From 3855110a6d28c0b9bf5ada8f395ec7c06d21fcb3 Mon Sep 17 00:00:00 2001
From: Torben Wiedemann <7226617+TWiedemann@users.noreply.github.com>
Date: Wed, 29 Jan 2025 15:37:27 +0100
Subject: [PATCH 01/14] Add map_word and parabolic_subgroup(_with_projection)
 and corresponding exports

---
 experimental/LieAlgebras/src/WeylGroup.jl     | 94 ++++++++++++++++++
 experimental/LieAlgebras/src/exports.jl       |  4 +
 .../LieAlgebras/test/WeylGroup-test.jl        | 95 ++++++++++++++++++-
 3 files changed, 192 insertions(+), 1 deletion(-)

diff --git a/experimental/LieAlgebras/src/WeylGroup.jl b/experimental/LieAlgebras/src/WeylGroup.jl
index f433fb8de75e..7992ad2e1f60 100644
--- a/experimental/LieAlgebras/src/WeylGroup.jl
+++ b/experimental/LieAlgebras/src/WeylGroup.jl
@@ -257,3 +257,97 @@ function isomorphism(::Type{PermGroup}, W::WeylGroup; set_properties::Bool=true)
 
   return MapFromFunc(W, G, iso, isoinv)
 end
+
+@doc raw"""
+    map_word(w::WeylGroupElem, genimgs::Vector; init = nothing)
+
+If `init` is `nothing` and `word(w) = [UInt(`$i_1$`), ..., UInt(`$i_n$`)]`, then return the product $R_1 R_2 \cdots R_n$
+with $R_j =$ `genimgs[`$i_j$`]`.
+Otherwise return the product $xR_1 R_2 \cdots R_n$ where $x =$ `init`.
+
+The length of `genimgs` must be equal to the rank of the parent of `w`.
+In particular, it must not be empty.
+If `w` is the trivial element, then `init` is returned if it is different
+from `nothing`, and otherwise `one(genimgs[1])` is returned.
+
+See also: [`map_word(::Union{FPGroupElem, SubFPGroupElem}, ::Vector)`](@ref),
+[`map_word(::Union{PcGroupElem, SubPcGroupElem}, ::Vector)`](@ref).
+Note that `map_word(::WeylGroupElem)` does not provide a `genimgs_inv` parameter
+because the generators of a Weyl group are always self-inverse.
+
+# Examples
+```jldoctest
+julia> W = weyl_group(:B, 3); imgs = [2, 3, 5];
+
+julia> map_word(one(W), imgs)
+1
+
+julia> map_word(W([1]), imgs)
+2
+
+julia> map_word(W([1]), imgs, init=7)
+14
+
+julia> map_word(W([1,2,1]), imgs)
+12
+
+julia> map_word(W([2,1,2]), imgs) # W([2,1,2]) == W([1,2,1])
+12
+
+julia> map_word(W([3, 2, 1, 3, 2, 3]), imgs)
+2250
+```
+"""
+function map_word(w::WeylGroupElem, genimgs::Vector; init=nothing)
+  @req length(genimgs) == number_of_generators(parent(w)) "Length of vector of images does not equal rank of Weyl group"
+  return Oscar.map_word([Int(i) for i in word(w)], genimgs, init=init)
+end
+
+@doc raw"""
+    parabolic_subgroup(W::WeylGroup, vec::Vector{<:Integer}, w::WeylGroupElem) -> WeylGroup, Map{WeylGroup, WeylGroup}
+
+Returns two objects: The subgroup `U` of `W` generated by `[inv(w)*u*w for u in gens(W)[vec]]`, whose generators are set to be this vector and which is itself a `WeylGroup`, and the embedding of `U` into `W`. The elements of `vec` must be pairwise distinct integers in `1:number_of_generators(W)` and `vec` must be non-empty.
+"""
+function parabolic_subgroup(W::WeylGroup, vec::Vector{<:Integer}, w::WeylGroupElem=one(W))
+  @req length(unique(vec)) == length(vec) "Elements of vector are not pairwise distinct"
+  cm = cartan_matrix(root_system(W))[vec, vec]
+  para = weyl_group(cm)
+  genimgs = map(x -> inv(w) * x * w, gens(W)[vec])
+  emb = function(u::WeylGroupElem)
+    return map_word(u, genimgs)
+  end
+  return para, MapFromFunc(para, W, emb)
+end
+
+@doc raw"""
+    parabolic_subgroup_with_projection(W::WeylGroup, vec::Vector{<:Integer}) -> WeylGroup, Map{WeylGroup, WeylGroup}, Map{WeylGroup, WeylGroup}
+
+Returns a triple `(U, emb, proj)` that describes a factor of `W`, that is, a product of irreducible factors.
+Here the parabolic subgroup `U` and its embedding into `W` are as returned by [`parabolic_subgroup`](@ref)`(W, vec)`
+and `proj` is the projection map from `W` onto `U`.
+If `U` is not a factor of `W`, an error occurs.
+"""
+function parabolic_subgroup_with_projection(W::WeylGroup, vec::Vector{<:Integer}; check_factor::Bool=true)
+  if check_factor
+    # Check that every generator in gens(W)[vec] commutes with every other generator.
+    # In other words, vec describes a union of irreducible components of the Coxeter diagram.
+    cm = cartan_matrix(root_system(W))
+    for i in vec
+      for j in setdiff(1:number_of_generators(W), vec)
+        @req coxeter_matrix_entry_from_cartan_matrix(cm, i, j) == 2 "Input vector does not describe a factor of the Weyl group, so there is no projection homomorphism"
