various improvements for characters (#2519)

* various improvements for characters

- adjusted the print functions for character tables to the rules
  from "Details on printing in Oscar"
  (and adjusted the manual examples and tests)

- document the output of the text/plain method for `show`
  (addresses issue #2482)

- fixed the text/latex output in the case that the irrationalities
  are shown via a legend (and that there are several lines in the legend)

- added more documentation and doctests

- changed `character_field` for Brauer characters:
  The field is now a *finite field* in this case.

- use the `GAPWrap` approach for more GAP functions

* move functions, do not copy them ...

* further improvements

- fixed a test for `labelled_matrix_formatted`
- renamed `group_class_function` to `class_function`
- added `natural_character` for matrix groups in positive characteristic
  and for group homomorphisms
- admit `IntegerUnion` for the `characteristic` of character tables,
  not just `Int`

* fixed doctests

* adjustments/fixes

- adjusted an `is_prime_power_with_data` call
  (The ordering of return values of `is_prime_power_with_data`
  has been changed in Nemocas/Nemo.jl/pull/1484.)

- adjusted the type of a (perhaps empty) list
  (This seemed to work at least up to Julia 1.8.5.)

- fixed some doctests

docs/src/Groups/group_characters.md

@@ -81,12 +81,14 @@ of ``G`` (or the `j`-th conjugacy class of ``p``-regular elements
 in the case of Brauer tables).
 
 Ordinary and ``p``-modular Brauer tables in OSCAR are distinguished by
-the field `characteristic`; its value is `0` for ordinary tables and
-``p`` otherwise.
+their [`characteristic(tbl::GAPGroupCharacterTable)`](@ref);
+its value is `0` for ordinary tables and ``p`` otherwise.
 
 ```@docs
 GAPGroupCharacterTable
 character_table
+Base.show(io::IO, ::MIME"text/plain", tbl::GAPGroupCharacterTable)
+characteristic(tbl::GAPGroupCharacterTable)
 Base.mod(tbl::GAPGroupCharacterTable, p::Int)
 all_character_table_names
 ```
@@ -101,16 +103,19 @@ indicator
 is_faithful(chi::GAPGroupClassFunction)
 is_rational(chi::GAPGroupClassFunction)
 is_irreducible(chi::GAPGroupClassFunction)
-schur_index
+schur_index(chi::GAPGroupClassFunction, recurse::Bool = true)
 det(chi::GAPGroupClassFunction)
 order(chi::GAPGroupClassFunction)
+order_field_of_definition(chi::GAPGroupClassFunction)
 ```
 
 ## Attributes of character tables
 
