Add natural_gset method for permutation, matrix & Weyl groups

examples/BlockSys.jl

@@ -206,7 +206,7 @@ function block_klueners(C::Oscar.GaloisGrp.GaloisCtx{Hecke.qAdicRootCtx},d) # ch
   cyc = gap_perm([findfirst(y -> y == x, Rts) for x = map(frobenius,Rts)])
   n = degree(C.f)
   divexact(n,d) # catch errors fast
-  Cyc = map(collect,orbits(gset(sub(symmetric_group(n),[cyc])[1]))) 
+  Cyc = map(collect,orbits(natural_gset(sub(symmetric_group(n),[cyc])[1]))) 
   l = length(Cyc)
   Z_values = Dict(zip([i for i in 1:l],length.(Cyc)))
   T = Tree(0)

examples/GSets_timing.jl

@@ -4,7 +4,7 @@
   # natural constructions (determined by the types of the seeds)
   G = symmetric_group(10)
 
-  Omega = gset(G)
+  Omega = natural_gset(G)
   @benchmark order(stabilizer(Omega, 1)[1])
   @benchmark order(stabilizer(Omega, Set([1, 2]))[1])
   @benchmark order(stabilizer(Omega, [1, 2])[1])

experimental/LieAlgebras/src/LieAlgebras.jl

@@ -75,6 +75,7 @@ import ..Oscar:
   kernel,
   lower_central_series,
   matrix,
+  natural_gset,
   normalizer,
   number_of_generators,
   ngens,

experimental/LieAlgebras/src/WeylGroup.jl

@@ -597,6 +597,28 @@ end
 ###############################################################################
 # G-set functionality (can get moved to src/ once `isomorphism(PermGroup, ::WeylGroup)` gets moved)
 
+"""
+    natural_gset(W::WeylGroup)
+
+Return the G-set `Omega` that consists of the closure of root space elements
+under the natural action of `W`.
+
+!!! note
+    This function is only implemented for finite Weyl groups, that is root systems of classical type.
+
+# Examples
+```jldoctest
+julia> W = weyl_group(:A, 2);
+
+julia> length(natural_gset(W))
+6
+```
+"""
+function natural_gset(W::WeylGroup)
+  @req is_finite(W) "only implemented for finite Weyl groups"
+  return gset(W, roots(root_system(W)); closed=true)
+end
+
 Base.:^(rw::Union{RootSpaceElem,WeightLatticeElem}, x::WeylGroupElem) = rw * x
 
 function gset_by_type(W::WeylGroup, Omega, ::Type{RootSpaceElem}; closed::Bool=false)

experimental/LieAlgebras/test/WeylGroup-test.jl

@@ -422,8 +422,7 @@
     @test is_semiregular(Omega)
 
     W = weyl_group([(:A, 2), (:B, 3)])
-    pts = roots(root_system(W))
-    Omega = gset(W, pts)
+    Omega = natural_gset(W)
     @test length(Omega) == 24
     @test length(orbits(Omega)) == 3
     @test !is_transitive(Omega)
@@ -446,29 +445,28 @@
 
     # orbit
     W = weyl_group([(:A, 2), (:B, 3)])
-    pts = roots(root_system(W))
-    Omega = gset(W, pts)
+    Omega = natural_gset(W)
     orbs = orbits(Omega)
     @test length(orbs) == 3
-    @test length(orbit(Omega, pts[2])) == 6
+    @test length(orbit(Omega, Omega[2])) == 6
     @test sort(map(length, orbs)) == [6, 6, 12]
 
     # action homomorphism
     W = weyl_group(:A, 2)
-    pts = roots(root_system(W))
-    Omega = gset(W, pts)
+    Omega = natural_gset(W)
     acthom = action_homomorphism(Omega)
     @test order(image(acthom)[1]) == order(W)
 
     # all_blocks
     bl = all_blocks(Omega)
     @test length(bl) == 4
-    @test Set([pts[1], pts[5]]) in bl
+    @test Set([Omega[1], Omega[5]]) in bl
 
