  
  [1X72 [33X[0;0YClass Functions[133X[101X
  
  [33X[0;0YThis  chapter describes operations for [13Xclass functions of finite groups[113X. For
  operations concerning [13Xcharacter tables[113X, see Chapter [14X71[114X.[133X
  
  [33X[0;0YSeveral  examples in this chapter require the [5XGAP[105X Character Table Library to
  be available. If it is not yet loaded then we load it now.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "ctbllib" );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X72.1 [33X[0;0YWhy Class Functions?[133X[101X
  
  [33X[0;0YIn  principle  it  is possible to represent group characters or more general
  class  functions  by  the  plain  lists  of  their  values, and in fact many
  operations  for  class  functions  work  with  plain lists of class function
  values. But this has two disadvantages.[133X
  
  [33X[0;0YFirst,  it  is  then necessary to regard a values list explicitly as a class
  function  of a particular character table, by supplying this character table
  as  an  argument.  In practice this means that with this setup, the user has
  the task to put the objects into the right context. For example, forming the
  scalar  product  or  the tensor product of two class functions or forming an
  induced  class  function  or  a  conjugate  class  function then needs three
  arguments  in  this  case;  this is particularly inconvenient in cases where
  infix  operations  cannot be used because of the additional argument, as for
  tensor products and induced class functions.[133X
  
  [33X[0;0YSecond,  when one says that [21X[22Xχ[122X is a character of a group [22XG[122X[121X then this object [22Xχ[122X
  carries  a  lot  of  information.  [22Xχ[122X  has  certain  properties such as being
  irreducible  or  not.  Several  subgroups of [22XG[122X are related to [22Xχ[122X, such as the
  kernel  and  the  centre  of  [22Xχ[122X.  Other  attributes  of  characters  are the
  determinant  and the central character. This knowledge cannot be stored in a
  plain list.[133X
  
  [33X[0;0YFor  dealing  with  a  group  together  with  its characters, and maybe also
  subgroups  and their characters, it is desirable that [5XGAP[105X keeps track of the
  interpretation  of  characters.  On  the  other  hand,  for using characters
  without  accessing  their  groups, such as characters of tables from the [5XGAP[105X
  table  library,  dealing  just  with  values  lists  is often sufficient. In
  particular,  if  one deals with incomplete character tables then it is often
  necessary to specify the arguments explicitly, for example one has to choose
  a fusion map or power map from a set of possibilities.[133X
  
  [33X[0;0YThe  main idea behind class function objects is that a class function object
  is  equal to its values list in the sense of [2X\=[102X ([14X31.11-1[114X), so class function
  objects  can  be used wherever their values lists can be used, but there are
  operations  for  class  function  objects  that do not work just with values
  lists.  [5XGAP[105X library functions prefer to return class function objects rather
  than  returning just values lists, for example [2XIrr[102X ([14X71.8-2[114X) lists consist of
  class  function  objects,  and  [2XTrivialCharacter[102X  ([14X72.7-1[114X)  returns  a class
  function object.[133X
  
  [33X[0;0YHere is an [13Xexample[113X that shows both approaches. First we define some groups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS4:= SymmetricGroup( 4 );;  SetName( S4, "S4" );[127X[104X
    [4X[25Xgap>[125X [27XD8:= SylowSubgroup( S4, 2 );; SetName( D8, "D8" );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YWe do some computations using the functions described later in this Chapter,
  first with class function objects.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XirrS4:= Irr( S4 );;[127X[104X
    [4X[25Xgap>[125X [27XirrD8:= Irr( D8 );;[127X[104X
    [4X[25Xgap>[125X [27Xchi:= irrD8[4];[127X[104X
    [4X[28XCharacter( CharacterTable( D8 ), [ 1, -1, 1, -1, 1 ] )[128X[104X
    [4X[25Xgap>[125X [27Xchi * chi;[127X[104X
    [4X[28XCharacter( CharacterTable( D8 ), [ 1, 1, 1, 1, 1 ] )[128X[104X
    [4X[25Xgap>[125X [27Xind:= chi ^ S4;[127X[104X
    [4X[28XCharacter( CharacterTable( S4 ), [ 3, -1, -1, 0, 1 ] )[128X[104X
    [4X[25Xgap>[125X [27XList( irrS4, x -> ScalarProduct( x, ind ) );[127X[104X
    [4X[28X[ 0, 1, 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27Xdet:= Determinant( ind );[127X[104X
    [4X[28XCharacter( CharacterTable( S4 ), [ 1, 1, 1, 1, 1 ] )[128X[104X
    [4X[25Xgap>[125X [27Xcent:= CentralCharacter( ind );[127X[104X
    [4X[28XClassFunction( CharacterTable( S4 ), [ 1, -2, -1, 0, 2 ] )[128X[104X
    [4X[25Xgap>[125X [27Xrest:= Restricted( cent, D8 );[127X[104X
    [4X[28XClassFunction( CharacterTable( D8 ), [ 1, -2, -1, -1, 2 ] )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we repeat these calculations with plain lists of character values. Here
  we need the character tables in some places.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XtS4:= CharacterTable( S4 );;[127X[104X
    [4X[25Xgap>[125X [27XtD8:= CharacterTable( D8 );;[127X[104X
    [4X[25Xgap>[125X [27Xchi:= ValuesOfClassFunction( irrD8[4] );[127X[104X
    [4X[28X[ 1, -1, 1, -1, 1 ][128X[104X
    [4X[25Xgap>[125X [27XTensored( [ chi ], [ chi ] )[1];[127X[104X
    [4X[28X[ 1, 1, 1, 1, 1 ][128X[104X
    [4X[25Xgap>[125X [27Xind:= InducedClassFunction( tD8, chi, tS4 );[127X[104X
    [4X[28XClassFunction( CharacterTable( S4 ), [ 3, -1, -1, 0, 1 ] )[128X[104X
    [4X[25Xgap>[125X [27XList( Irr( tS4 ), x -> ScalarProduct( tS4, x, ind ) );[127X[104X
    [4X[28X[ 0, 1, 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27Xdet:= DeterminantOfCharacter( tS4, ind );[127X[104X
    [4X[28XClassFunction( CharacterTable( S4 ), [ 1, 1, 1, 1, 1 ] )[128X[104X
    [4X[25Xgap>[125X [27Xcent:= CentralCharacter( tS4, ind );[127X[104X
    [4X[28XClassFunction( CharacterTable( S4 ), [ 1, -2, -1, 0, 2 ] )[128X[104X
    [4X[25Xgap>[125X [27Xrest:= Restricted( tS4, cent, tD8 );[127X[104X
    [4X[28XClassFunction( CharacterTable( D8 ), [ 1, -2, -1, -1, 2 ] )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIf  one  deals with character tables from the [5XGAP[105X table library then one has
  no  access  to their groups, but often the tables provide enough information
  for  computing  induced or restricted class functions, symmetrizations etc.,
  because  the  relevant  class  fusions  and  power  maps are often stored on
  library  tables.  In these cases it is possible to use the tables instead of
  the   groups  as  arguments.  (If  necessary  information  is  not  uniquely
  determined by the tables then an error is signalled.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs5 := CharacterTable( "A5.2" );; irrs5 := Irr( s5  );;[127X[104X
    [4X[25Xgap>[125X [27Xm11:= CharacterTable( "M11"  );; irrm11:= Irr( m11 );;[127X[104X
    [4X[25Xgap>[125X [27Xchi:= TrivialCharacter( s5 );[127X[104X
    [4X[28XCharacter( CharacterTable( "A5.2" ), [ 1, 1, 1, 1, 1, 1, 1 ] )[128X[104X
    [4X[25Xgap>[125X [27Xchi ^ m11;[127X[104X
    [4X[28XCharacter( CharacterTable( "M11" ), [ 66, 10, 3, 2, 1, 1, 0, 0, 0, 0 [128X[104X
    [4X[28X ] )[128X[104X
    [4X[25Xgap>[125X [27XDeterminant( irrs5[4] );[127X[104X
    [4X[28XCharacter( CharacterTable( "A5.2" ), [ 1, 1, 1, 1, -1, -1, -1 ] )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFunctions   that   compute  [13Xnormal[113X  subgroups  related  to  characters  have
  counterparts  that return the list of class positions corresponding to these
  groups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XClassPositionsOfKernel( irrs5[2] );[127X[104X
    [4X[28X[ 1, 2, 3, 4 ][128X[104X
    [4X[25Xgap>[125X [27XClassPositionsOfCentre( irrs5[2] );[127X[104X
    [4X[28X[ 1, 2, 3, 4, 5, 6, 7 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNon-normal  subgroups  cannot  be described this way, so for example inertia
  subgroups  (see [2XInertiaSubgroup[102X  ([14X72.8-13[114X))  can  in general not be computed
  from character tables without access to their groups.[133X
  
  [1X72.1-1 IsClassFunction[101X
  
  [29X[2XIsClassFunction[102X( [3Xobj[103X ) [32X Category
  
  [33X[0;0YA [13Xclass function[113X (in characteristic [22Xp[122X) of a finite group [22XG[122X is a map from the
  set  of  ([22Xp[122X-regular)  elements  in  [22XG[122X  to  the  field of cyclotomics that is
  constant on conjugacy classes of [22XG[122X.[133X
  
  [33X[0;0YEach class function in [5XGAP[105X is represented by an [13Ximmutable list[113X, where at the
  [22Xi[122X-th  position  the value on the [22Xi[122X-th conjugacy class of the character table
  of [22XG[122X is stored. The ordering of the conjugacy classes is the one used in the
  underlying  character  table. Note that if the character table has access to
  its underlying group then the ordering of conjugacy classes in the group and
  in  the  character table may differ (see [14X71.6[114X); class functions always refer
  to the ordering of classes in the character table.[133X
  
  [33X[0;0Y[13XClass  function  objects[113X  in  [5XGAP[105X  are  not just plain lists, they store the
  character   table   of   the   group   [22XG[122X   as   value   of   the   attribute
  [2XUnderlyingCharacterTable[102X ([14X72.2-1[114X). The group [22XG[122X itself is accessible only via
  the  character  table and thus only if the character table stores its group,
  as  value  of the attribute [2XUnderlyingGroup[102X ([14X71.6-1[114X). The reason for this is
  that many computations with class functions are possible without using their
  groups, for example class functions of character tables in the [5XGAP[105X character
  table library do in general not have access to their underlying groups.[133X
  
  [33X[0;0YThere  are  (at  least)  two  reasons  why  class  functions  in [5XGAP[105X are [13Xnot[113X
  implemented  as  mappings.  First, we want to distinguish class functions in
  different  characteristics,  for  example to be able to define the Frobenius
  character  of  a  given  Brauer  character;  viewed as mappings, the trivial
  characters  in  all  characteristics  coprime  to  the order of [22XG[122X are equal.
  Second,  the product of two class functions shall be again a class function,
  whereas the product of general mappings is defined as composition.[133X
  
  [33X[0;0YA further argument is that the typical operations for mappings such as [2XImage[102X
  ([14X32.4-6[114X) and [2XPreImage[102X ([14X32.5-6[114X) play no important role for class functions.[133X
  
  
  [1X72.2 [33X[0;0YBasic Operations for Class Functions[133X[101X
  
  [33X[0;0YBasic  operations for class functions are [2XUnderlyingCharacterTable[102X ([14X72.2-1[114X),
  [2XValuesOfClassFunction[102X   ([14X72.2-2[114X),   and   the  basic  operations  for  lists
  (see [14X21.2[114X).[133X
  
  [1X72.2-1 UnderlyingCharacterTable[101X
  
  [29X[2XUnderlyingCharacterTable[102X( [3Xpsi[103X ) [32X attribute
  
  [33X[0;0YFor  a  class  function [3Xpsi[103X of the group [22XG[122X, say, the character table of [22XG[122X is
  stored  as value of [2XUnderlyingCharacterTable[102X. The ordering of entries in the
  list  [3Xpsi[103X  (see [2XValuesOfClassFunction[102X  ([14X72.2-2[114X))  refers  to the ordering of
  conjugacy classes in this character table.[133X
  
  [33X[0;0YIf  [3Xpsi[103X is an ordinary class function then the underlying character table is
  the  ordinary character table of [22XG[122X (see [2XOrdinaryCharacterTable[102X ([14X71.8-4[114X)), if
  [3Xpsi[103X  is  a  class  function  in  characteristic  [22Xp  ≠  0[122X then the underlying
  character  table  is  the  [22Xp[122X-modular  Brauer  table  of  [22XG[122X  (see [2XBrauerTable[102X
  ([14X71.3-2[114X)).  So the underlying characteristic of [3Xpsi[103X can be read off from the
  underlying character table.[133X
  