+      end;
+    end
+  end
+
+  factor, emb = parabolic_subgroup(W, vec)
+  # Generators of W are mapped to the corresponding generators of factor,
+  # or to 1 if there is no corresponding generator
+  proj_gen_imgs = [one(factor) for _ in 1:number_of_generators(W)]
+  for i in 1:length(vec)
+    proj_gen_imgs[vec[i]] = gen(factor, i)
+  end
+  proj = function(w::WeylGroupElem)
+    return map_word(w, proj_gen_imgs)
+  end
+  return factor, emb, MapFromFunc(W, factor, proj)
+end
\ No newline at end of file
diff --git a/experimental/LieAlgebras/src/exports.jl b/experimental/LieAlgebras/src/exports.jl
index 9cac5995d5f7..4a25582e4493 100644
--- a/experimental/LieAlgebras/src/exports.jl
+++ b/experimental/LieAlgebras/src/exports.jl
@@ -51,3 +51,7 @@ export tensor_power
 export tensor_product_decomposition
 export trivial_module
 export universal_enveloping_algebra
+
+export parabolic_subgroup
+export parabolic_subgroup_with_projection
+export map_word
\ No newline at end of file
diff --git a/experimental/LieAlgebras/test/WeylGroup-test.jl b/experimental/LieAlgebras/test/WeylGroup-test.jl
index 39b25bb94953..0bfbf5f16639 100644
--- a/experimental/LieAlgebras/test/WeylGroup-test.jl
+++ b/experimental/LieAlgebras/test/WeylGroup-test.jl
@@ -135,4 +135,97 @@
       end
     end
   end
-end
+
+  @testset "WeylGroup parabolic subgroup test for $(Wname)" for (Wname, W, vec, check_proj) in [
+    ("A1", weyl_group(:A, 1), [1], true),
+    ("A5", weyl_group(:A, 5), [1, 5, 3], false),
+    ("B2", weyl_group(:B, 2), [2, 1], false),
+    ("F4", weyl_group(:F, 4), [2, 3], false),
+    ("A5+E8+D4", weyl_group((:A, 5), (:E, 8), (:D, 4)), [6:13; 1:5], true),
+    (
+      "A3 with non-canonical ordering of simple roots",
+      weyl_group(root_system([2 -1 -1; -1 2 0; -1 0 2])),
+      [2, 3], false
+    ),
+    (
+      "B4 with non-canonical ordering of simple roots",
+      weyl_group(root_system([2 -1 -1 0; -1 2 0 -1; -2 0 2 0; 0 -1 0 2])),
+      [2, 4], false
+    ),
+    (
+      "complicated case 1",
+      begin
+        cm = cartan_matrix((:A, 3), (:C, 3), (:E, 6), (:G, 2))
+        for _ in 1:50
+          i, j = rand(1:nrows(cm), 2)
+          if i != j
+            swap_rows!(cm, i, j)
+            swap_cols!(cm, i, j)
+          end
+        end
+        weyl_group(cm)
+      end,
+      unique(rand(1:14, 5)), false
+    ),
+    (
+      "complicated case 2",
+      begin
+        cm = cartan_matrix((:F, 4), (:B, 2), (:E, 7), (:G, 2))
+        for _ in 1:50
+          i, j = rand(1:nrows(cm), 2)
+          if i != j
+            swap_rows!(cm, i, j)
+            swap_cols!(cm, i, j)
+          end
+        end
+        weyl_group(root_system(cm))
+      end,
+      unique(rand(1:15, 6)), false
+    ),
+  ]
+    for k in 1:4
+      if k == 1
+        # In the first run, test the standard parabolics and projections
+        if check_proj
+          para, emb, proj = parabolic_subgroup_with_projection(W, vec)
+        else
+          para, emb = parabolic_subgroup(W, vec)
+        end
+        genimgs = [gen(W, i) for i in vec] # Desired images of gens(para) in W
+      else
+        # On subsequent runs, conjugate by random elements
+        conj = rand(W)
+        para, emb = parabolic_subgroup(W, vec, conj)
+        genimgs = [inv(conj) * gen(W, vec[i]) * conj for i in 1:length(vec)]
+      end
+      # Test that emb maps gens(para) to genimgs
+      for i in 1:length(vec)
+        @test emb(gen(para, i)) == genimgs[i]
+      end
+      # Test that emb is a homomorphism
+      for _ in 1:5
+        p1 = rand(para)
+        p2 = rand(para)
+        @test emb(p1) * emb(p2) == emb(p1 * p2)
+      end
+      # Test proj
+      if k == 1 && check_proj
+        # Test that proj maps gens(para) to gens(W)[vec]
+        for i in 1:length(vec)
+          @test proj(gen(W, vec[i])) == gen(para, i)
+        end
+        # Test that proj is a homomorphism
+        for _ in 1:5
+          w1 = rand(W)
+          w2 = rand(W)
+          @test proj(w1) * proj(w2) == proj(w1 * w2)
+        end
+        # Test that proj is the left-inverse of emb
+        for _ in 1:5
+          p = rand(para)
+          @test proj(emb(p)) == p
+        end
+      end
+    end
+  end
+end
\ No newline at end of file