     # blocks
     bl = blocks(Omega)
     @test length(bl) == 3
-    @test elements(bl) == map(Set, [[pts[1], pts[4]], [pts[6], pts[3]], [pts[5], pts[2]]])
+    @test elements(bl) ==
+      map(Set, [[Omega[1], Omega[4]], [Omega[5], Omega[2]], [Omega[6], Omega[3]]])
     @test length(orbits(bl)) == 1
   end
 end

src/Groups/gsets.jl

@@ -219,13 +219,43 @@ end
 
 ## (add more such actions: on sets of sets, on sets of tuples, ...)
 
-## natural action of a permutation group on the integers 1, ..., degree
-gset(G::PermGroup) = gset(G, 1:G.deg; closed = true)
+#############################################################################
+##
+##  natural method with implicit action function
+
+"""
+    natural_gset(G::PermGroup)
+
+Return the G-set `Omega` that consists of integers 1, ..., degree
+under the natural action of `G`.
+
+# Examples
+```jldoctest
+julia> G = symmetric_group(4);
+
+julia> length(natural_gset(G))
+4
+```
+"""
+natural_gset(G::PermGroup) = gset(G, 1:G.deg; closed = true)
+
+"""
+    natural_gset(G::MatrixGroup{T, MT}) where {MT, T <: FinFieldElem}
+
+Return the G-set `Omega` that consists of vectors under the 
+natural action of `G` over a finite field.
 
-## natural action of a matrix group over a finite field on vectors
-function gset(G::MatrixGroup{T, MT}) where T <: FinFieldElem where MT
-    V = free_module(base_ring(G), degree(G))
-    return gset(G, collect(V); closed = true)
+# Examples
+```jldoctest
+julia> G = matrix_group(GF(2), 2);
+
+julia> length(natural_gset(G))
+4
+```
+"""
+function natural_gset(G::MatrixGroup{T, MT}) where {MT, T <: FinFieldElem}
+  V = free_module(base_ring(G), degree(G))
+  return gset(G, collect(V); closed = true)
 end
 
 
@@ -511,7 +541,7 @@ Return the vector of transitive G-sets in `Omega`.
 julia> G = sylow_subgroup(symmetric_group(6), 2)[1]
 Permutation group of degree 6 and order 16
 
-julia> orbs = orbits(gset(G));
+julia> orbs = orbits(natural_gset(G));
 
 julia> map(collect, orbs)
 2-element Vector{Vector{Int64}}:
@@ -548,7 +578,7 @@ julia> map(length, orbs)
  2
 ```
 """
-@attr Vector{GSetByElements{PermGroup, Int}} orbits(G::PermGroup) = orbits(gset(G))
+@attr Vector{GSetByElements{PermGroup, Int}} orbits(G::PermGroup) = orbits(natural_gset(G))
 
 
 """
@@ -570,7 +600,7 @@ If `check` is `false` then it is not checked whether `omega` is in `Omega`.
 
 # Examples
 ```jldoctest
-julia> Omega = gset(symmetric_group(4));
+julia> Omega = natural_gset(symmetric_group(4));
 
 julia> stabilizer(Omega)
 (Permutation group of degree 4 and order 6, Hom: permutation group -> Sym(4))
@@ -910,7 +940,7 @@ To also obtain a conjugating element use [`is_conjugate_with_data`](@ref).
 julia> G = sylow_subgroup(symmetric_group(6), 2)[1]
 Permutation group of degree 6 and order 16
 
-julia> Omega = gset(G);
+julia> Omega = natural_gset(G);
 
 julia> is_conjugate(Omega, 1, 2)
 true
@@ -936,7 +966,7 @@ If the conjugating element `g` is not needed, use [`is_conjugate`](@ref).
 julia> G = sylow_subgroup(symmetric_group(6), 2)[1]
 Permutation group of degree 6 and order 16
 
-julia> Omega = gset(G);
+julia> Omega = natural_gset(G);
 
 julia> is_conjugate_with_data(Omega, 1, 2)
 (true, (1,2))
@@ -996,7 +1026,7 @@ An exception is thrown if this action is not transitive.
 