  [1X72.2-2 ValuesOfClassFunction[101X
  
  [29X[2XValuesOfClassFunction[102X( [3Xpsi[103X ) [32X attribute
  
  [33X[0;0Yis  the  list  of values of the class function [3Xpsi[103X, the [22Xi[122X-th entry being the
  value  on  the  [22Xi[122X-th  conjugacy  class  of  the  underlying  character table
  (see [2XUnderlyingCharacterTable[102X ([14X72.2-1[114X)).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= SymmetricGroup( 4 );[127X[104X
    [4X[28XSym( [ 1 .. 4 ] )[128X[104X
    [4X[25Xgap>[125X [27Xpsi:= TrivialCharacter( g );[127X[104X
    [4X[28XCharacter( CharacterTable( Sym( [ 1 .. 4 ] ) ), [ 1, 1, 1, 1, 1 ] )[128X[104X
    [4X[25Xgap>[125X [27XUnderlyingCharacterTable( psi );[127X[104X
    [4X[28XCharacterTable( Sym( [ 1 .. 4 ] ) )[128X[104X
    [4X[25Xgap>[125X [27XValuesOfClassFunction( psi );[127X[104X
    [4X[28X[ 1, 1, 1, 1, 1 ][128X[104X
    [4X[25Xgap>[125X [27XIsList( psi );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xpsi[1];[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27XLength( psi );[127X[104X
    [4X[28X5[128X[104X
    [4X[25Xgap>[125X [27XIsBound( psi[6] );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XConcatenation( psi, [ 2, 3 ] );[127X[104X
    [4X[28X[ 1, 1, 1, 1, 1, 2, 3 ][128X[104X
  [4X[32X[104X
  
  
  [1X72.3 [33X[0;0YComparison of Class Functions[133X[101X
  
  [33X[0;0YWith  respect  to  [2X\=[102X  ([14X31.11-1[114X)  and  [2X\<[102X  ([14X31.11-1[114X), class functions behave
  equally  to  their  lists of values (see [2XValuesOfClassFunction[102X ([14X72.2-2[114X)). So
  two  class  functions  are  equal  if  and only if their lists of values are
  equal,  no  matter  whether  they  are class functions of the same character
  table,  of  the  same  group  but  w.r.t. different  class  ordering,  or of
  different groups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgrps:= Filtered( AllSmallGroups( 8 ), g -> not IsAbelian( g ) );[127X[104X
    [4X[28X[ <pc group of size 8 with 3 generators>, [128X[104X
    [4X[28X  <pc group of size 8 with 3 generators> ][128X[104X
    [4X[25Xgap>[125X [27Xt1:= CharacterTable( grps[1] );  SetName( t1, "t1" );[127X[104X
    [4X[28XCharacterTable( <pc group of size 8 with 3 generators> )[128X[104X
    [4X[25Xgap>[125X [27Xt2:= CharacterTable( grps[2] );  SetName( t2, "t2" );[127X[104X
    [4X[28XCharacterTable( <pc group of size 8 with 3 generators> )[128X[104X
    [4X[25Xgap>[125X [27Xirr1:= Irr( grps[1] );[127X[104X
    [4X[28X[ Character( t1, [ 1, 1, 1, 1, 1 ] ), [128X[104X
    [4X[28X  Character( t1, [ 1, -1, -1, 1, 1 ] ), [128X[104X
    [4X[28X  Character( t1, [ 1, -1, 1, 1, -1 ] ), [128X[104X
    [4X[28X  Character( t1, [ 1, 1, -1, 1, -1 ] ), [128X[104X
    [4X[28X  Character( t1, [ 2, 0, 0, -2, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27Xirr2:= Irr( grps[2] );[127X[104X
    [4X[28X[ Character( t2, [ 1, 1, 1, 1, 1 ] ), [128X[104X
    [4X[28X  Character( t2, [ 1, -1, -1, 1, 1 ] ), [128X[104X
    [4X[28X  Character( t2, [ 1, -1, 1, 1, -1 ] ), [128X[104X
    [4X[28X  Character( t2, [ 1, 1, -1, 1, -1 ] ), [128X[104X
    [4X[28X  Character( t2, [ 2, 0, 0, -2, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27Xirr1 = irr2;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsSSortedList( irr1 );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xirr1[1] < irr1[2];[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xirr1[2] < irr1[3];[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X72.4 [33X[0;0YArithmetic Operations for Class Functions[133X[101X
  
  [33X[0;0YClass  functions  are  [13Xrow vectors[113X of cyclotomics. The [13Xadditive[113X behaviour of
  class  functions  is  defined such that they are equal to the plain lists of
  class function values except that the results are represented again as class
  functions  whenever this makes sense. The [13Xmultiplicative[113X behaviour, however,
  is different. This is motivated by the fact that the tensor product of class
  functions  is  a more interesting operation than the vector product of plain
  lists. (Another candidate for a multiplication of compatible class functions
  would  have  been  the  inner product, which is implemented via the function
  [2XScalarProduct[102X  ([14X72.8-5[114X).  In  terms  of  filters,  the  arithmetic  of class
  functions  is  based on the decision that they lie in [2XIsGeneralizedRowVector[102X
  ([14X21.12-1[114X),   with  additive  nesting  depth  [22X1[122X,  but  they  do  [13Xnot[113X  lie  in
  [2XIsMultiplicativeGeneralizedRowVector[102X ([14X21.12-2[114X).[133X
  
  [33X[0;0YMore specifically, the scalar multiple of a class function with a cyclotomic
  is  a  class function, and the sum and the difference of two class functions
  of   the   same  underlying  character  table  (see [2XUnderlyingCharacterTable[102X
  ([14X72.2-1[114X))  are  again  class  functions  of  this  table.  The  sum  and the
  difference  of  a class function and a list that is [13Xnot[113X a class function are
  plain lists, as well as the sum and the difference of two class functions of
  different character tables.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= SymmetricGroup( 4 );;  tbl:= CharacterTable( g );;[127X[104X
    [4X[25Xgap>[125X [27XSetName( tbl, "S4" );  irr:= Irr( g );[127X[104X
    [4X[28X[ Character( S4, [ 1, -1, 1, 1, -1 ] ), [128X[104X
    [4X[28X  Character( S4, [ 3, -1, -1, 0, 1 ] ), [128X[104X
    [4X[28X  Character( S4, [ 2, 0, 2, -1, 0 ] ), [128X[104X
    [4X[28X  Character( S4, [ 3, 1, -1, 0, -1 ] ), [128X[104X
    [4X[28X  Character( S4, [ 1, 1, 1, 1, 1 ] ) ][128X[104X
    [4X[25Xgap>[125X [27X2 * irr[5];[127X[104X
    [4X[28XCharacter( S4, [ 2, 2, 2, 2, 2 ] )[128X[104X
    [4X[25Xgap>[125X [27Xirr[1] / 7;[127X[104X
    [4X[28XClassFunction( S4, [ 1/7, -1/7, 1/7, 1/7, -1/7 ] )[128X[104X
    [4X[25Xgap>[125X [27Xlincomb:= irr[3] + irr[1] - irr[5];[127X[104X
    [4X[28XVirtualCharacter( S4, [ 2, -2, 2, -1, -2 ] )[128X[104X
    [4X[25Xgap>[125X [27Xlincomb:= lincomb + 2 * irr[5];[127X[104X
    [4X[28XVirtualCharacter( S4, [ 4, 0, 4, 1, 0 ] )[128X[104X
    [4X[25Xgap>[125X [27XIsCharacter( lincomb );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xlincomb;[127X[104X
    [4X[28XCharacter( S4, [ 4, 0, 4, 1, 0 ] )[128X[104X
    [4X[25Xgap>[125X [27Xirr[5] + 2;[127X[104X
    [4X[28X[ 3, 3, 3, 3, 3 ][128X[104X
    [4X[25Xgap>[125X [27Xirr[5] + [ 1, 2, 3, 4, 5 ];[127X[104X
    [4X[28X[ 2, 3, 4, 5, 6 ][128X[104X
    [4X[25Xgap>[125X [27Xzero:= 0 * irr[1];[127X[104X
    [4X[28XVirtualCharacter( S4, [ 0, 0, 0, 0, 0 ] )[128X[104X
    [4X[25Xgap>[125X [27Xzero + Z(3);[127X[104X
    [4X[28X[ Z(3), Z(3), Z(3), Z(3), Z(3) ][128X[104X
    [4X[25Xgap>[125X [27Xirr[5] + TrivialCharacter( DihedralGroup( 8 ) );[127X[104X
    [4X[28X[ 2, 2, 2, 2, 2 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe product of two class functions of the same character table is the tensor
  product  (pointwise  product)  of these class functions. Thus the set of all
  class  functions  of  a  fixed  group  forms  a ring, and for any field [22XF[122X of
  cyclotomics, the [22XF[122X-span of a given set of class functions forms an algebra.[133X
  
  [33X[0;0YThe  product of two class functions of [13Xdifferent[113X tables and the product of a
  class  function  and a list that is [13Xnot[113X a class function are not defined, an
  error  is  signalled  in  these  cases.  Note  that  in  this respect, class
  functions  behave differently from their values lists, for which the product
  is defined as the standard scalar product.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtens:= irr[3] * irr[4];[127X[104X
    [4X[28XCharacter( S4, [ 6, 0, -2, 0, 0 ] )[128X[104X
    [4X[25Xgap>[125X [27XValuesOfClassFunction( irr[3] ) * ValuesOfClassFunction( irr[4] );[127X[104X
    [4X[28X4[128X[104X
  [4X[32X[104X
  
  [33X[0;0YClass  functions  without zero values are invertible, the [13Xinverse[113X is defined
  pointwise.  As a consequence, for example groups of linear characters can be
  formed.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtens / irr[1];[127X[104X
    [4X[28XCharacter( S4, [ 6, 0, -2, 0, 0 ] )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YOther  (somewhat  strange)  implications  of  the  definition  of arithmetic
  operations  for  class  functions,  together  with the general rules of list
  arithmetic  (see [14X21.11[114X),  apply  to  the case of products involving [13Xlists[113X of
  class  functions.  No  inverse  of  the  list of irreducible characters as a
  matrix  is  defined; if one is interested in the inverse matrix then one can
  compute it from the matrix of class function values.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XInverse( List( irr, ValuesOfClassFunction ) );[127X[104X
    [4X[28X[ [ 1/24, 1/8, 1/12, 1/8, 1/24 ], [ -1/4, -1/4, 0, 1/4, 1/4 ], [128X[104X
    [4X[28X  [ 1/8, -1/8, 1/4, -1/8, 1/8 ], [ 1/3, 0, -1/3, 0, 1/3 ], [128X[104X
    [4X[28X  [ -1/4, 1/4, 0, -1/4, 1/4 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YAlso the product of a class function with a list of class functions is [13Xnot[113X a
  vector-matrix product but the list of pointwise products.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xirr[1] * irr{ [ 1 .. 3 ] };[127X[104X
    [4X[28X[ Character( S4, [ 1, 1, 1, 1, 1 ] ), [128X[104X
    [4X[28X  Character( S4, [ 3, 1, -1, 0, -1 ] ), [128X[104X
    [4X[28X  Character( S4, [ 2, 0, 2, -1, 0 ] ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YAnd  the  product  of two lists of class functions is [13Xnot[113X the matrix product
  but the sum of the pointwise products.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xirr * irr;[127X[104X
    [4X[28XCharacter( S4, [ 24, 4, 8, 3, 4 ] )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe [13Xpowering[113X operator [2X\^[102X ([14X31.12-1[114X) has several meanings for class functions.
  The power of a class function by a nonnegative integer is clearly the tensor
  power.  The  power  of  a  class  function by an element that normalizes the
  underlying  group  or  by  a  Galois  automorphism  is  the  conjugate class
  function.  (As  a consequence, the application of the permutation induced by
  such  an  action  cannot  be  denoted  by  [2X\^[102X ([14X31.12-1[114X); instead one can use
  [2XPermuted[102X  ([14X21.20-18[114X).)  The  power  of  a  class  function  by  a group or a
  character  table  is  the  induced  class function (see [2XInducedClassFunction[102X
  ([14X72.9-3[114X)).  The  power  of  a group element by a class function is the class
  function value at (the conjugacy class containing) this element.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xirr[3] ^ 3;[127X[104X
    [4X[28XCharacter( S4, [ 8, 0, 8, -1, 0 ] )[128X[104X
    [4X[25Xgap>[125X [27Xlin:= LinearCharacters( DerivedSubgroup( g ) );[127X[104X
    [4X[28X[ Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, 1, 1, 1 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( Alt( [ 1 .. 4 ] ) ),[128X[104X
    [4X[28X  [ 1, 1, E(3), E(3)^2 ] ),[128X[104X
    [4X[28X  Character( CharacterTable( Alt( [ 1 .. 4 ] ) ),[128X[104X
    [4X[28X  [ 1, 1, E(3)^2, E(3) ] ) ][128X[104X
    [4X[25Xgap>[125X [27XList( lin, chi -> chi ^ (1,2) );[127X[104X
    [4X[28X[ Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, 1, 1, 1 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( Alt( [ 1 .. 4 ] ) ),[128X[104X
    [4X[28X  [ 1, 1, E(3)^2, E(3) ] ),[128X[104X
    [4X[28X  Character( CharacterTable( Alt( [ 1 .. 4 ] ) ),[128X[104X
    [4X[28X  [ 1, 1, E(3), E(3)^2 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XOrbit( GaloisGroup( CF(3) ), lin[2] );[127X[104X
    [4X[28X[ Character( CharacterTable( Alt( [ 1 .. 4 ] ) ),[128X[104X
    [4X[28X  [ 1, 1, E(3), E(3)^2 ] ),[128X[104X
    [4X[28X  Character( CharacterTable( Alt( [ 1 .. 4 ] ) ),[128X[104X
    [4X[28X  [ 1, 1, E(3)^2, E(3) ] ) ][128X[104X
    [4X[25Xgap>[125X [27Xlin[1]^g;[127X[104X
    [4X[28XCharacter( S4, [ 2, 0, 2, 2, 0 ] )[128X[104X
    [4X[25Xgap>[125X [27X(1,2,3)^lin[2];[127X[104X
    [4X[28XE(3)[128X[104X
  [4X[32X[104X
  
