compute eigenvalues of representations of PSL(2,109) (PEC supplied)

.gitignore

@@ -0,0 +1,2 @@
+log
+data/*rep*

Manifest.toml

@@ -299,11 +299,17 @@ git-tree-sha1 = "928b8ca9b2791081dc71a51c55347c27c618760f"
 uuid = "77ba4419-2d1f-58cd-9bb1-8ffee604a2e3"
 version = "0.3.3"
 
+[[Nemo]]
+deps = ["AbstractAlgebra", "BinaryProvider", "InteractiveUtils", "Libdl", "LinearAlgebra", "Markdown", "Random", "Test"]
+git-tree-sha1 = "0db7e2b72bd67770d61ae2af18376a78d817816c"
+uuid = "2edaba10-b0f1-5616-af89-8c11ac63239a"
+version = "0.15.1"
+
 [[OpenBLAS_jll]]
 deps = ["CompilerSupportLibraries_jll", "Libdl", "Pkg"]
-git-tree-sha1 = "2ee3e636e94b9fd95fa8364d5cba2e20dae16609"
+git-tree-sha1 = "1887096f6897306a4662f7c5af936da7d5d1a062"
 uuid = "4536629a-c528-5b80-bd46-f80d51c5b363"
-version = "0.3.9+2"
+version = "0.3.9+4"
 
 [[OpenSpecFun_jll]]
 deps = ["CompilerSupportLibraries_jll", "Libdl", "Pkg"]

PSL.jl

@@ -0,0 +1,80 @@
+using Nemo
+using DelimitedFiles
+
+include("src/nemo_utils.jl")
+
+function parse_evalZ(arg, expr_str)
+    ex = Meta.parse(expr_str)
+    return @eval begin
+        let Z=$arg
+	    $ex
+	end
+    end
+end
+
+function parse_evalzz(arg, expr_str)
+    ex = Meta.parse(expr_str)
+    return @eval begin
+        let zz=$arg
+	    $ex
+	end
+    end
+end
+
+function load_discrete_repr(i, q=109; CC=AcbField(128))
+    ζ = root_of_unity(CC, (q+1)÷2)
+	degree = q-1
+
+	ra = readdlm("data/Discrete reps PSL(2, $q)/discrete_rep_$(i)_a.txt", ',', String)
+	a = matrix(CC, [CC(parse_evalZ(ζ, s)) for s in ra[1:degree, 1:degree]])
+
+	rb = readdlm("data/Discrete reps PSL(2, $q)/discrete_rep_$(i)_b.txt", ',', String)
+    b = matrix(CC, [CC(parse_evalZ(ζ, s)) for s in rb[1:degree, 1:degree]])
+
+    return a,b
+end
+
+
+function load_principal_repr(i, q=109; CC=AcbField(128))
+    ζ = root_of_unity(CC, (q+1)÷2)
+	degree = q+1
+
+	ra = readdlm("data/Principal reps PSL(2, $q)/principal_rep_$(i)_a.txt", ',', String)
+    a = matrix(CC, [CC(parse_evalzz(ζ, s)) for s in ra[1:degree, 1:degree]])
+
+	rb = readdlm("data/Principal reps PSL(2, $q)/principal_rep_$(i)_b.txt", ',', String)
+	b = matrix(CC, [CC(parse_evalzz(ζ, s)) for s in rb[1:degree, 1:degree]])
+
+    return a,b
+end
+
+# for i in 0:27
+# 	try
+# 		a,b = load_principal_repr(i)
+# 		adjacency = sum([[a^i for i in 1:4]; [b^i for i in 1:4]])
+# 		M = parent(adjacency)
+#
+# 		# X = M(rand(base_ring, size(adjacency)))
+#
+# 		# @time ev = eigvals(X*adjacency*inv(X))
+# 		@time evc = eigvals(adjacency)
+# 		ev = sort(real.(first.(evc)), lt=<, rev=true)
+# 		@info "Principal Series Representation $i" ev[1:4]
+# 	catch ex
+# 		@error "Principal Series Representation $i : failed"
+# 		ex isa InterruptException && throw(ex)
+# 	end
+# end
+
+for i in 1:27
+	try
+		a,b = load_discrete_repr(i)
+		adjacency = sum([[a^i for i in 1:4]; [b^i for i in 1:4]])
+		@time evc = eigvals(adjacency)
+		ev = sort(real.(first.(evc)), lt=<, rev=true)
+		@info "Discrete Series Representation $i" ev[1:4]
+	catch ex
+		@error "Discrete Series Representation $i : failed"
+		ex isa InterruptException && rethrow(ex)
+	end
+end

Project.toml

@@ -5,6 +5,7 @@ DelimitedFiles = "8bb1440f-4735-579b-a4ab-409b98df4dab"
 GroupRings = "0befed6a-bd73-11e8-1e41-a1190947c2f5"
 Groups = "5d8bd718-bd84-11e8-3b40-ad14f4a32557"
 Logging = "56ddb016-857b-54e1-b83d-db4d58db5568"
+Nemo = "2edaba10-b0f1-5616-af89-8c11ac63239a"
 PropertyT = "03b72c93-0167-51e2-8a1e-eb4ff1fb940d"
 RamanujanGraphs = "e7bd6bc6-b6b8-11e9-1ec2-2f89442c0d6c"
 SCS = "c946c3f1-0d1f-5ce8-9dea-7daa1f7e2d13"

src/PSL.jl

@@ -1,19 +0,0 @@
-using RamanujanGraphs
-using RamanujanGraphs.LightGraphs
-using Arpack
-
-Γ, eigenvalues = let q = 109
-      a = RamanujanGraphs.PSL₂{q}([  0  1
-                                   108 11])
-      b = RamanujanGraphs.PSL₂{q}([57  2
-                                   52 42])
-
-      S = unique([[a^i for i in 1:4]; [b^i for i in 1:4]])
-
-      @info "Generating set S of $(eltype(S))" S
-      @time Γ, verts, vlabels, elabels = RamanujanGraphs.cayley_graph((q^3 - q)÷2, S)
-      @assert all(degree(Γ,i) == length(S) for i in vertices(Γ))
-      A = adjacency_matrix(Γ)
-      @time eigenvalues, _ = eigs(A, nev=5)
-      @show Γ eigenvalues
-end

src/nemo_utils.jl

@@ -0,0 +1,22 @@
+import Base.reim
+reim(x::Nemo.acb) = reim(convert(ComplexF64, x))
+
+function root_of_unity(CC::AcbField, p, k=1)
+    @assert p > 0
+    res = zero(CC)
+    ccall((:acb_unit_root, Nemo.libarb), Cvoid, (Ref{acb}, Culong, Clong), res, p, prec(CC))
+    return res^k
+end
+
+import Base.adjoint
+function Base.adjoint(m::acb_mat)
+    res = zero(m)
+    ccall((:acb_mat_conjugate_transpose, Nemo.libarb),
+        Cvoid, (Ref{acb_mat}, Ref{acb_mat}), res, m)
+    return res
+end
+
+using Random
+import Base.rand
+
+rand(rng::AbstractRNG, rs::Random.SamplerTrivial{AcbField}) = (CC = rs[]; CC(rand(Float64), rand(Float64)))