 # Examples
 ```jldoctest; filter = Main.Oscar.doctestfilter_hash_changes_in_1_13()
-julia> Omega = gset(sylow_subgroup(symmetric_group(4), 2)[1])
+julia> Omega = natural_gset(sylow_subgroup(symmetric_group(4), 2)[1])
 G-set of
   permutation group of degree 4 and order 8
   with seeds 1:4
@@ -1029,7 +1059,7 @@ An exception is thrown if this action is not transitive.
 
 # Examples
 ```jldoctest; filter = Main.Oscar.doctestfilter_hash_changes_in_1_13()
-julia> Omega = gset(transitive_group(8, 2))
+julia> Omega = natural_gset(transitive_group(8, 2))
 G-set of
   permutation group of degree 8
   with seeds 1:8
@@ -1062,7 +1092,7 @@ An exception is thrown if this action is not transitive.
 
 # Examples
 ```jldoctest; filter = Main.Oscar.doctestfilter_hash_changes_in_1_13()
-julia> Omega = gset(transitive_group(8, 2))
+julia> Omega = natural_gset(transitive_group(8, 2))
 G-set of
   permutation group of degree 8
   with seeds 1:8
@@ -1095,7 +1125,7 @@ An exception is thrown if this action is not transitive.
 
 # Examples
 ```jldoctest; filter = Main.Oscar.doctestfilter_hash_changes_in_1_13()
-julia> Omega = gset(transitive_group(8, 2))
+julia> Omega = natural_gset(transitive_group(8, 2))
 G-set of
   permutation group of degree 8
   with seeds 1:8
@@ -1131,10 +1161,10 @@ An exception is thrown if the group action is not transitive on `Omega`.
 
 # Examples
 ```jldoctest
-julia> G = symmetric_group(4); Omega = gset(G); rank_action(Omega)  # 4-transitive
+julia> G = symmetric_group(4); Omega = natural_gset(G); rank_action(Omega)  # 4-transitive
 2
 
-julia> G = dihedral_group(PermGroup, 8); Omega = gset(G); rank_action(Omega)  # not 2-transitive
+julia> G = dihedral_group(PermGroup, 8); Omega = natural_gset(G); rank_action(Omega)  # not 2-transitive
 3
 ```
 """
@@ -1157,10 +1187,10 @@ The output is `0` if the group acts intransitively on `Omega`.
 
 # Examples
 ```jldoctest
-julia> G = mathieu_group(24); Omega = gset(G); transitivity(Omega)
+julia> G = mathieu_group(24); Omega = natural_gset(G); transitivity(Omega)
 5
 
-julia> G = symmetric_group(4); Omega = gset(G); transitivity(Omega)
+julia> G = symmetric_group(4); Omega = natural_gset(G); transitivity(Omega)
 4
 ```
 """
@@ -1181,7 +1211,7 @@ In other word, this tests if `Omega` consists of precisely one orbit.
 julia> G = sylow_subgroup(symmetric_group(6), 2)[1]
 Permutation group of degree 6 and order 16
 
-julia> Omega = gset(G);
+julia> Omega = natural_gset(G);
 
 julia> is_transitive(Omega)
 false
@@ -1208,9 +1238,9 @@ julia> G = symmetric_group(6);
 julia> H = sub(G, [G([2, 3, 4, 5, 6, 1])])[1]
 Permutation group of degree 6
 
-julia> OmegaG = gset(G);
+julia> OmegaG = natural_gset(G);
 
-julia> OmegaH = gset(H);
+julia> OmegaH = natural_gset(H);
 
 julia> is_regular(OmegaH)
 true
@@ -1234,7 +1264,7 @@ julia> G = symmetric_group(6);
 julia> H = sub(G, [G([2, 3, 1, 5, 6, 4])])[1]
 Permutation group of degree 6
 
-julia> OmegaH = gset(H);
+julia> OmegaH = natural_gset(H);
 
 julia> is_semiregular(H)
 true
@@ -1256,14 +1286,14 @@ the action is transitive and admits no nontrivial block systems.
 