  [1X72.4-1 Characteristic[101X
  
  [29X[2XCharacteristic[102X( [3Xchi[103X ) [32X attribute
  
  [33X[0;0YThe  [13Xcharacteristic[113X  of  class  functions  is  zero,  as  for  all  list  of
  cyclotomics.  For  class  functions  of a [22Xp[122X-modular character table, such as
  Brauer  characters,  the  prime  [22Xp[122X  is given by the [2XUnderlyingCharacteristic[102X
  ([14X71.9-5[114X) value of the character table.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCharacteristic( irr[1] );[127X[104X
    [4X[28X0[128X[104X
    [4X[25Xgap>[125X [27Xirrmod2:= Irr( g, 2 );[127X[104X
    [4X[28X[ Character( BrauerTable( Sym( [ 1 .. 4 ] ), 2 ), [ 1, 1 ] ), [128X[104X
    [4X[28X  Character( BrauerTable( Sym( [ 1 .. 4 ] ), 2 ), [ 2, -1 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XCharacteristic( irrmod2[1] );[127X[104X
    [4X[28X0[128X[104X
    [4X[25Xgap>[125X [27XUnderlyingCharacteristic( UnderlyingCharacterTable( irrmod2[1] ) );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [1X72.4-2 ComplexConjugate[101X
  
  [29X[2XComplexConjugate[102X( [3Xchi[103X ) [32X attribute
  [29X[2XGaloisCyc[102X( [3Xchi[103X, [3Xk[103X ) [32X operation
  [29X[2XPermuted[102X( [3Xchi[103X, [3Xpi[103X ) [32X method
  
  [33X[0;0YThe  operations  [2XComplexConjugate[102X,  [2XGaloisCyc[102X,  and  [2XPermuted[102X return a class
  function  when  they are called with a class function; The complex conjugate
  of a class function that is known to be a (virtual) character is again known
  to  be  a (virtual) character, and applying an arbitrary Galois automorphism
  to an ordinary (virtual) character yields a (virtual) character.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XComplexConjugate( lin[2] );[127X[104X
    [4X[28XCharacter( CharacterTable( Alt( [ 1 .. 4 ] ) ),[128X[104X
    [4X[28X [ 1, 1, E(3)^2, E(3) ] )[128X[104X
    [4X[25Xgap>[125X [27XGaloisCyc( lin[2], 5 );[127X[104X
    [4X[28XCharacter( CharacterTable( Alt( [ 1 .. 4 ] ) ),[128X[104X
    [4X[28X [ 1, 1, E(3)^2, E(3) ] )[128X[104X
    [4X[25Xgap>[125X [27XPermuted( lin[2], (2,3,4) );[127X[104X
    [4X[28XClassFunction( CharacterTable( Alt( [ 1 .. 4 ] ) ),[128X[104X
    [4X[28X [ 1, E(3)^2, 1, E(3) ] )[128X[104X
  [4X[32X[104X
  
  [1X72.4-3 Order[101X
  
  [29X[2XOrder[102X( [3Xchi[103X ) [32X attribute
  
  [33X[0;0YBy  definition of [2XOrder[102X ([14X31.10-10[114X) for arbitrary monoid elements, the return
  value  of [2XOrder[102X ([14X31.10-10[114X) for a character must be its multiplicative order.
  The   [13Xdeterminantal   order[113X   (see [2XDeterminantOfCharacter[102X  ([14X72.8-18[114X))  of  a
  character [3Xchi[103X can be computed as [10XOrder( Determinant( [3Xchi[103X[10X ) )[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xdet:= Determinant( irr[3] );[127X[104X
    [4X[28XCharacter( S4, [ 1, -1, 1, 1, -1 ] )[128X[104X
    [4X[25Xgap>[125X [27XOrder( det );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  
  [1X72.5 [33X[0;0YPrinting Class Functions[133X[101X
  
  [1X72.5-1 ViewObj[101X
  
  [29X[2XViewObj[102X( [3Xchi[103X ) [32X method
  
  [33X[0;0YThe  default  [2XViewObj[102X  ([14X6.3-5[114X)  methods for class functions print one of the
  strings   [10X"ClassFunction"[110X,  [10X"VirtualCharacter"[110X,  [10X"Character"[110X  (depending  on
  whether  the class function is known to be a character or virtual character,
  see [2XIsCharacter[102X  ([14X72.8-1[114X),  [2XIsVirtualCharacter[102X  ([14X72.8-2[114X)),  followed  by the
  [2XViewObj[102X  ([14X6.3-5[114X)  output for the underlying character table (see [14X71.13[114X), and
  the  list  of  values.  The  table is chosen (and not the group) in order to
  distinguish   class   functions   of   different  underlying  characteristic
  (see [2XUnderlyingCharacteristic[102X ([14X71.9-5[114X)).[133X
  
  [1X72.5-2 PrintObj[101X
  
  [29X[2XPrintObj[102X( [3Xchi[103X ) [32X method
  
  [33X[0;0YThe  default  [2XPrintObj[102X  ([14X6.3-5[114X)  method for class functions does the same as
  [2XViewObj[102X  ([14X6.3-5[114X),  except  that  the  character table is is [2XPrint[102X ([14X6.3-4[114X)-ed
  instead of [2XView[102X ([14X6.3-3[114X)-ed.[133X
  
  [33X[0;0Y[13XNote[113X  that  if  a  class  function  is  shown  only  with one of the strings
  [10X"ClassFunction"[110X,  [10X"VirtualCharacter"[110X,  it  may still be that it is in fact a
  character;  just  this was not known at the time when the class function was
  printed.[133X
  
  [33X[0;0YIn  order  to  reduce the space that is needed to print a class function, it
  may be useful to give a name (see [2XName[102X ([14X12.8-2[114X)) to the underlying character
  table.[133X
  
  [1X72.5-3 Display[101X
  
  [29X[2XDisplay[102X( [3Xchi[103X ) [32X method
  
  [33X[0;0YThe  default  [2XDisplay[102X  ([14X6.3-6[114X) method for a class function [3Xchi[103X calls [2XDisplay[102X
  ([14X6.3-6[114X) for its underlying character table (see [14X71.13[114X), with [3Xchi[103X as the only
  entry in the [10Xchars[110X list of the options record.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xchi:= TrivialCharacter( CharacterTable( "A5" ) );[127X[104X
    [4X[28XCharacter( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] )[128X[104X
    [4X[25Xgap>[125X [27XDisplay( chi );[127X[104X
    [4X[28XA5[128X[104X
    [4X[28X[128X[104X
    [4X[28X     2  2  2  .  .  .[128X[104X
    [4X[28X     3  1  .  1  .  .[128X[104X
    [4X[28X     5  1  .  .  1  1[128X[104X
    [4X[28X[128X[104X
    [4X[28X       1a 2a 3a 5a 5b[128X[104X
    [4X[28X    2P 1a 1a 3a 5b 5a[128X[104X
    [4X[28X    3P 1a 2a 1a 5b 5a[128X[104X
    [4X[28X    5P 1a 2a 3a 1a 1a[128X[104X
    [4X[28X[128X[104X
    [4X[28XY.1     1  1  1  1  1[128X[104X
  [4X[32X[104X
  
  
  [1X72.6 [33X[0;0YCreating Class Functions from Values Lists[133X[101X
  
  [1X72.6-1 ClassFunction[101X
  
  [29X[2XClassFunction[102X( [3Xtbl[103X, [3Xvalues[103X ) [32X operation
  [29X[2XClassFunction[102X( [3XG[103X, [3Xvalues[103X ) [32X operation
  
  [33X[0;0YIn the first form, [2XClassFunction[102X returns the class function of the character
  table [3Xtbl[103X with values given by the list [3Xvalues[103X of cyclotomics. In the second
  form,  [3XG[103X  must  be a group, and the class function of its ordinary character
  table is returned.[133X
  
  [33X[0;0YNote that [3Xtbl[103X determines the underlying characteristic of the returned class
  function (see [2XUnderlyingCharacteristic[102X ([14X71.9-5[114X)).[133X
  
  [1X72.6-2 VirtualCharacter[101X
  
  [29X[2XVirtualCharacter[102X( [3Xtbl[103X, [3Xvalues[103X ) [32X operation
  [29X[2XVirtualCharacter[102X( [3XG[103X, [3Xvalues[103X ) [32X operation
  
  [33X[0;0Y[2XVirtualCharacter[102X   returns  the  virtual  character  (see [2XIsVirtualCharacter[102X
  ([14X72.8-2[114X))  of  the  character  table  [3Xtbl[103X or the group [3XG[103X, respectively, with
  values given by the list [3Xvalues[103X.[133X
  
  [33X[0;0YIt  is  [13Xnot[113X  checked  whether  the  given  values  really describe a virtual
  character.[133X
  
  [1X72.6-3 Character[101X
  
  [29X[2XCharacter[102X( [3Xtbl[103X, [3Xvalues[103X ) [32X operation
  [29X[2XCharacter[102X( [3XG[103X, [3Xvalues[103X ) [32X operation
  
  [33X[0;0Y[2XCharacter[102X  returns the character (see [2XIsCharacter[102X ([14X72.8-1[114X)) of the character
  table  [3Xtbl[103X  or  the  group  [3XG[103X,  respectively,  with values given by the list
  [3Xvalues[103X.[133X
  
  [33X[0;0YIt is [13Xnot[113X checked whether the given values really describe a character.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= DihedralGroup( 8 );  tbl:= CharacterTable( g );[127X[104X
    [4X[28X<pc group of size 8 with 3 generators>[128X[104X
    [4X[28XCharacterTable( <pc group of size 8 with 3 generators> )[128X[104X
    [4X[25Xgap>[125X [27XSetName( tbl, "D8" );[127X[104X
    [4X[25Xgap>[125X [27Xphi:= ClassFunction( g, [ 1, -1, 0, 2, -2 ] );[127X[104X
    [4X[28XClassFunction( D8, [ 1, -1, 0, 2, -2 ] )[128X[104X
    [4X[25Xgap>[125X [27Xpsi:= ClassFunction( tbl,[127X[104X
    [4X[25X>[125X [27X             List( Irr( g ), chi -> ScalarProduct( chi, phi ) ) );[127X[104X
    [4X[28XClassFunction( D8, [ -3/8, 9/8, 5/8, 1/8, -1/4 ] )[128X[104X
    [4X[25Xgap>[125X [27Xchi:= VirtualCharacter( g, [ 0, 0, 8, 0, 0 ] );[127X[104X
    [4X[28XVirtualCharacter( D8, [ 0, 0, 8, 0, 0 ] )[128X[104X
    [4X[25Xgap>[125X [27Xreg:= Character( tbl, [ 8, 0, 0, 0, 0 ] );[127X[104X
    [4X[28XCharacter( D8, [ 8, 0, 0, 0, 0 ] )[128X[104X
  [4X[32X[104X
  
  [1X72.6-4 ClassFunctionSameType[101X
  
  [29X[2XClassFunctionSameType[102X( [3Xtbl[103X, [3Xchi[103X, [3Xvalues[103X ) [32X function
  