 ```@docs
 character_parameters
+class_names(tbl::GAPGroupCharacterTable)
 class_parameters
+conjugacy_classes(tbl::GAPGroupCharacterTable)
 decomposition_matrix
 identifier
 induced_cyclic(tbl::GAPGroupCharacterTable)
@@ -120,6 +125,7 @@ names_of_fusion_sources
 class_lengths
 orders_centralizers
 orders_class_representatives
+ordinary_table(tbl::GAPGroupCharacterTable)
 trivial_character(tbl::GAPGroupCharacterTable)
 ```
 
@@ -128,6 +134,8 @@ trivial_character(tbl::GAPGroupCharacterTable)
 ```@docs
 natural_character(G::PermGroup)
 natural_character(G::Union{MatrixGroup{QQFieldElem}, MatrixGroup{nf_elem}})
+natural_character(G::MatrixGroup{T, MT}) where T <: FinFieldElem where MT
+natural_character(rho::GAPGroupHomomorphism)
 trivial_character(G::GAPGroup)
 ```
 

experimental/GModule/GModule.jl

@@ -164,10 +164,6 @@ function __init__()
   set_verbose_level(:MinField, 0)
 end
 
-function Hecke.number_field(::QQField, chi::Oscar.GAPGroupClassFunction; cached::Bool = false)
-  return number_field(QQ, map(x->GAP.gap_to_julia(QQAbElem, x), chi.values), cached = cached)
-end
-
 function irreducible_modules(G::Oscar.GAPGroup)
   im = GAP.Globals.IrreducibleRepresentations(G.X)
   IM = GModule[] 
@@ -362,16 +358,16 @@ function Oscar.character(C::GModule{<:Any, <:Generic.FreeModule{nf_elem}})
   QQAb = abelian_closure(QQ)[1]
   K = cyclotomic_field(QQAb, Int(c))[1]
   fl, em = is_subfield(k, K)
-  return Oscar.group_class_function(group(C), [QQAb(em(preimage(mkK, x[2]))) for x = chr])
+  return Oscar.class_function(group(C), [QQAb(em(preimage(mkK, x[2]))) for x = chr])
 end
 
 function Oscar.character(C::GModule{<:Any, <:Generic.FreeModule{QQFieldElem}})
   QQAb = abelian_closure(QQ)[1]
-  return Oscar.group_class_function(group(C), [QQAb(x[2]) for x = _character(C)])
+  return Oscar.class_function(group(C), [QQAb(x[2]) for x = _character(C)])
 end
 
 function Oscar.character(C::GModule{<:Any, <:Generic.FreeModule{<:AbstractAlgebra.FieldElem}})
-  return Oscar.group_class_function(group(C), [base_ring(C)(x[2]) for x = _character(C)])
+  return Oscar.class_function(group(C), [base_ring(C)(x[2]) for x = _character(C)])
 end
 
 
@@ -1161,10 +1157,6 @@ function Oscar.abelian_group(M::Generic.FreeModule{fqPolyRepFieldElem})
   return A, MapFromFunc(A, M, to_M, to_A)
 end
 
-function Oscar.group(chi::Oscar.GAPGroupClassFunction)
-  return chi.table.GAPGroup
-end
-
 function Oscar.gmodule(chi::Oscar.GAPGroupClassFunction)
   f = GAP.Globals.IrreducibleAffordingRepresentation(chi.values)
   K = abelian_closure(QQ)[1]

experimental/GModule/Misc.jl

@@ -172,5 +172,4 @@ end
 Base.getindex(::QQField, a::QQAbElem) = number_field(QQ, a)
 Base.getindex(::QQField, a::Vector{QQAbElem}) = number_field(QQ, a)
 Base.getindex(::QQField, a::QQAbElem...) = number_field(QQ, [x for x =a])