From dac5ab099cf793414e277a22d79e7626932acdf1 Mon Sep 17 00:00:00 2001
From: Torben Wiedemann <7226617+TWiedemann@users.noreply.github.com>
Date: Thu, 30 Jan 2025 13:40:10 +0100
Subject: [PATCH 02/14] Fix small issues

---
 experimental/LieAlgebras/src/LieAlgebras.jl   |   1 +
 experimental/LieAlgebras/src/WeylGroup.jl     | 108 ++++++++++++++----
 experimental/LieAlgebras/src/exports.jl       |   6 +-
 .../LieAlgebras/test/WeylGroup-test.jl        |  16 +--
 4 files changed, 97 insertions(+), 34 deletions(-)

diff --git a/experimental/LieAlgebras/src/LieAlgebras.jl b/experimental/LieAlgebras/src/LieAlgebras.jl
index 8a80ebe3075b..0b4117b99c48 100644
--- a/experimental/LieAlgebras/src/LieAlgebras.jl
+++ b/experimental/LieAlgebras/src/LieAlgebras.jl
@@ -69,6 +69,7 @@ import ..Oscar:
   isomorphism,
   kernel,
   lower_central_series,
+  map_word,
   matrix,
   normalizer,
   number_of_generators,
diff --git a/experimental/LieAlgebras/src/WeylGroup.jl b/experimental/LieAlgebras/src/WeylGroup.jl
index 7992ad2e1f60..9263d22b87b0 100644
--- a/experimental/LieAlgebras/src/WeylGroup.jl
+++ b/experimental/LieAlgebras/src/WeylGroup.jl
@@ -259,20 +259,20 @@ function isomorphism(::Type{PermGroup}, W::WeylGroup; set_properties::Bool=true)
 end
 
 @doc raw"""
-    map_word(w::WeylGroupElem, genimgs::Vector; init = nothing)
+    map_word(w::WeylGroupElem, genimgs::Vector; genimgs_inv::Vector = genimgs, init = nothing)
 
-If `init` is `nothing` and `word(w) = [UInt(`$i_1$`), ..., UInt(`$i_n$`)]`, then return the product $R_1 R_2 \cdots R_n$
-with $R_j =$ `genimgs[`$i_j$`]`.
+If `init` is `nothing` and `word(w) = [`$i_1$`, ..., `$i_n$`]`,
+then return the product $R_1 R_2 \cdots R_n$ with $R_j =$ `genimgs[`$i_j$`]`.
 Otherwise return the product $xR_1 R_2 \cdots R_n$ where $x =$ `init`.
 
 The length of `genimgs` must be equal to the rank of the parent of `w`.
-In particular, it must not be empty.
 If `w` is the trivial element, then `init` is returned if it is different
-from `nothing`, and otherwise `one(genimgs[1])` is returned.
+from `nothing`, and otherwise `one(genimgs[1])` is returned if `genimgs` is non-empty.
+If `w` is trivial, `init` is nothing and `genimgs` is empty, an error occurs.
 
 See also: [`map_word(::Union{FPGroupElem, SubFPGroupElem}, ::Vector)`](@ref),
 [`map_word(::Union{PcGroupElem, SubPcGroupElem}, ::Vector)`](@ref).
-Note that `map_word(::WeylGroupElem)` does not provide a `genimgs_inv` parameter
+Note that `map_word(::WeylGroupElem)` accepts but ignores the `genimgs_inv` keyword argument
 because the generators of a Weyl group are always self-inverse.
 
 # Examples
@@ -285,7 +285,7 @@ julia> map_word(one(W), imgs)
 julia> map_word(W([1]), imgs)
 2
 
-julia> map_word(W([1]), imgs, init=7)
+julia> map_word(W([1]), imgs; init=7)
 14
 