  [33X[0;0YLet [3Xtbl[103X be a character table, [3Xchi[103X a class function object ([13Xnot[113X necessarily a
  class    function   of   [3Xtbl[103X),   and   [3Xvalues[103X   a   list   of   cyclotomics.
  [2XClassFunctionSameType[102X  returns  the class function [22Xψ[122X of [3Xtbl[103X with values list
  [3Xvalues[103X, constructed with [2XClassFunction[102X ([14X72.6-1[114X).[133X
  
  [33X[0;0YIf  [3Xchi[103X  is  known  to be a (virtual) character then [22Xψ[122X is also known to be a
  (virtual) character.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xh:= Centre( g );;[127X[104X
    [4X[25Xgap>[125X [27Xcentbl:= CharacterTable( h );;  SetName( centbl, "C2" );[127X[104X
    [4X[25Xgap>[125X [27XClassFunctionSameType( centbl, phi, [ 1, 1 ] );[127X[104X
    [4X[28XClassFunction( C2, [ 1, 1 ] )[128X[104X
    [4X[25Xgap>[125X [27XClassFunctionSameType( centbl, chi, [ 1, 1 ] );[127X[104X
    [4X[28XVirtualCharacter( C2, [ 1, 1 ] )[128X[104X
    [4X[25Xgap>[125X [27XClassFunctionSameType( centbl, reg, [ 1, 1 ] );[127X[104X
    [4X[28XCharacter( C2, [ 1, 1 ] )[128X[104X
  [4X[32X[104X
  
  
  [1X72.7 [33X[0;0YCreating Class Functions using Groups[133X[101X
  
  
  [1X72.7-1 [33X[0;0YTrivialCharacter[133X[101X
  
  [29X[2XTrivialCharacter[102X( [3Xtbl[103X ) [32X attribute
  [29X[2XTrivialCharacter[102X( [3XG[103X ) [32X attribute
  
  [33X[0;0Yis  the  [13Xtrivial  character[113X  of  the  group  [3XG[103X  or  its character table [3Xtbl[103X,
  respectively.  This  is  the  class  function with value equal to [22X1[122X for each
  class.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XTrivialCharacter( CharacterTable( "A5" ) );[127X[104X
    [4X[28XCharacter( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] )[128X[104X
    [4X[25Xgap>[125X [27XTrivialCharacter( SymmetricGroup( 3 ) );[127X[104X
    [4X[28XCharacter( CharacterTable( Sym( [ 1 .. 3 ] ) ), [ 1, 1, 1 ] )[128X[104X
  [4X[32X[104X
  
  [1X72.7-2 NaturalCharacter[101X
  
  [29X[2XNaturalCharacter[102X( [3XG[103X ) [32X attribute
  [29X[2XNaturalCharacter[102X( [3Xhom[103X ) [32X attribute
  
  [33X[0;0YIf  the  argument is a permutation group [3XG[103X then [2XNaturalCharacter[102X returns the
  (ordinary)  character  of the natural permutation representation of [3XG[103X on the
  set  of  moved points (see [2XMovedPoints[102X ([14X42.3-3[114X)), that is, the value on each
  class  is the number of points among the moved points of [3XG[103X that are fixed by
  any permutation in that class.[133X
  
  [33X[0;0YIf   the   argument  is  a  matrix  group  [3XG[103X  in  characteristic  zero  then
  [2XNaturalCharacter[102X  returns  the  (ordinary)  character  of the natural matrix
  representation  of  [3XG[103X,  that is, the value on each class is the trace of any
  matrix in that class.[133X
  
  [33X[0;0YIf  the  argument  is  a group homomorphism [3Xhom[103X whose image is a permutation
  group or a matrix group then [2XNaturalCharacter[102X returns the restriction of the
  natural character of the image of [3Xhom[103X to the preimage of [3Xhom[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNaturalCharacter( SymmetricGroup( 3 ) );[127X[104X
    [4X[28XCharacter( CharacterTable( Sym( [ 1 .. 3 ] ) ), [ 3, 1, 0 ] )[128X[104X
    [4X[25Xgap>[125X [27XNaturalCharacter( Group( [ [ 0, -1 ], [ 1, -1 ] ] ) );[127X[104X
    [4X[28XCharacter( CharacterTable( Group([ [ [ 0, -1 ], [ 1, -1 ] ] ]) ), [128X[104X
    [4X[28X[ 2, -1, -1 ] )[128X[104X
    [4X[25Xgap>[125X [27Xd8:= DihedralGroup( 8 );;  hom:= IsomorphismPermGroup( d8 );;[127X[104X
    [4X[25Xgap>[125X [27XNaturalCharacter( hom );[127X[104X
    [4X[28XCharacter( CharacterTable( <pc group of size 8 with 3 generators> ), [128X[104X
    [4X[28X[ 8, 0, 0, 0, 0 ] )[128X[104X
  [4X[32X[104X
  
  
  [1X72.7-3 [33X[0;0YPermutationCharacter[133X[101X
  
  [29X[2XPermutationCharacter[102X( [3XG[103X, [3XD[103X, [3Xopr[103X ) [32X operation
  [29X[2XPermutationCharacter[102X( [3XG[103X, [3XU[103X ) [32X operation
  
  [33X[0;0YCalled  with  a  group  [3XG[103X,  an  action domain or proper set [3XD[103X, and an action
  function  [3Xopr[103X (see Chapter [14X41[114X), [2XPermutationCharacter[102X returns the [13Xpermutation
  character[113X  of the action of [3XG[103X on [3XD[103X via [3Xopr[103X, that is, the value on each class
  is  the  number  of  points  in [3XD[103X that are fixed by an element in this class
  under the action [3Xopr[103X.[133X
  
  [33X[0;0YIf   the   arguments   are   a   group   [3XG[103X  and  a  subgroup  [3XU[103X  of  [3XG[103X  then
  [2XPermutationCharacter[102X returns the permutation character of the action of [3XG[103X on
  the right cosets of [3XU[103X via right multiplication.[133X
  
  [33X[0;0YTo  compute the permutation character of a [13Xtransitive permutation group[113X [3XG[103X on
  the  cosets of a point stabilizer [3XU[103X, the attribute [2XNaturalCharacter[102X ([14X72.7-2[114X)
  of [3XG[103X can be used instead of [10XPermutationCharacter( [3XG[103X[10X, [3XU[103X[10X )[110X.[133X
  
  [33X[0;0YMore  facilities concerning permutation characters are the transitivity test
  (see  Section [14X72.8[114X)  and  several  tools  for computing possible permutation
  characters (see [14X72.13[114X, [14X72.14[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPermutationCharacter( GL(2,2), AsSSortedList( GF(2)^2 ), OnRight );[127X[104X
    [4X[28XCharacter( CharacterTable( SL(2,2) ), [ 4, 2, 1 ] )[128X[104X
    [4X[25Xgap>[125X [27Xs3:= SymmetricGroup( 3 );;  a3:= DerivedSubgroup( s3 );;[127X[104X
    [4X[25Xgap>[125X [27XPermutationCharacter( s3, a3 );[127X[104X
    [4X[28XCharacter( CharacterTable( Sym( [ 1 .. 3 ] ) ), [ 2, 0, 2 ] )[128X[104X
  [4X[32X[104X
  
  
  [1X72.8 [33X[0;0YOperations for Class Functions[133X[101X
  
  [33X[0;0YIn  the description of the following operations, the optional first argument
  [3Xtbl[103X  is  needed  only  if  the  argument [3Xchi[103X is a plain list and not a class
  function  object.  In  this  case, [3Xtbl[103X must always be the character table of
  which [3Xchi[103X shall be regarded as a class function.[133X
  
  [1X72.8-1 IsCharacter[101X
  
  [29X[2XIsCharacter[102X( [[3Xtbl[103X, ][3Xchi[103X ) [32X property
  
  [33X[0;0YAn [13Xordinary character[113X of a group [22XG[122X is a class function of [22XG[122X whose values are
  the traces of a complex matrix representation of [22XG[122X.[133X
  
  [33X[0;0YA  [13XBrauer  character[113X of [22XG[122X in characteristic [22Xp[122X is a class function of [22XG[122X whose
  values  are  the  complex lifts of a matrix representation of [22XG[122X with image a
  finite field of characteristic [22Xp[122X.[133X
  
  [1X72.8-2 IsVirtualCharacter[101X
  
  [29X[2XIsVirtualCharacter[102X( [[3Xtbl[103X, ][3Xchi[103X ) [32X property
  
  [33X[0;0YA  [13Xvirtual  character[113X  is  a  class  function  that  can  be  written as the
  difference of two proper characters (see [2XIsCharacter[102X ([14X72.8-1[114X)).[133X
  
  [1X72.8-3 IsIrreducibleCharacter[101X
  
  [29X[2XIsIrreducibleCharacter[102X( [[3Xtbl[103X, ][3Xchi[103X ) [32X property
  
  [33X[0;0YA  character  is  [13Xirreducible[113X  if  it  cannot  be  written as the sum of two
  characters.  For  ordinary  characters  this can be checked using the scalar
  product   of   class  functions  (see [2XScalarProduct[102X  ([14X72.8-5[114X)).  For  Brauer
  characters there is no generic method for checking irreducibility.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS4:= SymmetricGroup( 4 );;  SetName( S4, "S4" );[127X[104X
    [4X[25Xgap>[125X [27Xpsi:= ClassFunction( S4, [ 1, 1, 1, -2, 1 ] );[127X[104X
    [4X[28XClassFunction( CharacterTable( S4 ), [ 1, 1, 1, -2, 1 ] )[128X[104X
    [4X[25Xgap>[125X [27XIsVirtualCharacter( psi );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsCharacter( psi );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xchi:= ClassFunction( S4, SizesCentralizers( CharacterTable( S4 ) ) );[127X[104X
    [4X[28XClassFunction( CharacterTable( S4 ), [ 24, 4, 8, 3, 4 ] )[128X[104X
    [4X[25Xgap>[125X [27XIsCharacter( chi );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsIrreducibleCharacter( chi );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsIrreducibleCharacter( TrivialCharacter( S4 ) );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X72.8-4 DegreeOfCharacter[101X
  
  [29X[2XDegreeOfCharacter[102X( [3Xchi[103X ) [32X attribute
  
  [33X[0;0Yis  the value of the character [3Xchi[103X on the identity element. This can also be
  obtained as [3Xchi[103X[10X[1][110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XList( Irr( S4 ), DegreeOfCharacter );[127X[104X
    [4X[28X[ 1, 3, 2, 3, 1 ][128X[104X
    [4X[25Xgap>[125X [27Xnat:= NaturalCharacter( S4 );[127X[104X
    [4X[28XCharacter( CharacterTable( S4 ), [ 4, 2, 0, 1, 0 ] )[128X[104X
    [4X[25Xgap>[125X [27Xnat[1];[127X[104X
    [4X[28X4[128X[104X
  [4X[32X[104X
  
  [1X72.8-5 ScalarProduct[101X
  
  [29X[2XScalarProduct[102X( [[3Xtbl[103X, ][3Xchi[103X, [3Xpsi[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Ythe  scalar  product  of  the  class  functions [3Xchi[103X and [3Xpsi[103X, which
            belong to the same character table [3Xtbl[103X.[133X
  
  [33X[0;0YIf  [3Xchi[103X  and [3Xpsi[103X are class function objects, the argument [3Xtbl[103X is not needed,
  but [3Xtbl[103X is necessary if at least one of [3Xchi[103X, [3Xpsi[103X is just a plain list.[133X
  
  [33X[0;0YThe  scalar  product  of  two  [13Xordinary[113X class functions [22Xχ[122X, [22Xψ[122X of a group [22XG[122X is
  defined as[133X
  
  [33X[0;0Y[22X( ∑_{g ∈ G} χ(g) ψ(g^{-1}) ) / |G|[122X.[133X
  
  [33X[0;0YFor two [13X[22Xp[122X-modular[113X class functions, the scalar product is defined as [22X( ∑_{g ∈
  S} χ(g) ψ(g^{-1}) ) / |G|[122X, where [22XS[122X is the set of [22Xp[122X-regular elements in [22XG[122X.[133X
  
  [1X72.8-6 MatScalarProducts[101X
  
  [29X[2XMatScalarProducts[102X( [[3Xtbl[103X, ][3Xlist[103X[, [3Xlist2[103X] ) [32X operation
  
  [33X[0;0YCalled  with  two lists [3Xlist[103X, [3Xlist2[103X of class functions of the same character
  table  (which  may  be given as the argument [3Xtbl[103X), [2XMatScalarProducts[102X returns
  the  matrix  of scalar products (see [2XScalarProduct[102X ([14X72.8-5[114X)) More precisely,
  this matrix contains in the [22Xi[122X-th row the list of scalar products of [22X[3Xlist2[103X[i][122X
  with the entries of [3Xlist[103X.[133X
  
