change cartan_symmetrizer and orientation of G2

experimental/LieAlgebras/src/CartanMatrix.jl

@@ -72,7 +72,7 @@ function cartan_matrix(fam::Symbol, rk::Int)
   elseif fam == :F
     mat = matrix(ZZ, [2 -1 0 0; -1 2 -1 0; 0 -2 2 -1; 0 0 -1 2])
   elseif fam == :G
-    mat = matrix(ZZ, [2 -3; -1 2])
+    mat = matrix(ZZ, [2 -1; -3 2])
   else
     error("Unreachable")
   end
@@ -169,6 +169,9 @@ where $a_{ij}$ are the entries of the Cartan matrix `gcm`.
 
 If `check=true` the function will verify that `gcm` is indeed a generalized Cartan matrix.
 
+!!! warning
+    Currently only Cartan matrices of finite type are supported.
+
 # Examples
 ```jldoctest
 julia> cartan_symmetrizer(cartan_matrix(:B, 2))
@@ -178,69 +181,37 @@ julia> cartan_symmetrizer(cartan_matrix(:B, 2))
 ```
 """
 function cartan_symmetrizer(gcm::ZZMatrix; check::Bool=true)
-  check && @req is_cartan_matrix(gcm) "not a Cartan matrix"
-  rk = nrows(gcm)
-  diag = ones(ZZRingElem, rk)
-
-  # used for traversal
-  undone = trues(rk)
-  plan = zeros(Int, rk) # roots planned sorted asc grouped by component
-  head = 0
-  tail = 0
-
-  # we collect roots of the same length
-  # once we know if they are short or long we scale approriately
-  while any(undone)
-    if head == tail
-      head += 1
-      plan[head] = findfirst(undone)::Int
-      undone[plan[head]] = false
-    end
-
-    prev = head
-    i = plan[head]
-    for j in 1:rk
-      if i == j
-        continue
+  ct, ord = cartan_type_with_ordering(gcm; check=check)
+
+  i = 1
+  d = ones(ZZRingElem, length(ord))
+  for (fam, rk) in ct
+    if fam == :A
+      i += rk
+    elseif fam == :B
+      for j in i:(i + rk - 2)
+        d[j] = 2
       end
-
-      if !undone[j] || is_zero_entry(gcm, i, j)
-        continue
-      end
-
-      head += 1
-      plan[head] = j
-      undone[j] = false
-
-      if diag[i] * gcm[i, j] == diag[j] * gcm[j, i]
-        continue
-      elseif gcm[i, j] == gcm[j, i]
-        diag[i] = lcm(diag[i], diag[j])
-        diag[j] = diag[i]
-        continue
-      end
-
-      if gcm[j, i] < -1
-        tail += 1
-        v = -gcm[j, i]
-        while tail < head
-          diag[plan[tail]] *= v
-          tail += 1
-        end
-      end
-      if gcm[i, j] < -1
-        diag[j] *= -gcm[i, j]
-        tail = head - 1
-      end
-    end
-
-    # we found new roots, meaning we are done with this component of the root system
-    if prev == head
-      tail = head
+      i += rk
+    elseif fam == :C
+      i += rk
+      d[i-1] = 2
+    elseif fam == :D
+      i += rk
+    elseif fam == :E
+      i += rk
+    elseif fam == :F
+      d[i] = d[i+1] = 2
+      i += rk
+    elseif fam == :G
+      d[i] = 3
+      i += rk
+    else
+      error("unreachable")
     end
   end
-
-  return diag
+  
+  return invpermute!(d, ord)
 end
 
 @doc raw"""
@@ -366,10 +337,10 @@ function cartan_type_with_ordering(gcm::ZZMatrix; check::Bool=true)
         push!(ord, v0, v1)
       elseif gcm[v0, v1] == -3
         push!(type, (:G, 2))
-        push!(ord, v0, v1)
+        push!(ord, v1, v0)
       elseif gcm[v1, v0] == -3
         push!(type, (:G, 2))
-        push!(ord, v1, v0)
+        push!(ord, v0, v1)
       else
         error("unreachable")
       end

