Lie algebras: more progress (#4053)

docs/oscar_references.bib

@@ -1766,6 +1766,18 @@ @Misc{MNP24
   url           = {https://gap-packages.github.io/tomlib/}
 }
 
+@Article{MP82,
+  author        = {Moody, R. V. and Patera, J.},
+  title         = {Fast recursion formula for weight multiplicities},
+  journal       = {Bull. Amer. Math. Soc. (N.S.)},
+  fjournal      = {American Mathematical Society. Bulletin. New Series},
+  volume        = {7},
+  number        = {1},
+  pages         = {237--242},
+  year          = {1982},
+  doi           = {10.1090/S0273-0979-1982-15021-2}
+}
+
 @Article{MR20,
   author        = {Markwig, Thomas and Ren, Yue},
   title         = {Computing tropical varieties over fields with valuation},

experimental/LieAlgebras/src/CoxeterGroup.jl

@@ -7,26 +7,32 @@ Return the Coxeter matrix $m$ associated to the Cartan matrix `gcm`. If there is
 then this will be expressed by $m_{ij} = 0$ (instead of the usual convention $m_{ij} = \infty$).
 The keyword argument `check` can be set to `false` to skip verification whether `gcm` is indeed a generalized Cartan matrix.
 """
-function coxeter_from_cartan_matrix(gcm; check::Bool=true)
+function coxeter_from_cartan_matrix(gcm::ZZMatrix; check::Bool=true)
   @req !check || is_cartan_matrix(gcm) "requires a generalized Cartan matrix"
 
-  cm = identity_matrix(gcm)
   rk, _ = size(gcm)
-  for i in 1:rk, j in 1:rk
-    if i == j
-      continue
-    end
+  cm = matrix(
+    ZZ, [coxeter_matrix_entry_from_cartan_matrix(gcm, i, j) for i in 1:rk, j in 1:rk]
+  )
 
-    if gcm[i, j] == 0
-      cm[i, j] = 2
-    elseif gcm[i, j] == -1
-      cm[i, j] = 3
-    elseif gcm[i, j] == -2
-      cm[i, j] = 4
-    elseif gcm[i, j] == -3
-      cm[i, j] = 6
-    end
+  return cm
+end
+
+function coxeter_matrix_entry_from_cartan_matrix(gcm::ZZMatrix, i::Int, j::Int)
+  if i == j
+    return 1
   end
 
-  return cm
+  d = gcm[i, j] * gcm[j, i]
+  if d == 0
+    return 2
+  elseif d == 1
+    return 3
+  elseif d == 2
+    return 4
+  elseif d == 3
+    return 6
+  else
+    return 0
+  end
 end

experimental/LieAlgebras/src/LieAlgebraModule.jl

@@ -1401,7 +1401,7 @@ end
     dim_of_simple_module([T = Int], L::LieAlgebra{C}, hw::Vector{<:IntegerUnion}) -> T
 
 Compute the dimension of the simple module of the Lie algebra `L` with highest weight `hw`
- using Weyl's dimension formula.
+using Weyl's dimension formula.
 The return value is of type `T`.
 