 julia> map_word(W([1,2,1]), imgs)
@@ -298,22 +298,57 @@ julia> map_word(W([3, 2, 1, 3, 2, 3]), imgs)
 2250
 ```
 """
-function map_word(w::WeylGroupElem, genimgs::Vector; init=nothing)
-  @req length(genimgs) == number_of_generators(parent(w)) "Length of vector of images does not equal rank of Weyl group"
-  return Oscar.map_word([Int(i) for i in word(w)], genimgs, init=init)
+function map_word(
+  w::WeylGroupElem, genimgs::Vector; genimgs_inv::Vector=genimgs, init=nothing
+)
+  @req length(genimgs) == number_of_generators(parent(w)) begin
+    "Length of vector of images does not equal rank of Weyl group"
+  end
+  return map_word(Int.(word(w)), genimgs; init=init)
 end
 
 @doc raw"""
-    parabolic_subgroup(W::WeylGroup, vec::Vector{<:Integer}, w::WeylGroupElem) -> WeylGroup, Map{WeylGroup, WeylGroup}
+    parabolic_subgroup(W::WeylGroup, vec::Vector{<:Integer}, w::WeylGroupElem=one(W)) -> WeylGroup, Map{WeylGroup, WeylGroup}
+
+Return a Weyl group `P` and an embedding $f:P\to W$ such that $f(P)$ is
+the subgroup `U` of `W` generated by `[inv(w)*u*w for u in gens(W)[vec]]`.
+Further, `f` maps `gen(P, i)` to `inv(w)*gen(W, vec[i])*w`.
+The elements of `vec` must be pairwise distinct integers in
+`1:number_of_generators(W)` and `vec` must be non-empty.
+
+# Examples
+```jldoctest
+julia> W = weyl_group(:B, 3)
+Weyl group
+  of root system of rank 3
+    of type B3
+
+julia> P1, f1 = parabolic_subgroup(W, [1, 2])
+(Weyl group of root system of type A2, Map: P1 -> W)
+
+julia> f1(P1[1] * P1[2]) == W[1] * W[2]
+true
+
+julia> P2, f2 = parabolic_subgroup(W, [1, 2], W[1])
+(Weyl group of root system of type A2, Map: P2 -> W)
+
+julia> f2(P2[1]) == W[1] && f2(P2[2]) == W[1] * W[2] * W[1]
+true
+
+julia> P3, f3 = parabolic_subgroup(W, [1,3,2])
+(Weyl group of root system of type B3 (non-canonical ordering), Map: P3 -> W)
 
-Returns two objects: The subgroup `U` of `W` generated by `[inv(w)*u*w for u in gens(W)[vec]]`, whose generators are set to be this vector and which is itself a `WeylGroup`, and the embedding of `U` into `W`. The elements of `vec` must be pairwise distinct integers in `1:number_of_generators(W)` and `vec` must be non-empty.
+julia> f3(P3[2]) == W[3]
+true
+```
 """
 function parabolic_subgroup(W::WeylGroup, vec::Vector{<:Integer}, w::WeylGroupElem=one(W))
-  @req length(unique(vec)) == length(vec) "Elements of vector are not pairwise distinct"
+  @req allunique(vec) "Elements of vector are not pairwise distinct"
+  @req all(i -> 1 <= i <= number_of_generators(W), vec) "Invalid indices"
   cm = cartan_matrix(root_system(W))[vec, vec]
   para = weyl_group(cm)
   genimgs = map(x -> inv(w) * x * w, gens(W)[vec])
-  emb = function(u::WeylGroupElem)
+  emb = function (u::WeylGroupElem)
     return map_word(u, genimgs)
   end
   return para, MapFromFunc(para, W, emb)
@@ -322,20 +357,47 @@ end
 @doc raw"""
     parabolic_subgroup_with_projection(W::WeylGroup, vec::Vector{<:Integer}) -> WeylGroup, Map{WeylGroup, WeylGroup}, Map{WeylGroup, WeylGroup}
 
-Returns a triple `(U, emb, proj)` that describes a factor of `W`, that is, a product of irreducible factors.
-Here the parabolic subgroup `U` and its embedding into `W` are as returned by [`parabolic_subgroup`](@ref)`(W, vec)`
-and `proj` is the projection map from `W` onto `U`.
-If `U` is not a factor of `W`, an error occurs.
+Return a triple `(P, emb, proj)` that describes a factor of `W`, that is,
+a product of irreducible factors.
+Here `P, emb = `[`parabolic_subgroup`](@ref)`(W, vec)`
+and `proj` is the projection map from `W` onto `P`,
+which is a left-inverse of `emb`.
+If `P` is not a factor of `W`, an error occurs.
+
+# Examples
+```jldoctest
+julia> W = weyl_group([(:A, 3), (:B, 3)])