  [33X[0;0YIf  only  one  list [3Xlist[103X of class functions is given then a lower triangular
  matrix  of  scalar  products is returned, containing (for [22Xj ≤ i[122X) in the [22Xi[122X-th
  row in column [22Xj[122X the value [10XScalarProduct[110X[22X( [3Xtbl[103X, [3Xlist[103X[j], [3Xlist[103X[i] )[122X.[133X
  
  [1X72.8-7 Norm[101X
  
  [29X[2XNorm[102X( [[3Xtbl[103X, ][3Xchi[103X ) [32X attribute
  
  [33X[0;0YFor an ordinary class function [3Xchi[103X of the group [22XG[122X, say, we have [22X[3Xchi[103X = ∑_{χ ∈
  Irr(G)}  a_χ χ[122X, with complex coefficients [22Xa_χ[122X. The [13Xnorm[113X of [3Xchi[103X is defined as
  [22X∑_{χ ∈ Irr(G)} a_χ overline{a_χ}[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtbl:= CharacterTable( "A5" );;[127X[104X
    [4X[25Xgap>[125X [27XScalarProduct( TrivialCharacter( tbl ), Sum( Irr( tbl ) ) );[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27XScalarProduct( tbl, [ 1, 1, 1, 1, 1 ], Sum( Irr( tbl ) ) );[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27Xtbl2:= tbl mod 2;  [127X[104X
    [4X[28XBrauerTable( "A5", 2 )[128X[104X
    [4X[25Xgap>[125X [27Xchi:= Irr( tbl2 )[1];[127X[104X
    [4X[28XCharacter( BrauerTable( "A5", 2 ), [ 1, 1, 1, 1 ] )[128X[104X
    [4X[25Xgap>[125X [27XScalarProduct( chi, chi );[127X[104X
    [4X[28X3/4[128X[104X
    [4X[25Xgap>[125X [27XScalarProduct( tbl2, [ 1, 1, 1, 1 ], [ 1, 1, 1, 1 ] );[127X[104X
    [4X[28X3/4[128X[104X
    [4X[25Xgap>[125X [27Xchars:= Irr( tbl ){ [ 2 .. 4 ] };;[127X[104X
    [4X[25Xgap>[125X [27Xchars:= Set( Tensored( chars, chars ) );;[127X[104X
    [4X[25Xgap>[125X [27XMatScalarProducts( Irr( tbl ), chars );[127X[104X
    [4X[28X[ [ 0, 0, 0, 1, 1 ], [ 1, 1, 0, 0, 1 ], [ 1, 0, 1, 0, 1 ], [128X[104X
    [4X[28X  [ 0, 1, 0, 1, 1 ], [ 0, 0, 1, 1, 1 ], [ 1, 1, 1, 1, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27XMatScalarProducts( tbl, chars );[127X[104X
    [4X[28X[ [ 2 ], [ 1, 3 ], [ 1, 2, 3 ], [ 2, 2, 1, 3 ], [ 2, 1, 2, 2, 3 ], [128X[104X
    [4X[28X  [ 2, 3, 3, 3, 3, 5 ] ][128X[104X
    [4X[25Xgap>[125X [27XList( chars, Norm );[127X[104X
    [4X[28X[ 2, 3, 3, 3, 3, 5 ][128X[104X
  [4X[32X[104X
  
  [1X72.8-8 ConstituentsOfCharacter[101X
  
  [29X[2XConstituentsOfCharacter[102X( [[3Xtbl[103X, ][3Xchi[103X ) [32X attribute
  
  [33X[0;0Yis  the set of irreducible characters that occur in the decomposition of the
  (virtual) character [3Xchi[103X with nonzero coefficient.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xnat:= NaturalCharacter( S4 );[127X[104X
    [4X[28XCharacter( CharacterTable( S4 ), [ 4, 2, 0, 1, 0 ] )[128X[104X
    [4X[25Xgap>[125X [27XConstituentsOfCharacter( nat );[127X[104X
    [4X[28X[ Character( CharacterTable( S4 ), [ 1, 1, 1, 1, 1 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( S4 ), [ 3, 1, -1, 0, -1 ] ) ][128X[104X
  [4X[32X[104X
  
  [1X72.8-9 KernelOfCharacter[101X
  
  [29X[2XKernelOfCharacter[102X( [[3Xtbl[103X, ][3Xchi[103X ) [32X attribute
  
  [33X[0;0YFor  a class function [3Xchi[103X of the group [22XG[122X, say, [2XKernelOfCharacter[102X returns the
  normal subgroup of [22XG[122X that is formed by those conjugacy classes for which the
  value  of [3Xchi[103X equals the degree of [3Xchi[103X. If the underlying character table of
  [3Xchi[103X   does   not   store   the   group   [22XG[122X   then  an  error  is  signalled.
  (See [2XClassPositionsOfKernel[102X  ([14X72.8-10[114X)  for  a  way  to  handle  the  kernel
  implicitly, by listing the positions of conjugacy classes in the kernel.)[133X
  
  [33X[0;0YThe  returned  group  is  the kernel of any representation of [22XG[122X that affords
  [3Xchi[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XList( Irr( S4 ), KernelOfCharacter );[127X[104X
    [4X[28X[ Alt( [ 1 .. 4 ] ), Group(()), Group([ (1,2)(3,4), (1,4)(2,3) ]),[128X[104X
    [4X[28X  Group(()), Group([ (), (1,2), (1,2)(3,4), (1,2,3), (1,2,3,4) ]) ][128X[104X
  [4X[32X[104X
  
  [1X72.8-10 ClassPositionsOfKernel[101X
  
  [29X[2XClassPositionsOfKernel[102X( [3Xchi[103X ) [32X attribute
  
  [33X[0;0Yis  the list of positions of those conjugacy classes that form the kernel of
  the  character  [3Xchi[103X,  that is, those positions with character value equal to
  the character degree.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XList( Irr( S4 ), ClassPositionsOfKernel );[127X[104X
    [4X[28X[ [ 1, 3, 4 ], [ 1 ], [ 1, 3 ], [ 1 ], [ 1, 2, 3, 4, 5 ] ][128X[104X
  [4X[32X[104X
  
  [1X72.8-11 CentreOfCharacter[101X
  
  [29X[2XCentreOfCharacter[102X( [[3Xtbl[103X, ][3Xchi[103X ) [32X attribute
  
  [33X[0;0YFor  a  character  [3Xchi[103X  of  the  group [22XG[122X, say, [2XCentreOfCharacter[102X returns the
  [13Xcentre[113X  of  [3Xchi[103X, that is, the normal subgroup of all those elements of [22XG[122X for
  which  the  quotient  of  the value of [3Xchi[103X by the degree of [3Xchi[103X is a root of
  unity.[133X
  
  [33X[0;0YIf  the underlying character table of [3Xpsi[103X does not store the group [22XG[122X then an
  error  is  signalled.  (See [2XClassPositionsOfCentre[102X  ([14X72.8-12[114X)  for  a way to
  handle  the centre implicitly, by listing the positions of conjugacy classes
  in the centre.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XList( Irr( S4 ), CentreOfCharacter );[127X[104X
    [4X[28X[ Group([ (), (1,2), (1,2)(3,4), (1,2,3), (1,2,3,4) ]), Group(()), [128X[104X
    [4X[28X  Group([ (1,2)(3,4), (1,4)(2,3) ]), Group(()), Group([ (), (1,2), (1,[128X[104X
    [4X[28X   2)(3,4), (1,2,3), (1,2,3,4) ]) ][128X[104X
  [4X[32X[104X
  
  [1X72.8-12 ClassPositionsOfCentre[101X
  
  [29X[2XClassPositionsOfCentre[102X( [3Xchi[103X ) [32X attribute
  
  [33X[0;0Yis  the list of positions of classes forming the centre of the character [3Xchi[103X
  (see [2XCentreOfCharacter[102X ([14X72.8-11[114X)).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XList( Irr( S4 ), ClassPositionsOfCentre );[127X[104X
    [4X[28X[ [ 1, 2, 3, 4, 5 ], [ 1 ], [ 1, 3 ], [ 1 ], [ 1, 2, 3, 4, 5 ] ][128X[104X
  [4X[32X[104X
  
  [1X72.8-13 InertiaSubgroup[101X
  
  [29X[2XInertiaSubgroup[102X( [[3Xtbl[103X, ][3XG[103X, [3Xchi[103X ) [32X operation
  
  [33X[0;0YLet  [3Xchi[103X  be a character of the group [22XH[122X, say, and [3Xtbl[103X the character table of
  [22XH[122X;  if  the argument [3Xtbl[103X is not given then the underlying character table of
  [3Xchi[103X  (see [2XUnderlyingCharacterTable[102X  ([14X72.2-1[114X))  is used instead. Furthermore,
  let [3XG[103X be a group that contains [22XH[122X as a normal subgroup.[133X
  
  [33X[0;0Y[2XInertiaSubgroup[102X  returns  the stabilizer in [3XG[103X of [3Xchi[103X, w.r.t. the action of [3XG[103X
  on the classes of [22XH[122X via conjugation. In other words, [2XInertiaSubgroup[102X returns
  the group of all those elements [22Xg ∈ [3XG[103X[122X that satisfy [22X[3Xchi[103X^g = [3Xchi[103X[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xder:= DerivedSubgroup( S4 );[127X[104X
    [4X[28XAlt( [ 1 .. 4 ] )[128X[104X
    [4X[25Xgap>[125X [27XList( Irr( der ), chi -> InertiaSubgroup( S4, chi ) );[127X[104X
    [4X[28X[ S4, Alt( [ 1 .. 4 ] ), Alt( [ 1 .. 4 ] ), S4 ][128X[104X
  [4X[32X[104X
  
  [1X72.8-14 CycleStructureClass[101X
  
  [29X[2XCycleStructureClass[102X( [[3Xtbl[103X, ][3Xchi[103X, [3Xclass[103X ) [32X operation
  
  [33X[0;0YLet  [3Xpermchar[103X  be  a  permutation  character, and [3Xclass[103X be the position of a
  conjugacy  class  of  the  character  table of [3Xpermchar[103X. [2XCycleStructureClass[102X
  returns a list describing the cycle structure of each element in class [3Xclass[103X
  in  the  underlying  permutation  representation,  in the same format as the
  result of [2XCycleStructurePerm[102X ([14X42.4-2[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xnat:= NaturalCharacter( S4 );[127X[104X
    [4X[28XCharacter( CharacterTable( S4 ), [ 4, 2, 0, 1, 0 ] )[128X[104X
    [4X[25Xgap>[125X [27XList( [ 1 .. 5 ], i -> CycleStructureClass( nat, i ) );[127X[104X
    [4X[28X[ [  ], [ 1 ], [ 2 ], [ , 1 ], [ ,, 1 ] ][128X[104X
  [4X[32X[104X
  
  [1X72.8-15 IsTransitive[101X
  
  [29X[2XIsTransitive[102X( [[3Xtbl[103X, ][3Xchi[103X ) [32X property
  
  [33X[0;0YFor a permutation character [3Xchi[103X of the group [22XG[122X that corresponds to an action
  on  the  [22XG[122X-set [22XΩ[122X (see [2XPermutationCharacter[102X ([14X72.7-3[114X)), [2XIsTransitive[102X ([14X41.10-1[114X)
  returns [9Xtrue[109X if the action of [22XG[122X on [22XΩ[122X is transitive, and [9Xfalse[109X otherwise.[133X
  
  [1X72.8-16 Transitivity[101X
  
  [29X[2XTransitivity[102X( [[3Xtbl[103X, ][3Xchi[103X ) [32X attribute
  
  [33X[0;0YFor a permutation character [3Xchi[103X of the group [22XG[122X that corresponds to an action
  on the [22XG[122X-set [22XΩ[122X (see [2XPermutationCharacter[102X ([14X72.7-3[114X)), [2XTransitivity[102X returns the
  maximal   nonnegative  integer  [22Xk[122X  such  that  the  action  of  [22XG[122X  on  [22XΩ[122X  is
  [22Xk[122X-transitive.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsTransitive( nat );  Transitivity( nat );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X4[128X[104X
    [4X[25Xgap>[125X [27XTransitivity( 2 * TrivialCharacter( S4 ) );[127X[104X
    [4X[28X0[128X[104X
  [4X[32X[104X
  