experimental/LieAlgebras/src/DynkinDiagram.jl

@@ -100,7 +100,7 @@ function show_dynkin_diagram(fam::Symbol, rk::Int, labels::AbstractVector{Int})
     D = "$(labels[1]) - $(labels[2]) >=> $(labels[3]) - $(labels[4])"
   elseif fam == :G
     @assert rk == 2
-    D = "$(labels[1]) <<< $(labels[2])"
+    D = "$(labels[1]) >>> $(labels[2])"
   end
   isempty(D) && error("Unreachable")
   print(D)

experimental/LieAlgebras/test/CartanMatrix-test.jl

@@ -104,31 +104,38 @@
     @test cartan_symmetrizer(cartan_matrix(:E, 7)) == [1, 1, 1, 1, 1, 1, 1]
     @test cartan_symmetrizer(cartan_matrix(:E, 8)) == [1, 1, 1, 1, 1, 1, 1, 1]
     @test cartan_symmetrizer(cartan_matrix(:F, 4)) == [2, 2, 1, 1]
-    @test cartan_symmetrizer(cartan_matrix(:G, 2)) == [1, 3]
+    @test cartan_symmetrizer(cartan_matrix(:G, 2)) == [3, 1]
 
     # affine type
-    @test cartan_symmetrizer(ZZ[2 -2; -2 2]) == [1, 1] # A1~1
-    @test cartan_symmetrizer(ZZ[2 -2 0; -1 2 -1; 0 -2 2]) == [1, 2, 1] # D3~2
-    @test cartan_symmetrizer(ZZ[2 -4; -1 2]) == [1, 4] # A1~2
+    #@test cartan_symmetrizer(ZZ[2 -2; -2 2]) == [1, 1] # A1~1
+    #@test cartan_symmetrizer(ZZ[2 -2 0; -1 2 -1; 0 -2 2]) == [1, 2, 1] # D3~2
+    #@test cartan_symmetrizer(ZZ[2 -4; -1 2]) == [1, 4] # A1~2
 
     # hyperbolic type
-    @test cartan_symmetrizer(ZZ[2 -2; -3 2]) == [3, 2]
+    #@test cartan_symmetrizer(ZZ[2 -2; -3 2]) == [3, 2]
   end
 
   @testset "cartan_bilinear_form" begin
     function test_cartan_bilinear_form(cm::ZZMatrix)
       rk = nrows(cm)
-      for _ in 1:50
-        i, j = rand(1:rk, 2)
-        if i != j
-          swap_rows!(cm, i, j)
-          swap_cols!(cm, i, j)
+      for _ in 1:10
+        for _ in 1:10
+          i, j = rand(1:rk, 2)
+          if i != j
+            swap_rows!(cm, i, j)
+            swap_cols!(cm, i, j)
+          end
+        end
+
+        bil = cartan_bilinear_form(cm)
+        if !is_symmetric(bil)
+          println(cm)
+          println(bil)
+          break
         end
+        @test is_symmetric(bil) == true
+        @test all(cm[i, j] == div(2 * bil[i, j], bil[i, i]) for i in 1:rk, j in 1:rk)
       end
-
-      bil = cartan_bilinear_form(cm)
-      @test is_symmetric(bil) == true
-      @test all(cm[i, j] == div(2 * bil[i, j], bil[i, i]) for i in 1:rk, j in 1:rk)
     end
 
     # irreducible matrices
@@ -148,9 +155,9 @@
     test_cartan_bilinear_form(cartan_matrix((:F, 4), (:B, 2), (:E, 7), (:G, 2)))
 
     # affine type
-    @test cartan_bilinear_form(ZZ[2 -2; -2 2]) == ZZ[2 -2; -2 2]
-    @test cartan_bilinear_form(ZZ[2 -2 0; -1 2 -1; 0 -2 2]) == ZZ[2 -2 0; -2 4 -2; 0 -2 2]
-    @test cartan_bilinear_form(ZZ[2 -4; -1 2]) == ZZ[2 -4; -4 8]
+    #@test cartan_bilinear_form(ZZ[2 -2; -2 2]) == ZZ[2 -2; -2 2]
+    #@test cartan_bilinear_form(ZZ[2 -2 0; -1 2 -1; 0 -2 2]) == ZZ[2 -2 0; -2 4 -2; 0 -2 2]
+    #@test cartan_bilinear_form(ZZ[2 -4; -1 2]) == ZZ[2 -4; -4 8]
   end
 
   @testset "cartan_type_with_ordering" begin