-Base.getindex(::QQField, chi::Oscar.GAPGroupClassFunction) = number_field(QQ, chi)
 

experimental/SymmetricIntersections/src/representations.jl

@@ -183,7 +183,7 @@ function character_representation(RR::RepRing{S, T}, f::GAPGroupHomomorphism{T,
     b = f(representative(h))
     push!(coll, Q(trace(b)))
   end
-  c = Oscar.group_class_function(character_table_underlying_group(RR), coll)
+  c = Oscar.class_function(character_table_underlying_group(RR), coll)
   return c
 end
 

src/GAP/wrappers.jl

@@ -14,50 +14,75 @@ using GAP
 
 GAP.@wrap AlgExtElm(x::GapObj, y::GAP.Obj)::GapObj
 GAP.@wrap AlgebraicExtension(x::GapObj, y::GapObj)::GapObj
+GAP.@wrap AntiSymmetricParts(x::GapObj, y::GapObj, z::GapInt)::GapObj
 GAP.@wrap AsList(x::GapObj)::GapObj
 GAP.@wrap AsSet(x::GapObj)::GapObj
+GAP.@wrap AtlasIrrationality(x::GapObj)::GAP.Obj
 GAP.@wrap Basis(x::GapObj)::GapObj
 GAP.@wrap Basis(x::GapObj, y::GapObj)::GapObj
 GAP.@wrap BasisNC(x::GapObj, y::GapObj)::GapObj
 GAP.@wrap CanonicalBasis(x::GapObj)::GapObj
+GAP.@wrap CentreOfCharacter(x::GapObj, y::GapObj)::GapObj
 GAP.@wrap CF(x::Any, y::Any)::GapObj
 GAP.@wrap CF(x::Any)::GapObj
 GAP.@wrap CHAR_FFE_DEFAULT(x::Any)::GapInt
 GAP.@wrap Characteristic(x::Any)::GapInt
+GAP.@wrap CharacterParameters(x::GapObj)::GapObj
+GAP.@wrap CharacterTable(x::GapObj)::GapObj
+GAP.@wrap CharacterTable(x::GapObj, y::GAP.Obj)::GapObj
+GAP.@wrap CharacterTableWreathSymmetric(x::GapObj, y::GapInt)::GapObj
+GAP.@wrap ClassFunction(x::GapObj, y::GapObj)::GapObj
+GAP.@wrap ClassNames(x::GapObj)::GapObj
+GAP.@wrap ClassMultiplicationCoefficient(x::GapObj, y::Int, z::Int, t::Int)::GAP.Obj
+GAP.@wrap ClassPositionsOfCentre(x::GapObj)::GapObj
+GAP.@wrap ClassPositionsOfPCore(x::GapObj, y::GAP.Obj)::GapObj
+GAP.@wrap ClassParameters(x::GapObj)::GapObj
 GAP.@wrap Coefficients(x::Any, y::Any)::GapObj
 GAP.@wrap CoefficientsFamily(x::GapObj)::GapObj
 GAP.@wrap CoefficientsOfUnivariatePolynomial(x::GapObj)::GapObj
 GAP.@wrap CoeffsCyc(x::GAP.Obj, y::Int)::GapObj
+GAP.@wrap ComputedPowerMaps(x::GapObj)::GapObj
 GAP.@wrap Conductor(x::Any)::GapInt
+GAP.@wrap ConjugacyClasses(x::GapObj)::GapObj
 GAP.@wrap CycList(x::GapObj)::GapInt
 GAP.@wrap CyclotomicPol(x::Int)::GapObj
 GAP.@wrap Decomposition(x::GapObj, y::GapObj, z::GAP.Obj)::GapObj
+GAP.@wrap DecompositionMatrix(x::GapObj)::GapObj
 GAP.@wrap DefiningPolynomial(x::GapObj)::GapObj
 GAP.@wrap DegreeFFE(x::Any)::Int
 GAP.@wrap DegreeOfLaurentPolynomial(x::GapObj)::GapInt
 GAP.@wrap DegreeOverPrimeField(x::GapObj)::Int
 GAP.@wrap DenominatorCyc(x::Any)::GapInt
 GAP.@wrap DenominatorRat(x::Any)::GapInt
+GAP.@wrap DescriptionOfRootOfUnity(x::Any)::GapObj
+GAP.@wrap DeterminantOfCharacter(x::GapObj)::GapObj
 GAP.@wrap Dimension(x::GapObj)::Int
 GAP.@wrap E(x::Any)::GapInt
