support character_field for a vector of characters (#4411)

src/GAP/wrappers.jl

@@ -101,6 +101,7 @@ GAP.@wrap FamilyObj(x::GAP.Obj)::GapObj
 GAP.@wrap FamilyPcgs(x::GAP.Obj)::GapObj
 GAP.@wrap fhmethsel(x::GapObj)::GAP.Obj
 GAP.@wrap Field(x::Any)::GapObj
+GAP.@wrap Flat(x::GapObj)::GapObj
 GAP.@wrap FreeAbelianGroup(x::Int)::GapObj
 GAP.@wrap FreeGeneratorsOfFpGroup(x::GapObj)::GapObj
 GAP.@wrap FreeGroupOfFpGroup(x::GapObj)::GapObj

src/Groups/group_characters.jl

@@ -2529,11 +2529,11 @@ multiplicities_eigenvalues(chi::GAPGroupClassFunction, i::Int) = multiplicities_
 
 
 function Base.:*(n::IntegerUnion, chi::GAPGroupClassFunction)
-    return GAPGroupClassFunction(parent(chi), n * GapObj(chi))
+    return GAPGroupClassFunction(parent(chi), GapObj(n) * GapObj(chi))
 end
 
 function Base.:^(chi::GAPGroupClassFunction, n::IntegerUnion)
-    return GAPGroupClassFunction(parent(chi), GapObj(chi) ^ n)
+    return GAPGroupClassFunction(parent(chi), GapObj(chi) ^ GapObj(n))
 end
 
 function Base.:^(chi::GAPGroupClassFunction, tbl::GAPGroupCharacterTable)
@@ -2828,6 +2828,7 @@ end
 
 @doc raw"""
     character_field(chi::GAPGroupClassFunction)
+    character_field(l::Vector{GAPGroupClassFunction})
 
 If `chi` is an ordinary character then
 return the pair `(F, phi)` where `F` is a number field that is generated
@@ -2839,6 +2840,10 @@ return the pair `(F, phi)` where `F` is the finite field that is generated
 by the `p`-modular reductions of the values of `chi`,
 and `phi` is the identity map on `F`.
 
+If a nonempty vector `l` of characters is given then `(F, phi)` is returned
+such that `F` is the smallest field that contains the character fields
+of the entries of `l`.
+
 # Examples
 ```jldoctest
 julia> t = character_table("A5");
@@ -2851,6 +2856,9 @@ julia> flds_2 = map(character_field, mod(t, 2));
 
 julia> println([degree(x[1]) for x in flds_2])
 [1, 2, 2, 1]
+
+julia> degree(character_field(collect(t))[1])
+2
 ```
 """
 function character_field(chi::GAPGroupClassFunction)
@@ -2864,8 +2872,31 @@ function character_field(chi::GAPGroupClassFunction)
       return (F, identity_map(F))
     end
 
-    values = GapObj(chi)  # a list of GAP cyclotomics
+    values = GapObj(chi)::GapObj
     gapfield = GAPWrap.Field(values)
+    return _character_field(gapfield)
+end
+
+function character_field(l::Vector{GAPGroupClassFunction})
+    @req length(l) > 0 "need at least one class function"
+    p = characteristic(l[1])
+    @req all(chi -> characteristic(chi) == p, l) "all entries must have the same characteristic"
+
+    if p != 0
+      # Brauer characters, construct a finite field
+      orders = [order_field_of_definition(chi) for chi in l]
+      exps = [is_prime_power_with_data(q)[2] for q in orders]
+      e = lcm(exps)
+      F = GF(p, e)
+      return (F, identity_map(F))
+    end
+
+    values = GapObj(l, recursive = true)::GapObj
+    gapfield = GAPWrap.Field(GAPWrap.Flat(values))
+    return _character_field(gapfield)
+end
+
+function _character_field(gapfield::GapObj)
     N = GAPWrap.Conductor(gapfield)
     FF, _ = abelian_closure(QQ)
     if GAPWrap.IsCyclotomicField(gapfield)

test/Groups/group_characters.jl

@@ -828,6 +828,10 @@ end
   @test tr == t[end]
   @test tr == trivial_character(g)
   @test !is_faithful(tr)
+  @test 2 * tr == tr + tr
+  @test ZZ(2) * tr == tr + tr
+  @test tr^2 == tr
+  @test tr^ZZ(2) == tr
   re = regular_character(g)
   @test coordinates(re) == degree.(t)
   re = regular_character(t)
@@ -1052,6 +1056,7 @@ end
   tbl = character_table("A5")
   @test [degree(character_field(chi)[1]) for chi in tbl] == [1, 2, 2, 1, 1]
   @test characteristic(character_field(tbl[1])[1]) == 0
+  @test degree(character_field(collect(tbl))[1]) == 2
 
   for id in [ "C5", "A5" ]   # cyclotomic and non-cyclotomic number fields
     for chi in character_table(id)
@@ -1108,6 +1113,7 @@ end
   @test [order(character_field(chi)[1]) for chi in modtbl] == [2, 4, 4, 2]
   @test order_field_of_definition(Int, modtbl[1]) isa Int
   @test_throws ArgumentError order_field_of_definition(ordtbl[1])
+  @test degree(character_field(collect(modtbl))[1]) == 2
 end
 
 @testset "Schur index" begin