+Weyl group
+  of root system of rank 6
+    of type A3 x B3
+
+julia> P1, f1, p1 = parabolic_subgroup_with_projection(W, [1,2,3])
+(Weyl group of root system of type A3, Map: P1 -> W, Map: W -> P1)
+
+julia> p1(W[1]*W[4]*W[2]*W[6]) == P1[1] * P1[2]
+true
+
+julia> P2, f2, p2 = parabolic_subgroup_with_projection(W, [4,6,5])
+(Weyl group of root system of type B3 (non-canonical ordering), Map: P2 -> W, Map: W -> P2)
+
+julia> p2(W[5]) == P2[3]
+true
+```
 """
-function parabolic_subgroup_with_projection(W::WeylGroup, vec::Vector{<:Integer}; check_factor::Bool=true)
+function parabolic_subgroup_with_projection(
+  W::WeylGroup, vec::Vector{<:Integer}; check_factor::Bool=true
+)
   if check_factor
     # Check that every generator in gens(W)[vec] commutes with every other generator.
     # In other words, vec describes a union of irreducible components of the Coxeter diagram.
     cm = cartan_matrix(root_system(W))
     for i in vec
       for j in setdiff(1:number_of_generators(W), vec)
-        @req coxeter_matrix_entry_from_cartan_matrix(cm, i, j) == 2 "Input vector does not describe a factor of the Weyl group, so there is no projection homomorphism"
-      end;
+        @req is_zero_entry(cm, i, j) begin
+          "Input vector does not describe a factor of the Weyl group, \
+          so there is no projection homomorphism"
+        end
+      end
     end
   end
 
@@ -346,8 +408,8 @@ function parabolic_subgroup_with_projection(W::WeylGroup, vec::Vector{<:Integer}
   for i in 1:length(vec)
     proj_gen_imgs[vec[i]] = gen(factor, i)
   end
-  proj = function(w::WeylGroupElem)
+  proj = function (w::WeylGroupElem)
     return map_word(w, proj_gen_imgs)
   end
   return factor, emb, MapFromFunc(W, factor, proj)
-end
\ No newline at end of file
+end
diff --git a/experimental/LieAlgebras/src/exports.jl b/experimental/LieAlgebras/src/exports.jl
index 4a25582e4493..f162db6add27 100644
--- a/experimental/LieAlgebras/src/exports.jl
+++ b/experimental/LieAlgebras/src/exports.jl
@@ -40,6 +40,8 @@ export killing_matrix
 export lie_algebra
 export lower_central_series
 export matrix_repr_basis
+export parabolic_subgroup
+export parabolic_subgroup_with_projection
 export show_dynkin_diagram
 export simple_module
 export special_linear_lie_algebra
@@ -51,7 +53,3 @@ export tensor_power
 export tensor_product_decomposition
 export trivial_module
 export universal_enveloping_algebra
-
-export parabolic_subgroup
-export parabolic_subgroup_with_projection
-export map_word
\ No newline at end of file
diff --git a/experimental/LieAlgebras/test/WeylGroup-test.jl b/experimental/LieAlgebras/test/WeylGroup-test.jl
index 0bfbf5f16639..bee847954237 100644
--- a/experimental/LieAlgebras/test/WeylGroup-test.jl
+++ b/experimental/LieAlgebras/test/WeylGroup-test.jl
@@ -136,7 +136,9 @@
     end
   end
 
-  @testset "WeylGroup parabolic subgroup test for $(Wname)" for (Wname, W, vec, check_proj) in [
+  @testset "WeylGroup parabolic subgroup test for $(Wname)" for (
+    Wname, W, vec, check_proj
+  ) in [
     ("A1", weyl_group(:A, 1), [1], true),
     ("A5", weyl_group(:A, 5), [1, 5, 3], false),
     ("B2", weyl_group(:B, 2), [2, 1], false),
@@ -145,12 +147,12 @@
     (
       "A3 with non-canonical ordering of simple roots",
       weyl_group(root_system([2 -1 -1; -1 2 0; -1 0 2])),
-      [2, 3], false
+      [2, 3], false,
     ),
     (
       "B4 with non-canonical ordering of simple roots",
       weyl_group(root_system([2 -1 -1 0; -1 2 0 -1; -2 0 2 0; 0 -1 0 2])),
-      [2, 4], false
+      [2, 4], false,
     ),
     (
       "complicated case 1",
@@ -165,7 +167,7 @@
         end
         weyl_group(cm)
       end,