+GAP.@wrap EigenvaluesChar(x::GapObj, y::GAP.Obj)::GapObj
 GAP.@wrap Elements(x::GapObj)::GapObj
 GAP.@wrap ElementsFamily(x::GapObj)::GapObj
+GAP.@wrap ELMS_LIST(x::GapObj, y::GapObj)::GapObj
 GAP.@wrap ExtRepOfObj(x::GapObj)::GapObj
 GAP.@wrap ExtRepPolynomialRatFun(x::GapObj)::GapObj
 GAP.@wrap FamilyObj(x::GAP.Obj)::GapObj
 GAP.@wrap Field(x::Any)::GapObj
 GAP.@wrap FreeAbelianGroup(x::Int)::GapObj
 GAP.@wrap FreeGeneratorsOfFpGroup(x::GapObj)::GapObj
 GAP.@wrap FreeGroupOfFpGroup(x::GapObj)::GapObj
+GAP.@wrap FusionConjugacyClasses(x::GapObj, y::GapObj)::GapObj
 GAP.@wrap GaloisCyc(x::GAP.Obj, GapInt)::GAP.Obj
 GAP.@wrap GeneratorsOfField(x::GapObj)::GapObj
 GAP.@wrap GeneratorsOfGroup(x::GapObj)::GapObj
+GAP.@wrap GetFusionMap(x::GapObj, y::GapObj)::GapObj
 GAP.@wrap GF(x::Any, y::Any)::GapObj
 GAP.@wrap GF(x::Any)::GapObj
 GAP.@wrap GroupHomomorphismByFunction(x1, x2, x3)::GapObj
 GAP.@wrap GroupHomomorphismByFunction(x1, x2, x3, x4)::GapObj
 GAP.@wrap GroupHomomorphismByFunction(x1, x2, x3, x4, x5)::GapObj
+GAP.@wrap HasCharacterParameters(x::GapObj)::Bool
+GAP.@wrap HasClassParameters(x::GapObj)::Bool
 GAP.@wrap HasMaximalAbelianQuotient(x::Any)::Bool
+GAP.@wrap HasMaxes(x::GapObj)::Bool
 GAP.@wrap HasSize(x::Any)::Bool
 GAP.@wrap Image(x::Any, y::Any)::GapObj
 GAP.@wrap Image(x::Any)::GapObj
@@ -67,6 +92,10 @@ GAP.@wrap Indeterminate(x::GapObj)::GapObj
 GAP.@wrap Indeterminate(x::GapObj, y::GAP.Obj)::GapObj
 GAP.@wrap IndeterminateNumberOfUnivariateRationalFunction(x::GapObj)::Int
 GAP.@wrap IndeterminatesOfPolynomialRing(x::GapObj)::GapObj
+GAP.@wrap Indicator(x::GapObj, y::GapObj, z::GapInt)::GapObj
+GAP.@wrap InducedClassFunctionsByFusionMap(x::GapObj, y::GapObj, z::GapObj, t::GapObj)::GapObj
+GAP.@wrap InducedClassFunction(x::GapObj, y::GapObj)::GapObj
+GAP.@wrap InducedCyclic(x::GapObj)::GapObj
 GAP.@wrap INT_FFE_DEFAULT(x::Any)::GapInt
 GAP.@wrap IntFFE(x::Any)::GapInt
 GAP.@wrap Inverse(x::GapObj)::GapObj
@@ -171,14 +200,18 @@ GAP.@wrap IsZero(x::Any)::Bool
 GAP.@wrap IsZmodnZObj(x::Any)::Bool
 GAP.@wrap IsZmodnZObjNonprimeCollection(x::Any)::Bool
 GAP.@wrap Iterator(x::Any)::GapObj
+GAP.@wrap KernelOfCharacter(x::GapObj, y::GapObj)::GapObj
 GAP.@wrap LargestMovedPoint(x::Any)::Int
 GAP.@wrap LeftActingDomain(x::GapObj)::GapObj
 GAP.@wrap LinearCombination(x::GapObj, y::GapObj)::GapObj
 GAP.@wrap MatScalarProducts(x::GapObj, y::GapObj, z::GapObj)::GapObj
+GAP.@wrap Maxes(x::GapObj)::GapObj
 GAP.@wrap MinimalPolynomial(x::GapObj, y::GAP.Obj)::GapObj
 GAP.@wrap MinimalGeneratingSet(x::GapObj)::GapObj
 GAP.@wrap mod(x::Any, y::Any)::GAP.Obj
+GAP.@wrap NamesOfFusionSources(x::GapObj)::GapObj
 GAP.@wrap NextIterator(x::GapObj)::Any
+GAP.@wrap NormalSubgroupClasses(x::GapObj, y::GAP.Obj)::GapObj
 GAP.@wrap NrCols(x::GapObj)::Int
