) for s in S)
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Δs = PropertyT.laplacians(
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RG,
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S,
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x -> (gx = PropertyT.grading(ψ(x)); Set([gx, -gx])),
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)
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Δ, Δs
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end
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@testset "Adj correctness: genus=$genus" begin
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all_subtypes = (
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:A₁, :C₁, Symbol("A₁×A₁"), Symbol("C₁×C₁"), Symbol("A₁×C₁"), :A₂, :C₂
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)
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@test PropertyT.Adj(Δs, :A₂)[one(SpN)] == 384
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@test iszero(PropertyT.Adj(Δs, Symbol("A₁×A₁")))
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@test iszero(PropertyT.Adj(Δs, Symbol("C₁×C₁")))
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@testset "divisibility by 16" begin
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for subtype in all_subtypes
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subtype in (:A₁, :C₁) && continue
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@test isinteger(PropertyT.Adj(Δs, subtype)[one(SpN)] / 16)
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end
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end
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@test sum(PropertyT.Adj(Δs, subtype) for subtype in all_subtypes) == Δ^2
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end
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end
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end
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