fix _positive_direction so that Adj works for N=2

src/roots.jl

@@ -48,10 +48,8 @@ function isorthogonal(α::AbstractRoot{N}, β::AbstractRoot{M}) where {N,M}
 end
 
 function _positive_direction(α::Root{N}) where {N}
-    last = -1 / √2^(N - 1)
-    return Root{N,Float64}(
-        SVector(ntuple(i -> ifelse(i == N, last, (√2)^-i), N)),
-    )
+    v = α.coord + 1 / (N * 100) * rand(N)
+    return Root{N,Float64}(v / norm(v, 2))
 end
 
 function positive(roots::AbstractVector{<:Root{N}}) where {N}

test/graded_adj.jl

@@ -46,6 +46,28 @@
 
 
     @testset "Symplectic group" begin
+        @testset "Sp2(ℤ)" begin
+            genus = 2
+            halfradius = 1
+
+            SpN = MatrixGroups.SymplecticGroup{2genus}(Int8)
+
+            RSpN, S_sp, sizes_sp = PropertyT.group_algebra(SpN, halfradius=halfradius, twisted=true)
+
+            Δ, Δs = let RG = RSpN, S = S_sp, ψ = identity
+                Δ = RG(length(S)) - sum(RG(s) for s in S)
+                Δs = PropertyT.laplacians(
+                    RG,
+                    S,
+                    x -> (gx = PropertyT.grading(ψ(x)); Set([gx, -gx])),
+                )
+                Δ, Δs
+            end
+
+            sq = sum(Δᵢ^2 for Δᵢ in values(Δs))
+            @test PropertyT.Adj(Δs, :C₂) + sq == Δ^2
+        end
+
         genus = 3
         halfradius = 1