 GAP.@wrap NrConjugacyClasses(x::Any)::GapInt
 GAP.@wrap NrRows(x::GapObj)::Int
@@ -193,29 +226,40 @@ GAP.@wrap OnSets(x::GapObj, y::GapObj)::GapObj
 GAP.@wrap OnSetsSets(x::GapObj, y::GapObj)::GapObj
 GAP.@wrap OnTuples(x::GapObj, y::GapObj)::GapObj
 GAP.@wrap Order(x::Any)::GapInt
+GAP.@wrap OrthogonalComponents(x::GapObj, y::GapObj, z::GapInt)::GapObj
+GAP.@wrap OrthogonalDiscriminants(x::GapObj)::GapObj
 GAP.@wrap Permuted(x::GapObj, y::GapObj)::GapObj
 GAP.@wrap PolynomialByExtRep(x::GapObj, y::GapObj)::GapObj
 GAP.@wrap PolynomialRing(x::GapObj)::GapObj
 GAP.@wrap PolynomialRing(x::GapObj, y::Int)::GapObj
+GAP.@wrap PossibleClassFusions(x::GapObj, y::GapObj)::GapObj
 GAP.@wrap PrimePGroup(x::GapObj)::GapInt
 GAP.@wrap PrimitiveElement(x::GapObj)::GapObj
 GAP.@wrap Range(x::GapObj)::GapObj
 GAP.@wrap ReduceCoeffs(x::GapObj, y::GapObj)
 GAP.@wrap RelatorsOfFpGroup(x::GapObj)::GapObj
 GAP.@wrap Representative(x::GapObj)::GAP.Obj
 GAP.@wrap ScalarProduct(x::GapObj, y::GapObj, z::GapObj)::GAP.Obj
+GAP.@wrap SchurIndexByCharacter(x::GapObj, y::GapObj, z::GapObj)::GAP.Obj
 GAP.@wrap SetMaximalAbelianQuotient(x::Any, y::Any)::Nothing
 GAP.@wrap SetSize(x::Any, y::Any)::Nothing
 GAP.@wrap Size(x::Any)::GapInt
+GAP.@wrap SizeOfFieldOfDefinition(x::GapObj, y::GapInt)::GapInt
+GAP.@wrap SizesCentralizers(x::GapObj)::GapObj
+GAP.@wrap SizesConjugacyClasses(x::GapObj)::GapObj
 GAP.@wrap Source(x::GapObj)::GapObj
 GAP.@wrap Sqrt(x::Int64)::GAP.Obj
 GAP.@wrap StringViewObj(x::Any)::GapObj
+GAP.@wrap SymmetricParts(x::GapObj, y::GapObj, z::GapInt)::GapObj
+GAP.@wrap Symmetrizations(x::GapObj, y::GapObj, z::GapInt)::GapObj
+GAP.@wrap SymplecticComponents(x::GapObj, y::GapObj, z::GapInt)::GapObj
 GAP.@wrap UnderlyingElement(x::GapObj)::GapObj
 GAP.@wrap UnderlyingRingElement(x::GapObj)::GapObj
 GAP.@wrap UnivariatePolynomialByCoefficients(x::GapObj, y::GapObj, z::Int)::GapObj
 GAP.@wrap Value(x::GapObj, y::Any)::Any
 GAP.@wrap Value(x::GapObj, y::Any, z::Any)::Any
 GAP.@wrap Value(x::GapObj, y::GapObj, z::GapObj, a::Any)::Any
+GAP.@wrap ValuesOfClassFunction(x::GapObj)::GapObj
 GAP.@wrap Z(x::Any)::GAP.Obj
 GAP.@wrap Zero(x::Any)::GAP.Obj
 

src/Groups/MatrixDisplay.jl

@@ -419,8 +419,22 @@ function labelled_matrix_formatted(io::IO, mat::Matrix{String})
     end
 
     # footer (vector of strings)
-    for line in footer
-      write(io, line, "\n")
+    if length(footer) > 0
+      if TeX
+        write(io, "\n\\begin{array}{l}\n")
+        # Omit a separating empty line.
+        if footer[1] != ""
+          write(io, footer[1], " \\\\\n")
+        end
+        for line in footer[2:end]
+          write(io, line, " \\\\\n")
+        end
+        write(io, "\\end{array}\n")
+      else
+        for line in footer
+          write(io, line, "\n")
+        end
+      end
     end
 end