  [1X72.8-17 CentralCharacter[101X
  
  [29X[2XCentralCharacter[102X( [[3Xtbl[103X, ][3Xchi[103X ) [32X attribute
  
  [33X[0;0YFor  a  character  [3Xchi[103X  of  the  group  [22XG[122X, say, [2XCentralCharacter[102X returns the
  [13Xcentral character[113X of [3Xchi[103X.[133X
  
  [33X[0;0YThe  central  character  of  [22Xχ[122X is the class function [22Xω_χ[122X defined by [22Xω_χ(g) =
  |g^G| ⋅ χ(g)/χ(1)[122X for each [22Xg ∈ G[122X.[133X
  
  [1X72.8-18 DeterminantOfCharacter[101X
  
  [29X[2XDeterminantOfCharacter[102X( [[3Xtbl[103X, ][3Xchi[103X ) [32X attribute
  
  [33X[0;0Y[2XDeterminantOfCharacter[102X  returns  the  [13Xdeterminant character[113X of the character
  [3Xchi[103X.  This is defined to be the character obtained by taking the determinant
  of   representing   matrices   of  any  representation  affording  [3Xchi[103X;  the
  determinant can be computed using [2XEigenvaluesChar[102X ([14X72.8-19[114X).[133X
  
  [33X[0;0YIt   is   also   possible   to   call   [2XDeterminant[102X   ([14X24.4-4[114X)   instead  of
  [2XDeterminantOfCharacter[102X.[133X
  
  [33X[0;0YNote that the determinant character is well-defined for virtual characters.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCentralCharacter( TrivialCharacter( S4 ) );[127X[104X
    [4X[28XClassFunction( CharacterTable( S4 ), [ 1, 6, 3, 8, 6 ] )[128X[104X
    [4X[25Xgap>[125X [27XDeterminantOfCharacter( Irr( S4 )[3] );[127X[104X
    [4X[28XCharacter( CharacterTable( S4 ), [ 1, -1, 1, 1, -1 ] )[128X[104X
  [4X[32X[104X
  
  [1X72.8-19 EigenvaluesChar[101X
  
  [29X[2XEigenvaluesChar[102X( [[3Xtbl[103X, ][3Xchi[103X, [3Xclass[103X ) [32X operation
  
  [33X[0;0YLet  [3Xchi[103X  be  a  character  of the group [22XG[122X, say. For an element [22Xg ∈ G[122X in the
  [3Xclass[103X-th  conjugacy class, of order [22Xn[122X, let [22XM[122X be a matrix of a representation
  affording [3Xchi[103X.[133X
  
  [33X[0;0Y[2XEigenvaluesChar[102X  returns  the  list  of  length  [22Xn[122X  where  at position [22Xk[122X the
  multiplicity of [10XE[110X[22X(n)^k = exp(2 π i k / n)[122X as an eigenvalue of [22XM[122X is stored.[133X
  
  [33X[0;0YWe  have  [10X[3Xchi[103X[10X[  [3Xclass[103X[10X ] = List( [ 1 .. n ], k -> E(n)^k ) * EigenvaluesChar(
  [3Xtbl[103X[10X, [3Xchi[103X[10X, [3Xclass[103X[10X )[110X.[133X
  
  [33X[0;0YIt is also possible to call [2XEigenvalues[102X ([14X24.8-3[114X) instead of [2XEigenvaluesChar[102X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xchi:= Irr( CharacterTable( "A5" ) )[2];[127X[104X
    [4X[28XCharacter( CharacterTable( "A5" ), [128X[104X
    [4X[28X[ 3, -1, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ] )[128X[104X
    [4X[25Xgap>[125X [27XList( [ 1 .. 5 ], i -> Eigenvalues( chi, i ) );[127X[104X
    [4X[28X[ [ 3 ], [ 2, 1 ], [ 1, 1, 1 ], [ 0, 1, 1, 0, 1 ], [ 1, 0, 0, 1, 1 ] ][128X[104X
  [4X[32X[104X
  
  [1X72.8-20 Tensored[101X
  
  [29X[2XTensored[102X( [3Xchars1[103X, [3Xchars2[103X ) [32X operation
  
  [33X[0;0YLet  [3Xchars1[103X  and [3Xchars2[103X be lists of (values lists of) class functions of the
  same  character  table.  [2XTensored[102X returns the list of tensor products of all
  entries in [3Xchars1[103X with all entries in [3Xchars2[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xirra5:= Irr( CharacterTable( "A5" ) );;[127X[104X
    [4X[25Xgap>[125X [27Xchars1:= irra5{ [ 1 .. 3 ] };;  chars2:= irra5{ [ 2, 3 ] };;[127X[104X
    [4X[25Xgap>[125X [27XTensored( chars1, chars2 );[127X[104X
    [4X[28X[ Character( CharacterTable( "A5" ), [128X[104X
    [4X[28X    [ 3, -1, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [128X[104X
    [4X[28X    [ 3, -1, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [128X[104X
    [4X[28X    [ 9, 1, 0, -2*E(5)-E(5)^2-E(5)^3-2*E(5)^4, [128X[104X
    [4X[28X      -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 9, 1, 0, -1, -1 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 9, 1, 0, -1, -1 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [128X[104X
    [4X[28X    [ 9, 1, 0, -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4, [128X[104X
    [4X[28X      -2*E(5)-E(5)^2-E(5)^3-2*E(5)^4 ] ) ][128X[104X
  [4X[32X[104X
  
  
  [1X72.9 [33X[0;0YRestricted and Induced Class Functions[133X[101X
  
  [33X[0;0YFor  restricting  a  class  function  of  a  group [22XG[122X to a subgroup [22XH[122X and for
  inducing  a  class  function of [22XH[122X to [22XG[122X, the [13Xclass fusion[113X from [22XH[122X to [22XG[122X must be
  known (see [14X73.3[114X).[133X
  
  [33X[0;0YIf  [22XF[122X  is  the  factor  group  of [22XG[122X by the normal subgroup [22XN[122X then each class
  function  of [22XF[122X can be naturally regarded as a class function of [22XG[122X, with [22XN[122X in
  its kernel. For a class function of [22XF[122X, the corresponding class function of [22XG[122X
  is  called  the  [13Xinflated[113X  class  function. Restriction and inflation are in
  principle   the  same,  namely  indirection  of  a  class  function  by  the
  appropriate  fusion  map,  and  thus  no  extra operation is needed for this
  process. But note that contrary to the case of a subgroup fusion, the factor
  fusion  can  in  general not be computed from the groups [22XG[122X and [22XF[122X; either one
  needs  the natural homomorphism, or the factor fusion to the character table
  of  [22XF[122X  must  be stored on the table of [22XG[122X. This explains the different syntax
  for computing restricted and inflated class functions.[133X
  
  [33X[0;0YIn the following, the meaning of the optional first argument [3Xtbl[103X is the same
  as in Section [14X72.8[114X.[133X
  
  [1X72.9-1 RestrictedClassFunction[101X
  
  [29X[2XRestrictedClassFunction[102X( [[3Xtbl[103X, ][3Xchi[103X, [3Xtarget[103X ) [32X operation
  
  [33X[0;0YLet  [3Xchi[103X be a class function of the group [22XG[122X, say, and let [3Xtarget[103X be either a
  subgroup  [22XH[122X  of  [22XG[122X or an injective homomorphism from [22XH[122X to [22XG[122X or the character
  table  of  [3XH[103X.  Then  [2XRestrictedClassFunction[102X returns the class function of [22XH[122X
  obtained by restricting [3Xchi[103X to [22XH[122X.[133X
  
  [33X[0;0YIf  [3Xchi[103X  is a class function of a [13Xfactor group[113X [22XG[122Xof [22XH[122X, where [3Xtarget[103X is either
  the  group  [22XH[122X or a homomorphism from [22XH[122X to [22XG[122X or the character table of [22XH[122X then
  the  restriction  can  be  computed  in the case of the homomorphism; in the
  other  cases,  this  is  possible  only  if the factor fusion from [22XH[122X to [22XG[122X is
  stored on the character table of [22XH[122X.[133X
  
  [1X72.9-2 RestrictedClassFunctions[101X
  
  [29X[2XRestrictedClassFunctions[102X( [[3Xtbl[103X, ][3Xchars[103X, [3Xtarget[103X ) [32X operation
  
  [33X[0;0Y[2XRestrictedClassFunctions[102X is similar to [2XRestrictedClassFunction[102X ([14X72.9-1[114X), the
  only  difference is that it takes a list [3Xchars[103X of class functions instead of
  one class function, and returns the list of restricted class functions.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xa5:= CharacterTable( "A5" );;  s5:= CharacterTable( "S5" );;[127X[104X
    [4X[25Xgap>[125X [27XRestrictedClassFunction( Irr( s5 )[2], a5 );[127X[104X
    [4X[28XCharacter( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] )[128X[104X
    [4X[25Xgap>[125X [27XRestrictedClassFunctions( Irr( s5 ), a5 );[127X[104X
    [4X[28X[ Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 6, -2, 0, 1, 1 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 4, 0, 1, -1, -1 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 4, 0, 1, -1, -1 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 5, 1, -1, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 5, 1, -1, 0, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27Xhom:= NaturalHomomorphismByNormalSubgroup( S4, der );;[127X[104X
    [4X[25Xgap>[125X [27XRestrictedClassFunctions( Irr( Image( hom ) ), hom );[127X[104X
    [4X[28X[ Character( CharacterTable( S4 ), [ 1, 1, 1, 1, 1 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( S4 ), [ 1, -1, 1, 1, -1 ] ) ][128X[104X
  [4X[32X[104X
  
  
  [1X72.9-3 [33X[0;0YInducedClassFunction[133X[101X
  
  [29X[2XInducedClassFunction[102X( [[3Xtbl[103X, ][3Xchi[103X, [3XH[103X ) [32X operation
  [29X[2XInducedClassFunction[102X( [[3Xtbl[103X, ][3Xchi[103X, [3Xhom[103X ) [32X operation
  [29X[2XInducedClassFunction[102X( [[3Xtbl[103X, ][3Xchi[103X, [3Xsuptbl[103X ) [32X operation
  
  [33X[0;0YLet  [3Xchi[103X be a class function of the group [22XG[122X, say, and let [3Xtarget[103X be either a
  supergroup  [22XH[122X of [22XG[122X or an injective homomorphism from [22XH[122X to [22XG[122X or the character
  table  of  [3XH[103X.  Then  [2XInducedClassFunction[102X  returns  the  class function of [22XH[122X
  obtained by inducing [3Xchi[103X to [22XH[122X.[133X
  
  [1X72.9-4 InducedClassFunctions[101X
  
  [29X[2XInducedClassFunctions[102X( [[3Xtbl[103X, ][3Xchars[103X, [3Xtarget[103X ) [32X operation
  
  [33X[0;0Y[2XInducedClassFunctions[102X  is similar to [2XInducedClassFunction[102X ([14X72.9-3[114X), the only
  difference  is  that it takes a list [3Xchars[103X of class functions instead of one
  class function, and returns the list of induced class functions.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XInducedClassFunctions( Irr( a5 ), s5 );[127X[104X
    [4X[28X[ Character( CharacterTable( "A5.2" ), [ 2, 2, 2, 2, 0, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5.2" ), [ 6, -2, 0, 1, 0, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5.2" ), [ 6, -2, 0, 1, 0, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5.2" ), [ 8, 0, 2, -2, 0, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5.2" ), [ 10, 2, -2, 0, 0, 0, 0 ] ) ][128X[104X
  [4X[32X[104X
  
  [1X72.9-5 InducedClassFunctionsByFusionMap[101X
  
  [29X[2XInducedClassFunctionsByFusionMap[102X( [3Xsubtbl[103X, [3Xtbl[103X, [3Xchars[103X, [3Xfusionmap[103X ) [32X function
  
  [33X[0;0YLet [3Xsubtbl[103X and [3Xtbl[103X be two character tables of groups [22XH[122X and [22XG[122X, such that [22XH[122X is
  a  subgroup  of [22XG[122X, let [3Xchars[103X be a list of class functions of [3Xsubtbl[103X, and let
  [3Xfusionmap[103X  be a fusion map from [3Xsubtbl[103X to [3Xtbl[103X. The function returns the list
  of induced class functions of [3Xtbl[103X that correspond to [3Xchars[103X, w.r.t. the given
  fusion map.[133X
  