-      unique(rand(1:14, 5)), false
+      unique(rand(1:14, 5)), false,
     ),
     (
       "complicated case 2",
@@ -180,12 +182,12 @@
         end
         weyl_group(root_system(cm))
       end,
-      unique(rand(1:15, 6)), false
+      unique(rand(1:15, 6)), false,
     ),
   ]
     for k in 1:4
       if k == 1
-        # In the first run, test the standard parabolics and projections
+        # On the first run, test the standard parabolics and projections
         if check_proj
           para, emb, proj = parabolic_subgroup_with_projection(W, vec)
         else
@@ -228,4 +230,4 @@
       end
     end
   end
-end
\ No newline at end of file
+end

From bc81b5fc743b6aac95a2a506043553c5a7896516 Mon Sep 17 00:00:00 2001
From: Max Horn <max@quendi.de>
Date: Thu, 30 Jan 2025 15:05:01 +0100
Subject: [PATCH 03/14] Apply suggestions from code review
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

Co-authored-by: Lars Göttgens <lars.goettgens@gmail.com>
---
 experimental/LieAlgebras/src/WeylGroup.jl | 3 +--
 1 file changed, 1 insertion(+), 2 deletions(-)

diff --git a/experimental/LieAlgebras/src/WeylGroup.jl b/experimental/LieAlgebras/src/WeylGroup.jl
index 9263d22b87b0..77df4f1bd733 100644
--- a/experimental/LieAlgebras/src/WeylGroup.jl
+++ b/experimental/LieAlgebras/src/WeylGroup.jl
@@ -394,8 +394,7 @@ function parabolic_subgroup_with_projection(
     for i in vec
       for j in setdiff(1:number_of_generators(W), vec)
         @req is_zero_entry(cm, i, j) begin
-          "Input vector does not describe a factor of the Weyl group, \
-          so there is no projection homomorphism"
+          "Input vector does not describe a factor of the Weyl group, so there is no projection homomorphism"
         end
       end
     end

From 90a055e9e197c20bae0fec4a1c45a7734b5e6797 Mon Sep 17 00:00:00 2001
From: Torben Wiedemann <7226617+TWiedemann@users.noreply.github.com>
Date: Thu, 30 Jan 2025 20:23:20 +0100
Subject: [PATCH 04/14] Use map_word in isomorphism(PermGroup, ...)

---
 experimental/LieAlgebras/src/WeylGroup.jl | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/experimental/LieAlgebras/src/WeylGroup.jl b/experimental/LieAlgebras/src/WeylGroup.jl
index 77df4f1bd733..47393cd5450f 100644
--- a/experimental/LieAlgebras/src/WeylGroup.jl
+++ b/experimental/LieAlgebras/src/WeylGroup.jl
@@ -247,7 +247,7 @@ function isomorphism(::Type{PermGroup}, W::WeylGroup; set_properties::Bool=true)
   end
 
   iso = function (w::WeylGroupElem)
-    reduce(*, (gen(G, Int(i)) for i in word(w)); init=one(G))
+    map_word(w, gens(G); init=one(G))
   end
 
   isoinv = function (p::PermGroupElem)

From 1eb90771d2648297a92c0777e451ab98e4e6ea8d Mon Sep 17 00:00:00 2001
From: Torben Wiedemann <7226617+TWiedemann@users.noreply.github.com>
Date: Thu, 30 Jan 2025 20:26:46 +0100
Subject: [PATCH 05/14] Add ref to map_word(::WeylGroupElem) in
 map_word(::FPGroupElem) and map_word(::PcGroupElem)

---
 src/Groups/GAPGroups.jl | 6 ++++--
 1 file changed, 4 insertions(+), 2 deletions(-)

diff --git a/src/Groups/GAPGroups.jl b/src/Groups/GAPGroups.jl
index a876eeed8a43..1d5d5d0fef1b 100644
--- a/src/Groups/GAPGroups.jl
+++ b/src/Groups/GAPGroups.jl
@@ -2023,7 +2023,8 @@ If `init == nothing` and `genimgs` is empty, an error occurs.
 Thus the intended value for the empty word must be specified as `init`
 whenever it is possible that the elements in `genimgs` do not support `one`.
 
-See also: [`map_word(::Union{PcGroupElem, SubPcGroupElem}, ::Vector)`](@ref).
+See also: [`map_word(::Union{PcGroupElem, SubPcGroupElem}, ::Vector)`](@ref),
+[`map_word(::WeylGroupElem, ::Vector)](@ref).
 
 # Examples
 ```jldoctest
@@ -2106,7 +2107,8 @@ and $R_j =$ `genimgs[`$i_j$`]`$^{e_j}$.
 