 # Example
@@ -1431,11 +1431,65 @@ function dim_of_simple_module(L::LieAlgebra, hw::Vector{<:IntegerUnion})
 end
 
 @doc raw"""
-    dominant_character(L::LieAlgebra{C}, hw::Vector{Int}) -> Dict{Vector{Int}, Int}
+    dominant_weights([T,] L::LieAlgebra{C}, hw::Vector{<:IntegerUnion}) -> Vector{T}
+
+Computes the dominant weights occurring in the simple module of the Lie algebra `L` with highest weight `hw`,
+sorted ascendingly by the total height of roots needed to reach them from `hw`.
+
+When supplying `T = Vector{Int}`, the weights are returned as vectors of integers.
+
+See [MP82](@cite) for details and the implemented algorithm.
+
+# Example
+```jldoctest
+julia> L = lie_algebra(QQ, :B, 3);
+
+julia> dominant_weights(L, [1, 0, 3])
+7-element Vector{Vector{Int64}}:
+ [1, 0, 3]
+ [1, 1, 1]
+ [2, 0, 1]
+ [0, 0, 3]
+ [0, 1, 1]
+ [1, 0, 1]
+ [0, 0, 1]
+```
+"""
+function dominant_weights(T::Type, L::LieAlgebra, hw::Vector{<:IntegerUnion})
+  if has_root_system(L)
+    R = root_system(L)
+    return dominant_weights(T, R, hw)
+  else # TODO: remove branch once root system detection is implemented
+    @req is_dominant_weight(hw) "Not a dominant weight."
+    return first.(
+      sort!(
+        collect(
+          T(w) => d for (w, d) in
+          zip(
+            GAP.Globals.DominantWeights(
+              GAP.Globals.RootSystem(codomain(Oscar.iso_oscar_gap(L))),
+              GAP.Obj(hw; recursive=true),
+            )...,
+          )
+        ); by=last)
+    )
+  end
+end
+
+function dominant_weights(L::LieAlgebra, hw::Vector{<:IntegerUnion})
+  return dominant_weights(Vector{Int}, L, hw)
+end
+
+@doc raw"""
+    dominant_character(L::LieAlgebra{C}, hw::Vector{<:IntegerUnion}) -> Dict{Vector{Int}, Int}
 
 Computes the dominant weights occurring in the simple module of the Lie algebra `L` with highest weight `hw`,
 together with their multiplicities.
 
+The return type may change in the future.
+
+This function uses an optimized version of the Freudenthal formula, see [MP82](@cite) for details.
+
 # Example
 ```jldoctest
 julia> L = lie_algebra(QQ, :A, 3);
@@ -1448,19 +1502,31 @@ Dict{Vector{Int64}, Int64} with 4 entries:
   [0, 2, 0] => 1
 ```
 """
-function dominant_character(L::LieAlgebra, hw::Vector{Int})
-  @req is_dominant_weight(hw) "Not a dominant weight."
-  return Dict{Vector{Int},Int}(
-    Vector{Int}(w) => d for (w, d) in
-    zip(GAPWrap.DominantCharacter(codomain(Oscar.iso_oscar_gap(L)), GAP.Obj(hw))...)
-  )
+function dominant_character(L::LieAlgebra, hw::Vector{<:IntegerUnion})
+  if has_root_system(L)
+    R = root_system(L)
+    return dominant_character(R, hw)
+  else # TODO: remove branch once root system detection is implemented
+    @req is_dominant_weight(hw) "Not a dominant weight."
+    return Dict{Vector{Int},Int}(
+      Vector{Int}(w) => d for (w, d) in
+      zip(
+        GAPWrap.DominantCharacter(
+          codomain(Oscar.iso_oscar_gap(L)), GAP.Obj(hw; recursive=true)
+        )...,
+      )
+    )
+  end
 end
 
 @doc raw"""
-    character(L::LieAlgebra{C}, hw::Vector{Int}) -> Dict{Vector{Int}, Int}
+    character(L::LieAlgebra{C}, hw::Vector{<:IntegerUnion}) -> Dict{Vector{Int}, Int}
 
 Computes all weights occurring in the simple module of the Lie algebra `L` with highest weight `hw`,
 together with their multiplicities.
+This is achieved by acting with the Weyl group on the [`dominant_character`](@ref dominant_character(::LieAlgebra, ::Vector{<:IntegerUnion})).
+
+The return type may change in the future.
 