  [33X[0;0Y[2XInducedClassFunctionsByFusionMap[102X  is  the  function  that  does the work for
  [2XInducedClassFunction[102X ([14X72.9-3[114X) and [2XInducedClassFunctions[102X ([14X72.9-4[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfus:= PossibleClassFusions( a5, s5 );[127X[104X
    [4X[28X[ [ 1, 2, 3, 4, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27XInducedClassFunctionsByFusionMap( a5, s5, Irr( a5 ), fus[1] );[127X[104X
    [4X[28X[ Character( CharacterTable( "A5.2" ), [ 2, 2, 2, 2, 0, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5.2" ), [ 6, -2, 0, 1, 0, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5.2" ), [ 6, -2, 0, 1, 0, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5.2" ), [ 8, 0, 2, -2, 0, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5.2" ), [ 10, 2, -2, 0, 0, 0, 0 ] ) ][128X[104X
  [4X[32X[104X
  
  [1X72.9-6 InducedCyclic[101X
  
  [29X[2XInducedCyclic[102X( [3Xtbl[103X[, [3Xclasses[103X][, [3X"all"[103X] ) [32X operation
  
  [33X[0;0Y[2XInducedCyclic[102X  calculates characters induced up from cyclic subgroups of the
  ordinary character table [3Xtbl[103X to [3Xtbl[103X, and returns the strictly sorted list of
  the induced characters.[133X
  
  [33X[0;0YIf  the  string  [10X"all"[110X is specified then all irreducible characters of these
  subgroups  are  induced,  otherwise  only  the  permutation  characters  are
  calculated.[133X
  
  [33X[0;0YIf a list [3Xclasses[103X is specified then only those cyclic subgroups generated by
  these classes are considered, otherwise all classes of [3Xtbl[103X are considered.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XInducedCyclic( a5, "all" );[127X[104X
    [4X[28X[ Character( CharacterTable( "A5" ), [ 12, 0, 0, 2, 2 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [128X[104X
    [4X[28X    [ 12, 0, 0, E(5)^2+E(5)^3, E(5)+E(5)^4 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [128X[104X
    [4X[28X    [ 12, 0, 0, E(5)+E(5)^4, E(5)^2+E(5)^3 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 20, 0, -1, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 20, 0, 2, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 30, -2, 0, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 30, 2, 0, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 60, 0, 0, 0, 0 ] ) ][128X[104X
  [4X[32X[104X
  
  
  [1X72.10 [33X[0;0YReducing Virtual Characters[133X[101X
  
  [33X[0;0YThe  following operations are intended for the situation that one is given a
  list  of  virtual  characters  of a character table and is interested in the
  irreducible  characters  of  this  table.  The  idea  is  to compute virtual
  characters  of  small  norm  from  the  given ones, hoping to get eventually
  virtual characters of norm [22X1[122X.[133X
  
  [1X72.10-1 ReducedClassFunctions[101X
  
  [29X[2XReducedClassFunctions[102X( [[3Xtbl[103X, ][[3Xconstituents[103X, ][3Xreducibles[103X ) [32X operation
  
  [33X[0;0YLet [3Xreducibles[103X be a list of ordinary virtual characters of the group [22XG[122X, say.
  If  [3Xconstituents[103X  is  given  then it must also be a list of ordinary virtual
  characters  of  [22XG[122X, otherwise we have [3Xconstituents[103X equal to [3Xreducibles[103X in the
  following.[133X
  
  [33X[0;0Y[2XReducedClassFunctions[102X  returns  a  record with the components [10Xremainders[110X and
  [10Xirreducibles[110X,   both  lists  of  virtual  characters  of  [22XG[122X.  These  virtual
  characters are computed as follows.[133X
  
  [33X[0;0YLet [10Xrems[110X be the set of nonzero class functions obtained by subtraction of[133X
  
  
  [24X[33X[0;6Y∑_χ ( [[3Xreducibles[103X[i], χ] / [χ, χ] ) ⋅ χ[133X[124X
  
  [33X[0;0Yfrom  [22X[3Xreducibles[103X[i][122X,  where  the summation runs over [3Xconstituents[103X and [22X[χ, ψ][122X
  denotes  the  scalar  product  of [22XG[122X-class functions. Let [10Xirrs[110X be the list of
  irreducible characters in [10Xrems[110X.[133X
  
  [33X[0;0YWe project [10Xrems[110X into the orthogonal space of [10Xirrs[110X and all those irreducibles
  found  this  way until no new irreducibles arise. Then the [10Xirreducibles[110X list
  is  the  set of all found irreducible characters, and the [10Xremainders[110X list is
  the set of all nonzero remainders.[133X
  
  [1X72.10-2 ReducedCharacters[101X
  
  [29X[2XReducedCharacters[102X( [[3Xtbl[103X, ][3Xconstituents[103X, [3Xreducibles[103X ) [32X operation
  
  [33X[0;0Y[2XReducedCharacters[102X  is  similar  to [2XReducedClassFunctions[102X ([14X72.10-1[114X), the only
  difference  is  that  [3Xconstituents[103X and [3Xreducibles[103X are assumed to be lists of
  characters.  This means that only those scalar products must be formed where
  the  degree  of  the character in [3Xconstituents[103X does not exceed the degree of
  the character in [3Xreducibles[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtbl:= CharacterTable( "A5" );;[127X[104X
    [4X[25Xgap>[125X [27Xchars:= Irr( tbl ){ [ 2 .. 4 ] };;[127X[104X
    [4X[25Xgap>[125X [27Xchars:= Set( Tensored( chars, chars ) );;[127X[104X
    [4X[25Xgap>[125X [27Xred:= ReducedClassFunctions( chars );[127X[104X
    [4X[28Xrec( [128X[104X
    [4X[28X  irreducibles := [128X[104X
    [4X[28X    [ Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ), [128X[104X
    [4X[28X      Character( CharacterTable( "A5" ), [128X[104X
    [4X[28X        [ 3, -1, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ] ), [128X[104X
    [4X[28X      Character( CharacterTable( "A5" ), [128X[104X
    [4X[28X        [ 3, -1, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ] ), [128X[104X
    [4X[28X      Character( CharacterTable( "A5" ), [ 4, 0, 1, -1, -1 ] ), [128X[104X
    [4X[28X      Character( CharacterTable( "A5" ), [ 5, 1, -1, 0, 0 ] ) ], [128X[104X
    [4X[28X  remainders := [  ] )[128X[104X
  [4X[32X[104X
  
  [1X72.10-3 IrreducibleDifferences[101X
  
  [29X[2XIrreducibleDifferences[102X( [3Xtbl[103X, [3Xreducibles[103X, [3Xreducibles2[103X[, [3Xscprmat[103X] ) [32X function
  
  [33X[0;0Y[2XIrreducibleDifferences[102X  returns  the  list  of  irreducible characters which
  occur  as  difference  of  an  element  of  [3Xreducibles[103X  and  an  element  of
  [3Xreducibles2[103X,  where  these two arguments are lists of class functions of the
  character table [3Xtbl[103X.[133X
  
  [33X[0;0YIf  [3Xreducibles2[103X is the string [10X"triangle"[110X then the differences of elements in
  [3Xreducibles[103X are considered.[133X
  
  [33X[0;0YIf  [3Xscprmat[103X  is  not specified then it will be calculated, otherwise we must
  have  [10X[3Xscprmat[103X[10X = MatScalarProducts( [3Xtbl[103X[10X, [3Xreducibles[103X[10X, [3Xreducibles2[103X[10X )[110X or [10X[3Xscprmat[103X[10X
  = MatScalarProducts( [3Xtbl[103X[10X, [3Xreducibles[103X[10X )[110X, respectively.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIrreducibleDifferences( a5, chars, "triangle" );[127X[104X
    [4X[28X[ Character( CharacterTable( "A5" ), [128X[104X
    [4X[28X    [ 3, -1, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [128X[104X
    [4X[28X    [ 3, -1, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ] ) ][128X[104X
  [4X[32X[104X
  
  [1X72.10-4 LLL[101X
  
  [29X[2XLLL[102X( [3Xtbl[103X, [3Xcharacters[103X[, [3Xy[103X][, [3X"sort"[103X][, [3X"linearcomb"[103X] ) [32X function
  
  [33X[0;0Y[2XLLL[102X  calls  the  LLL algorithm (see [2XLLLReducedBasis[102X ([14X25.5-1[114X)) in the case of
  lattices  spanned  by  the  virtual  characters  [3Xcharacters[103X  of the ordinary
  character table [3Xtbl[103X (see [2XScalarProduct[102X ([14X72.8-5[114X)). By finding shorter vectors
  in  the  lattice  spanned by [3Xcharacters[103X, i.e., virtual characters of smaller
  norm, in some cases [2XLLL[102X is able to find irreducible characters.[133X
  
  [33X[0;0Y[2XLLL[102X  returns  a  record  with  at least components [10Xirreducibles[110X (the list of
  found  irreducible  characters),  [10Xremainders[110X  (a  list  of reducible virtual
  characters),  and  [10Xnorms[110X  (the  list of norms of the vectors in [10Xremainders[110X).
  [10Xirreducibles[110X  together with [10Xremainders[110X form a basis of the [22Xℤ[122X-lattice spanned
  by [3Xcharacters[103X.[133X
  
  [33X[0;0YNote  that  the vectors in the [10Xremainders[110X list are in general [13Xnot[113X orthogonal
  (see [2XReducedClassFunctions[102X  ([14X72.10-1[114X))  to  the  irreducible  characters  in
  [10Xirreducibles[110X.[133X
  
  [33X[0;0YOptional arguments of [2XLLL[102X are[133X
  
  [8X[3Xy[103X[8X[108X
        [33X[0;6Ycontrols   the   sensitivity  of  the  algorithm,  see [2XLLLReducedBasis[102X
        ([14X25.5-1[114X),[133X
  
  [8X[3X"sort"[103X[8X[108X
        [33X[0;6Y[2XLLL[102X  sorts  [3Xcharacters[103X  and  the  [10Xremainders[110X  component  of the result
        according to the degrees,[133X
  
  [8X[3X"linearcomb"[103X[8X[108X
        [33X[0;6Ythe  returned  record  contains  components [10Xirreddecomp[110X and [10Xreddecomp[110X,
        which  are decomposition matrices of [10Xirreducibles[110X and [10Xremainders[110X, with
        respect to [3Xcharacters[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs4:= CharacterTable( "Symmetric", 4 );;[127X[104X
    [4X[25Xgap>[125X [27Xchars:= [ [ 8, 0, 0, -1, 0 ], [ 6, 0, 2, 0, 2 ],[127X[104X
    [4X[25X>[125X [27X    [ 12, 0, -4, 0, 0 ], [ 6, 0, -2, 0, 0 ], [ 24, 0, 0, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X    [ 12, 0, 4, 0, 0 ], [ 6, 0, 2, 0, -2 ], [ 12, -2, 0, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X    [ 8, 0, 0, 2, 0 ], [ 12, 2, 0, 0, 0 ], [ 1, 1, 1, 1, 1 ] ];;[127X[104X
    [4X[25Xgap>[125X [27XLLL( s4, chars );[127X[104X
    [4X[28Xrec( [128X[104X
    [4X[28X  irreducibles := [128X[104X
    [4X[28X    [ Character( CharacterTable( "Sym(4)" ), [ 2, 0, 2, -1, 0 ] ), [128X[104X
    [4X[28X      Character( CharacterTable( "Sym(4)" ), [ 1, 1, 1, 1, 1 ] ), [128X[104X
    [4X[28X      Character( CharacterTable( "Sym(4)" ), [ 3, 1, -1, 0, -1 ] ), [128X[104X
    [4X[28X      Character( CharacterTable( "Sym(4)" ), [ 3, -1, -1, 0, 1 ] ), [128X[104X
    [4X[28X      Character( CharacterTable( "Sym(4)" ), [ 1, -1, 1, 1, -1 ] ) ], [128X[104X
    [4X[28X  norms := [  ], remainders := [  ] )[128X[104X
  [4X[32X[104X
  
  [1X72.10-5 Extract[101X
  
  [29X[2XExtract[102X( [3Xtbl[103X, [3Xreducibles[103X, [3Xgrammat[103X[, [3Xmissing[103X] ) [32X function
  
  [33X[0;0YLet  [3Xtbl[103X  be an ordinary character table, [3Xreducibles[103X a list of characters of
  [3Xtbl[103X,   and   [3Xgrammat[103X   the   matrix   of   scalar   products  of  [3Xreducibles[103X
  (see [2XMatScalarProducts[102X   ([14X72.8-6[114X)).   [2XExtract[102X   tries  to  find  irreducible
  characters  by  drawing  conclusions  out  of  the  scalar  products,  using
  combinatorial and backtrack means.[133X
  
  [33X[0;0YThe   optional  argument  [3Xmissing[103X  is  the  maximal  number  of  irreducible
  characters  that  occur  as  constituents  of  [3Xreducibles[103X.  Specification of
  [3Xmissing[103X may accelerate [2XExtract[102X.[133X
  