 If `init` is different from `nothing`, return $x g_{i_1}^{e_1} g_{i_2}^{e_2} \cdots g_{i_n}^{e_n}$ where $x =$ `init`.
 
-See also: [`map_word(::Union{FPGroupElem, SubFPGroupElem}, ::Vector)`](@ref).
+See also: [`map_word(::Union{FPGroupElem, SubFPGroupElem}, ::Vector)`](@ref),
+[`map_word(::WeylGroupElem, ::Vector)](@ref).
 
 # Examples
 ```jldoctest

From 23b6b1ca130bc6570a08b57f1da5f0db596b2217 Mon Sep 17 00:00:00 2001
From: Torben Wiedemann <7226617+TWiedemann@users.noreply.github.com>
Date: Mon, 3 Feb 2025 22:27:58 +0100
Subject: [PATCH 06/14] Adapt error message in
 parabolic_subgroup_with_projection

Co-authored-by: Max Horn <max@quendi.de>
---
 experimental/LieAlgebras/src/WeylGroup.jl | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/experimental/LieAlgebras/src/WeylGroup.jl b/experimental/LieAlgebras/src/WeylGroup.jl
index 47393cd5450f..39e67bbed96d 100644
--- a/experimental/LieAlgebras/src/WeylGroup.jl
+++ b/experimental/LieAlgebras/src/WeylGroup.jl
@@ -394,7 +394,7 @@ function parabolic_subgroup_with_projection(
     for i in vec
       for j in setdiff(1:number_of_generators(W), vec)
         @req is_zero_entry(cm, i, j) begin
-          "Input vector does not describe a factor of the Weyl group, so there is no projection homomorphism"
+          "Input vector must describe a direct factor of the Weyl group"
         end
       end
     end

From 00a8b5bef3c5f59385ae46ce33ed050f46ae3e42 Mon Sep 17 00:00:00 2001
From: Torben Wiedemann <7226617+TWiedemann@users.noreply.github.com>
Date: Mon, 3 Feb 2025 22:39:15 +0100
Subject: [PATCH 07/14] Use conj

Co-authored-by: Max Horn <max@quendi.de>
---
 experimental/LieAlgebras/src/WeylGroup.jl | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/experimental/LieAlgebras/src/WeylGroup.jl b/experimental/LieAlgebras/src/WeylGroup.jl
index 39e67bbed96d..f4381107e879 100644
--- a/experimental/LieAlgebras/src/WeylGroup.jl
+++ b/experimental/LieAlgebras/src/WeylGroup.jl
@@ -347,7 +347,7 @@ function parabolic_subgroup(W::WeylGroup, vec::Vector{<:Integer}, w::WeylGroupEl
   @req all(i -> 1 <= i <= number_of_generators(W), vec) "Invalid indices"
   cm = cartan_matrix(root_system(W))[vec, vec]
   para = weyl_group(cm)
-  genimgs = map(x -> inv(w) * x * w, gens(W)[vec])
+  genimgs = [conj(W[i], w) for i in vec]
   emb = function (u::WeylGroupElem)
     return map_word(u, genimgs)
   end

From 489da37c784626bc7e2e081c4b35f3dedeb43ea8 Mon Sep 17 00:00:00 2001
From: Torben Wiedemann <7226617+TWiedemann@users.noreply.github.com>
Date: Mon, 3 Feb 2025 22:43:29 +0100
Subject: [PATCH 08/14] Use conj in parabolic subgroup test

Co-authored-by: Max Horn <max@quendi.de>
---
 experimental/LieAlgebras/test/WeylGroup-test.jl | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/experimental/LieAlgebras/test/WeylGroup-test.jl b/experimental/LieAlgebras/test/WeylGroup-test.jl
index bee847954237..2d0faed577c4 100644
--- a/experimental/LieAlgebras/test/WeylGroup-test.jl
+++ b/experimental/LieAlgebras/test/WeylGroup-test.jl
@@ -196,9 +196,9 @@
         genimgs = [gen(W, i) for i in vec] # Desired images of gens(para) in W
       else
         # On subsequent runs, conjugate by random elements
-        conj = rand(W)
-        para, emb = parabolic_subgroup(W, vec, conj)
-        genimgs = [inv(conj) * gen(W, vec[i]) * conj for i in 1:length(vec)]
+        r = rand(W)
+        para, emb = parabolic_subgroup(W, vec, r)
+        genimgs = [conj(W[i], r) for i in vec]
       end
       # Test that emb maps gens(para) to genimgs
       for i in 1:length(vec)

From cad4863e41af594e43a7d27e51637c3834900897 Mon Sep 17 00:00:00 2001
From: Torben Wiedemann <7226617+TWiedemann@users.noreply.github.com>
Date: Mon, 3 Feb 2025 22:44:26 +0100
Subject: [PATCH 09/14] Clarify doc of map_word

Co-authored-by: Max Horn <max@quendi.de>
---
 experimental/LieAlgebras/src/WeylGroup.jl | 5 +++--
 1 file changed, 3 insertions(+), 2 deletions(-)

diff --git a/experimental/LieAlgebras/src/WeylGroup.jl b/experimental/LieAlgebras/src/WeylGroup.jl
index f4381107e879..d48be79ffb4d 100644
--- a/experimental/LieAlgebras/src/WeylGroup.jl
+++ b/experimental/LieAlgebras/src/WeylGroup.jl
@@ -272,8 +272,9 @@ If `w` is trivial, `init` is nothing and `genimgs` is empty, an error occurs.
 
 See also: [`map_word(::Union{FPGroupElem, SubFPGroupElem}, ::Vector)`](@ref),
 [`map_word(::Union{PcGroupElem, SubPcGroupElem}, ::Vector)`](@ref).
-Note that `map_word(::WeylGroupElem)` accepts but ignores the `genimgs_inv` keyword argument
-because the generators of a Weyl group are always self-inverse.
+Note that `map_word(::WeylGroupElem)` accepts the `genimgs_inv` keyword argument
+for consistency with other `map_word` methods, but ignores it because the
+generators of a Weyl group are always self-inverse.
 
 # Examples
 ```jldoctest

From f6b733fa59c9d5f914bec3485303aeefedece26d Mon Sep 17 00:00:00 2001
From: Torben Wiedemann <7226617+TWiedemann@users.noreply.github.com>
Date: Mon, 3 Feb 2025 22:47:25 +0100
Subject: [PATCH 10/14] Use fill for vector initialisation

Co-authored-by: Max Horn <max@quendi.de>
---
 experimental/LieAlgebras/src/WeylGroup.jl | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/experimental/LieAlgebras/src/WeylGroup.jl b/experimental/LieAlgebras/src/WeylGroup.jl
index d48be79ffb4d..f587b2549411 100644
--- a/experimental/LieAlgebras/src/WeylGroup.jl
+++ b/experimental/LieAlgebras/src/WeylGroup.jl
@@ -404,7 +404,7 @@ function parabolic_subgroup_with_projection(
   factor, emb = parabolic_subgroup(W, vec)
   # Generators of W are mapped to the corresponding generators of factor,
   # or to 1 if there is no corresponding generator
-  proj_gen_imgs = [one(factor) for _ in 1:number_of_generators(W)]
+  proj_gen_imgs = fill(one(factor), ngens(W))
   for i in 1:length(vec)
     proj_gen_imgs[vec[i]] = gen(factor, i)
   end

From 675603d4c3febbbac0185230e1425b4cb5ea8525 Mon Sep 17 00:00:00 2001
From: Torben Wiedemann <7226617+TWiedemann@users.noreply.github.com>
Date: Mon, 3 Feb 2025 22:50:01 +0100
Subject: [PATCH 11/14] Switch for loops