 # Example
 ```jldoctest
@@ -1476,29 +1542,37 @@ Dict{Vector{Int64}, Int64} with 10 entries:
   [1, -1, 1]  => 1
   [-1, 0, 1]  => 1
   [1, 0, -1]  => 1
-  [0, -1, 0]  => 1
   [2, 0, 0]   => 1
+  [0, -1, 0]  => 1
 ```
 """
-function character(L::LieAlgebra, hw::Vector{Int})
-  @req is_dominant_weight(hw) "Not a dominant weight."
-  dc = dominant_character(L, hw)
-  c = Dict{Vector{Int},Int}()
-  W = GAPWrap.WeylGroup(GAPWrap.RootSystem(codomain(Oscar.iso_oscar_gap(L))))
-  for (w, d) in dc
-    it = GAPWrap.WeylOrbitIterator(W, GAP.Obj(w))
-    while !GAPWrap.IsDoneIterator(it)
-      push!(c, Vector{Int}(GAPWrap.NextIterator(it)) => d)
+function character(L::LieAlgebra, hw::Vector{<:IntegerUnion})
+  if has_root_system(L)
+    R = root_system(L)
+    return character(R, hw)
+  else # TODO: remove branch once root system detection is implemented
+    @req is_dominant_weight(hw) "Not a dominant weight."
+    dc = dominant_character(L, hw)
+    c = Dict{Vector{Int},Int}()
+    W = GAPWrap.WeylGroup(GAPWrap.RootSystem(codomain(Oscar.iso_oscar_gap(L))))
+    for (w, d) in dc
+      it = GAPWrap.WeylOrbitIterator(W, GAP.Obj(w))
+      while !GAPWrap.IsDoneIterator(it)
+        push!(c, Vector{Int}(GAPWrap.NextIterator(it)) => d)
+      end
     end
+    return c
   end
-  return c
 end
 
 @doc raw"""
-    tensor_product_decomposition(L::LieAlgebra, hw1::Vector{Int}, hw2::Vector{Int}) -> MSet{Vector{Int}}
+    tensor_product_decomposition(L::LieAlgebra, hw1::Vector{<:IntegerUnion}, hw2::Vector{<:IntegerUnion}) -> MSet{Vector{Int}}
 
 Computes the decomposition of the tensor product of the simple modules of the Lie algebra `L` with highest weights `hw1` and `hw2`
 into simple modules with their multiplicities.
+This function uses Klimyk's formula (see [Hum72; Exercise 24.9](@cite)).
+
+The return type may change in the future.
 
 # Example
 ```jldoctest
@@ -1518,13 +1592,22 @@ MSet{Vector{Int64}} with 6 elements:
   [0, 3]
 ```
 """
-function tensor_product_decomposition(L::LieAlgebra, hw1::Vector{Int}, hw2::Vector{Int})
-  @req is_dominant_weight(hw1) && is_dominant_weight(hw2) "Both weights must be dominant."
-  return multiset(
-    Tuple{Vector{Vector{Int}},Vector{Int}}(
-      GAPWrap.DecomposeTensorProduct(
-        codomain(Oscar.iso_oscar_gap(L)), GAP.Obj(hw1), GAP.Obj(hw2)
-      ),
-    )...,
-  )
+function tensor_product_decomposition(
+  L::LieAlgebra, hw1::Vector{<:IntegerUnion}, hw2::Vector{<:IntegerUnion}
+)
+  if has_root_system(L)
+    R = root_system(L)
+    return tensor_product_decomposition(R, hw1, hw2)
+  else # TODO: remove branch once root system detection is implemented
+    @req is_dominant_weight(hw1) && is_dominant_weight(hw2) "Both weights must be dominant."
+    return multiset(
+      Tuple{Vector{Vector{Int}},Vector{Int}}(
+        GAPWrap.DecomposeTensorProduct(
+          codomain(Oscar.iso_oscar_gap(L)),
+          GAP.Obj(hw1; recursive=true),
+          GAP.Obj(hw2; recursive=true),
+        ),
+      )...,
+    )
+  end
 end

experimental/LieAlgebras/src/LieAlgebras.jl

@@ -21,6 +21,7 @@ import ..Oscar:
   _iso_oscar_gap,
   _vec,
   action,
+  add!,
   basis_matrix,
   basis,
   canonical_injection,
@@ -43,6 +44,7 @@ import ..Oscar:
   elem_type,
   expressify,
   exterior_power,
+  fp_group,
   gen,
   gens,
   height,
@@ -62,9 +64,11 @@ import ..Oscar:
   is_simple,
   is_solvable,
   is_welldefined,
+  isomorphism,
   kernel,
   lower_central_series,
   matrix,
+  neg!,
   normalizer,
   number_of_generators,
   ngens,
@@ -74,6 +78,7 @@ import ..Oscar:
   root,
   roots,
   sub,
+  sub!,
   symbols,
   symmetric_power,
   tensor_product,