 # Examples
 ```jldoctest
-julia> G = symmetric_group(4); Omega = gset(G);
+julia> G = symmetric_group(4); Omega = natural_gset(G);
 
 julia> is_primitive(Omega)
 true
 
 julia> H, _ = sub(G, [cperm([2, 3, 4])]);
 
-julia> is_primitive(gset(H))
+julia> is_primitive(natural_gset(H))
 false
 ```
 """

src/deprecations.jl

@@ -153,3 +153,7 @@ Base.@deprecate_binding in_linear_system is_in_linear_system
 @deprecate acting_domain(C::GroupCoset) acting_group(C)
 @deprecate acting_domain(Omega::GSet) acting_group(Omega)
 @deprecate grid_morphism lattice_homomorphism
+
+# deprecated for 1.4
+@deprecate gset(G::PermGroup) natural_gset(G)
+@deprecate gset(G::MatrixGroup{T, MT}) where {MT, T <: FinFieldElem} natural_gset(G)

src/exports.jl

@@ -1242,6 +1242,7 @@ export n_vertices
 export name
 export names_of_fusion_sources
 export natural_character
+export natural_gset
 export nef_cone
 export negative_coroot
 export negative_coroots

test/Groups/gsets.jl

@@ -2,7 +2,7 @@
 
   # natural constructions (determined by the types of the seeds)
   G = symmetric_group(6)
-  Omega = gset(G)
+  Omega = natural_gset(G)
   @test AbstractAlgebra.PrettyPrinting.repr_terse(Omega) == "G-set"
   @test isa(Omega, GSet)
   @test (@inferred length(Omega)) == 6
@@ -60,7 +60,7 @@
 
   # larger examples
   G = symmetric_group(100)
-  Omega = gset(G)
+  Omega = natural_gset(G)
   S1, _ = stabilizer(Omega, [1, 2, 3, 4, 5])
   @test order(S1) == factorial(big(95))
   S2, _ = stabilizer(Omega, (1, 2, 3, 4, 5))
@@ -258,9 +258,9 @@ end
 
   # transitivity
   @test transitivity(G8) == 1
-  @test transitivity(gset(G8)) == 1
+  @test transitivity(natural_gset(G8)) == 1
   @test transitivity(S4) == 4
-  @test transitivity(gset(S4)) == 4
+  @test transitivity(natural_gset(S4)) == 4
   @test_throws ArgumentError transitivity(S4, 1:3)
   @test transitivity(S4, 1:4) == 4
   @test transitivity(S4, 1:5) == 0
@@ -273,7 +273,7 @@ end
   G = general_linear_group(2, 3)
   V = free_module(base_ring(G), degree(G))
   v = gen(V, 1)
-  Omega = gset(G)
+  Omega = natural_gset(G)
   @test isa(Omega, GSet)
   @test length(Omega) == 9
   @test order(stabilizer(Omega, v)[1]) * length(orbit(Omega, v)) == order(G)
@@ -311,31 +311,31 @@ end
   @test is_semiregular(Omega)
 
   # orbit
-  Omega = gset(G)
+  Omega = natural_gset(G)
   v = gen(V, 1)
   @test length(orbit(Omega, v)) == length(Oscar.orbit_via_Julia(Omega, v))
 
   # permutation
-  Omega = gset(G)
+  Omega = natural_gset(G)
   g = gen(G, 1)
   pi = permutation(Omega, g)
   @test order(pi) == order(g)
   @test degree(parent(pi)) == length(Omega)
 
   # action homomorphism
-  Omega = gset(G)
+  Omega = natural_gset(G)
   acthom = action_homomorphism(Omega)
   @test pi == g^acthom
   @test has_preimage_with_preimage(acthom, pi)[1]
   @test order(image(acthom)[1]) == 48
 
   # is_conjugate
-  Omega = gset(G)
+  Omega = natural_gset(G)
   @test is_conjugate(Omega, gen(V, 1), gen(V, 2))
   @test ! is_conjugate(Omega, zero(V), gen(V, 1))
 
   # is_conjugate_with_data
-  Omega = gset(G)
+  Omega = natural_gset(G)
   rep = is_conjugate_with_data(Omega, gens(V)...)
   @test rep[1]
   @test gen(V, 1) * rep[2] == gen(V, 2)