Co-authored-by: Max Horn <max@quendi.de>
---
 experimental/LieAlgebras/src/WeylGroup.jl | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/experimental/LieAlgebras/src/WeylGroup.jl b/experimental/LieAlgebras/src/WeylGroup.jl
index f587b2549411..e417a73a0edb 100644
--- a/experimental/LieAlgebras/src/WeylGroup.jl
+++ b/experimental/LieAlgebras/src/WeylGroup.jl
@@ -392,8 +392,8 @@ function parabolic_subgroup_with_projection(
     # Check that every generator in gens(W)[vec] commutes with every other generator.
     # In other words, vec describes a union of irreducible components of the Coxeter diagram.
     cm = cartan_matrix(root_system(W))
-    for i in vec
-      for j in setdiff(1:number_of_generators(W), vec)
+    for i in setdiff(1:number_of_generators(W), vec)
+      for j in vec
         @req is_zero_entry(cm, i, j) begin
           "Input vector must describe a direct factor of the Weyl group"
         end

From 6687dc15e7e09e4c428b64548029771a22cfc6b1 Mon Sep 17 00:00:00 2001
From: Torben Wiedemann <7226617+TWiedemann@users.noreply.github.com>
Date: Tue, 4 Feb 2025 09:52:39 +0100
Subject: [PATCH 12/14] Document check

---
 experimental/LieAlgebras/src/WeylGroup.jl | 7 ++++---
 1 file changed, 4 insertions(+), 3 deletions(-)

diff --git a/experimental/LieAlgebras/src/WeylGroup.jl b/experimental/LieAlgebras/src/WeylGroup.jl
index e417a73a0edb..2d536f5e16dc 100644
--- a/experimental/LieAlgebras/src/WeylGroup.jl
+++ b/experimental/LieAlgebras/src/WeylGroup.jl
@@ -363,7 +363,8 @@ a product of irreducible factors.
 Here `P, emb = `[`parabolic_subgroup`](@ref)`(W, vec)`
 and `proj` is the projection map from `W` onto `P`,
 which is a left-inverse of `emb`.
-If `P` is not a factor of `W`, an error occurs.
+If `check = true`, then it is checked whether `vec` actually describes
+a union of irreducible components of the Dynkin diagram.
 
 # Examples
 ```jldoctest
@@ -386,9 +387,9 @@ true
 ```
 """
 function parabolic_subgroup_with_projection(
-  W::WeylGroup, vec::Vector{<:Integer}; check_factor::Bool=true
+  W::WeylGroup, vec::Vector{<:Integer}; check::Bool=true
 )
-  if check_factor
+  if check
     # Check that every generator in gens(W)[vec] commutes with every other generator.
     # In other words, vec describes a union of irreducible components of the Coxeter diagram.
     cm = cartan_matrix(root_system(W))

From 345cc85028c93312f739b5b915a2fe77ae4a2d5c Mon Sep 17 00:00:00 2001
From: Torben Wiedemann <7226617+TWiedemann@users.noreply.github.com>
Date: Tue, 4 Feb 2025 11:10:06 +0100
Subject: [PATCH 13/14] Update experimental/LieAlgebras/src/WeylGroup.jl
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

Co-authored-by: Lars Göttgens <lars.goettgens@gmail.com>
---
 experimental/LieAlgebras/src/WeylGroup.jl | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/experimental/LieAlgebras/src/WeylGroup.jl b/experimental/LieAlgebras/src/WeylGroup.jl
index 2d536f5e16dc..0e2fa062f96a 100644
--- a/experimental/LieAlgebras/src/WeylGroup.jl
+++ b/experimental/LieAlgebras/src/WeylGroup.jl
@@ -356,7 +356,7 @@ function parabolic_subgroup(W::WeylGroup, vec::Vector{<:Integer}, w::WeylGroupEl
 end
 
 @doc raw"""
-    parabolic_subgroup_with_projection(W::WeylGroup, vec::Vector{<:Integer}) -> WeylGroup, Map{WeylGroup, WeylGroup}, Map{WeylGroup, WeylGroup}
+    parabolic_subgroup_with_projection(W::WeylGroup, vec::Vector{<:Integer}; check::Bool=true) -> WeylGroup, Map{WeylGroup, WeylGroup}, Map{WeylGroup, WeylGroup}
 
 Return a triple `(P, emb, proj)` that describes a factor of `W`, that is,
 a product of irreducible factors.

From 834868a4c1eb85774104776ae4b93a2e3be0069e Mon Sep 17 00:00:00 2001
From: Torben Wiedemann <7226617+TWiedemann@users.noreply.github.com>
Date: Tue, 4 Feb 2025 11:10:52 +0100
Subject: [PATCH 14/14] Update experimental/LieAlgebras/src/WeylGroup.jl
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

Co-authored-by: Lars Göttgens <lars.goettgens@gmail.com>
---
 experimental/LieAlgebras/src/WeylGroup.jl | 1 +
 1 file changed, 1 insertion(+)

diff --git a/experimental/LieAlgebras/src/WeylGroup.jl b/experimental/LieAlgebras/src/WeylGroup.jl
index 0e2fa062f96a..9da551e098a2 100644
--- a/experimental/LieAlgebras/src/WeylGroup.jl
+++ b/experimental/LieAlgebras/src/WeylGroup.jl
@@ -363,6 +363,7 @@ a product of irreducible factors.
 Here `P, emb = `[`parabolic_subgroup`](@ref)`(W, vec)`
 and `proj` is the projection map from `W` onto `P`,
 which is a left-inverse of `emb`.
+
 If `check = true`, then it is checked whether `vec` actually describes
 a union of irreducible components of the Dynkin diagram.