  [33X[0;0Y[2XExtract[102X  returns a record [3Xext[103X with the components [10Xsolution[110X and [10Xchoice[110X, where
  the  value of [10Xsolution[110X is a list [22X[ M_1, ..., M_n ][122X of decomposition matrices
  [22XM_i[122X  (up to permutations of rows) with the property that [22XM_i^tr ⋅ X[122X is equal
  to  the sublist at the positions [3Xext[103X[10X.choice[i][110X of [3Xreducibles[103X, for a matrix [22XX[122X
  of  irreducible  characters; the value of [10Xchoice[110X is a list of length [22Xn[122X whose
  entries are lists of indices.[133X
  
  [33X[0;0YSo the [22Xj[122X-th column in each matrix [22XM_i[122X corresponds to [22X[3Xreducibles[103X[j][122X, and each
  row  in [22XM_i[122X corresponds to an irreducible character. [2XDecreased[102X ([14X72.10-7[114X) can
  be used to examine the solution for computable irreducibles.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs4:= CharacterTable( "Symmetric", 4 );;[127X[104X
    [4X[25Xgap>[125X [27Xred:= [ [ 5, 1, 5, 2, 1 ], [ 2, 0, 2, 2, 0 ], [ 3, -1, 3, 0, -1 ],[127X[104X
    [4X[25X>[125X [27X           [ 6, 0, -2, 0, 0 ], [ 4, 0, 0, 1, 2 ] ];;[127X[104X
    [4X[25Xgap>[125X [27Xgram:= MatScalarProducts( s4, red, red );[127X[104X
    [4X[28X[ [ 6, 3, 2, 0, 2 ], [ 3, 2, 1, 0, 1 ], [ 2, 1, 2, 0, 0 ], [128X[104X
    [4X[28X  [ 0, 0, 0, 2, 1 ], [ 2, 1, 0, 1, 2 ] ][128X[104X
    [4X[25Xgap>[125X [27Xext:= Extract( s4, red, gram, 5 );[127X[104X
    [4X[28Xrec( choice := [ [ 2, 5, 3, 4, 1 ] ], [128X[104X
    [4X[28X  solution := [128X[104X
    [4X[28X    [ [128X[104X
    [4X[28X      [ [ 1, 1, 0, 0, 2 ], [ 1, 0, 1, 0, 1 ], [ 0, 1, 0, 1, 0 ], [128X[104X
    [4X[28X          [ 0, 0, 1, 0, 1 ], [ 0, 0, 0, 1, 0 ] ] ] )[128X[104X
    [4X[25Xgap>[125X [27Xdec:= Decreased( s4, red, ext.solution[1], ext.choice[1] );[127X[104X
    [4X[28Xrec( [128X[104X
    [4X[28X  irreducibles := [128X[104X
    [4X[28X    [ Character( CharacterTable( "Sym(4)" ), [ 1, 1, 1, 1, 1 ] ), [128X[104X
    [4X[28X      Character( CharacterTable( "Sym(4)" ), [ 3, -1, -1, 0, 1 ] ), [128X[104X
    [4X[28X      Character( CharacterTable( "Sym(4)" ), [ 1, -1, 1, 1, -1 ] ), [128X[104X
    [4X[28X      Character( CharacterTable( "Sym(4)" ), [ 3, 1, -1, 0, -1 ] ), [128X[104X
    [4X[28X      Character( CharacterTable( "Sym(4)" ), [ 2, 0, 2, -1, 0 ] ) ], [128X[104X
    [4X[28X  matrix := [  ], remainders := [  ] )[128X[104X
  [4X[32X[104X
  
  [1X72.10-6 OrthogonalEmbeddingsSpecialDimension[101X
  
  [29X[2XOrthogonalEmbeddingsSpecialDimension[102X( [3Xtbl[103X, [3Xreducibles[103X, [3Xgrammat[103X[, [3X"positive"[103X], [3Xdim[103X ) [32X function
  
  [33X[0;0Y[2XOrthogonalEmbeddingsSpecialDimension[102X  is  a  variant of [2XOrthogonalEmbeddings[102X
  ([14X25.6-1[114X)  for  the  situation  that  [3Xtbl[103X  is  an  ordinary  character table,
  [3Xreducibles[103X  is a list of virtual characters of [3Xtbl[103X, [3Xgrammat[103X is the matrix of
  scalar  products (see [2XMatScalarProducts[102X ([14X72.8-6[114X)), and [3Xdim[103X is an upper bound
  for  the  number of irreducible characters of [3Xtbl[103X that occur as constituents
  of  [3Xreducibles[103X;  if  the  vectors  in  [3Xreducibles[103X  are  known  to  be proper
  characters  then  the  string  [10X"positive"[110X may be entered as fourth argument.
  (See [2XOrthogonalEmbeddings[102X ([14X25.6-1[114X) for information why this may help.)[133X
  
  [33X[0;0Y[2XOrthogonalEmbeddingsSpecialDimension[102X    first    uses   [2XOrthogonalEmbeddings[102X
  ([14X25.6-1[114X)  to  compute  all  orthogonal embeddings of [3Xgrammat[103X into a standard
  lattice  of dimension up to [3Xdim[103X, and then calls [2XDecreased[102X ([14X72.10-7[114X) in order
  to find irreducible characters of [3Xtbl[103X.[133X
  
  [33X[0;0Y[2XOrthogonalEmbeddingsSpecialDimension[102X  returns  a  record  with the following
  components.[133X
  
  [8X[10Xirreducibles[110X[8X[108X
        [33X[0;6Ya  list  of  found  irreducibles,  the  intersection  of  all lists of
        irreducibles   found   by   [2XDecreased[102X   ([14X72.10-7[114X),  for  all  possible
        embeddings, and[133X
  
  [8X[10Xremainders[110X[8X[108X
        [33X[0;6Ya list of remaining reducible virtual characters.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs6:= CharacterTable( "S6" );;[127X[104X
    [4X[25Xgap>[125X [27Xred:= InducedCyclic( s6, "all" );;[127X[104X
    [4X[25Xgap>[125X [27XAdd( red, TrivialCharacter( s6 ) );[127X[104X
    [4X[25Xgap>[125X [27Xlll:= LLL( s6, red );;[127X[104X
    [4X[25Xgap>[125X [27Xirred:= lll.irreducibles;[127X[104X
    [4X[28X[ Character( CharacterTable( "A6.2_1" ), [128X[104X
    [4X[28X    [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A6.2_1" ), [128X[104X
    [4X[28X    [ 9, 1, 0, 0, 1, -1, -3, -3, 1, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A6.2_1" ), [128X[104X
    [4X[28X    [ 16, 0, -2, -2, 0, 1, 0, 0, 0, 0, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XSet( Flat( MatScalarProducts( s6, irred, lll.remainders ) ) );[127X[104X
    [4X[28X[ 0 ][128X[104X
    [4X[25Xgap>[125X [27Xdim:= NrConjugacyClasses( s6 ) - Length( lll.irreducibles );[127X[104X
    [4X[28X8[128X[104X
    [4X[25Xgap>[125X [27Xrem:= lll.remainders;;  Length( rem );[127X[104X
    [4X[28X8[128X[104X
    [4X[25Xgap>[125X [27Xgram:= MatScalarProducts( s6, rem, rem );;  RankMat( gram );[127X[104X
    [4X[28X8[128X[104X
    [4X[25Xgap>[125X [27Xemb1:= OrthogonalEmbeddings( gram, 8 );[127X[104X
    [4X[28Xrec( norms := [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ], [128X[104X
    [4X[28X  solutions := [ [ 1, 2, 3, 7, 11, 12, 13, 15 ], [128X[104X
    [4X[28X      [ 1, 2, 4, 8, 10, 12, 13, 14 ], [ 1, 2, 5, 6, 9, 12, 13, 16 ] ],[128X[104X
    [4X[28X  vectors := [128X[104X
    [4X[28X    [ [ -1, 0, 1, 0, 1, 0, 1, 0 ], [ 1, 0, 0, 1, 0, 1, 0, 0 ], [128X[104X
    [4X[28X      [ 0, 1, 1, 0, 0, 0, 1, 1 ], [ 0, 1, 1, 0, 0, 0, 1, 0 ], [128X[104X
    [4X[28X      [ 0, 1, 1, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 1, 0 ], [128X[104X
    [4X[28X      [ 0, -1, 0, 0, 0, 0, 0, 1 ], [ 0, 1, 0, 0, 0, 0, 0, 0 ], [128X[104X
    [4X[28X      [ 0, 0, 1, 0, 0, 0, 1, 1 ], [ 0, 0, 1, 0, 0, 0, 0, 1 ], [128X[104X
    [4X[28X      [ 0, 0, 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, -1, 1, 0, 0, 0 ], [128X[104X
    [4X[28X      [ 0, 0, 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 1, 1 ], [128X[104X
    [4X[28X      [ 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 1 ] ] )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  the following example we temporarily decrease the line length limit from
  its default value [22X80[122X to [22X62[122X in order to get a nicer output format.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xemb2:= OrthogonalEmbeddingsSpecialDimension( s6, rem, gram, 8 );[127X[104X
    [4X[28Xrec( [128X[104X
    [4X[28X  irreducibles := [128X[104X
    [4X[28X    [ Character( CharacterTable( "A6.2_1" ), [128X[104X
    [4X[28X        [ 5, 1, -1, 2, -1, 0, 1, -3, -1, 1, 0 ] ), [128X[104X
    [4X[28X      Character( CharacterTable( "A6.2_1" ), [128X[104X
    [4X[28X        [ 5, 1, 2, -1, -1, 0, -3, 1, -1, 0, 1 ] ), [128X[104X
    [4X[28X      Character( CharacterTable( "A6.2_1" ), [128X[104X
    [4X[28X        [ 10, -2, 1, 1, 0, 0, -2, 2, 0, 1, -1 ] ), [128X[104X
    [4X[28X      Character( CharacterTable( "A6.2_1" ), [128X[104X
    [4X[28X        [ 10, -2, 1, 1, 0, 0, 2, -2, 0, -1, 1 ] ) ], [128X[104X
    [4X[28X  remainders := [128X[104X
    [4X[28X    [ VirtualCharacter( CharacterTable( "A6.2_1" ), [128X[104X
    [4X[28X        [ 0, 0, 3, -3, 0, 0, 4, -4, 0, 1, -1 ] ), [128X[104X
    [4X[28X      VirtualCharacter( CharacterTable( "A6.2_1" ), [128X[104X
    [4X[28X        [ 6, 2, 3, 0, 0, 1, 2, -2, 0, -1, -2 ] ), [128X[104X
    [4X[28X      VirtualCharacter( CharacterTable( "A6.2_1" ), [128X[104X
    [4X[28X        [ 10, 2, 1, 1, 2, 0, 2, 2, -2, -1, -1 ] ), [128X[104X
    [4X[28X      VirtualCharacter( CharacterTable( "A6.2_1" ), [128X[104X
    [4X[28X        [ 14, 2, 2, -1, 0, -1, 6, 2, 0, 0, -1 ] ) ] )[128X[104X
  [4X[32X[104X
  
  [1X72.10-7 Decreased[101X
  
  [29X[2XDecreased[102X( [3Xtbl[103X, [3Xchars[103X, [3Xdecompmat[103X[, [3Xchoice[103X] ) [32X function
  
  [33X[0;0YLet  [3Xtbl[103X  be an ordinary character table, [3Xchars[103X a list of virtual characters
  of  [3Xtbl[103X,  and [3Xdecompmat[103X a decomposition matrix, that is, a matrix [22XM[122X with the
  property  that  [22XM^tr  ⋅  X  =  [3Xchars[103X[122X holds, where [22XX[122X is a list of irreducible
  characters  of  [3Xtbl[103X.  [2XDecreased[102X tries to compute the irreducibles in [22XX[122X or at
  least some of them.[133X
  
  [33X[0;0YUsually  [2XDecreased[102X  is  applied  to  the  output  of  [2XExtract[102X  ([14X72.10-5[114X)  or
  [2XOrthogonalEmbeddings[102X    ([14X25.6-1[114X)   or   [2XOrthogonalEmbeddingsSpecialDimension[102X
  ([14X72.10-6[114X).   In   the  case  of  [2XExtract[102X  ([14X72.10-5[114X),  the  choice  component
  corresponding to the decomposition matrix must be entered as argument [3Xchoice[103X
  of [2XDecreased[102X.[133X
  
  [33X[0;0Y[2XDecreased[102X  returns  [9Xfail[109X  if  it  can  prove  that  no list [22XX[122X of irreducible
  characters  corresponding  to  the  arguments  exists;  otherwise  [2XDecreased[102X
  returns a record with the following components.[133X
  
  [8X[10Xirreducibles[110X[8X[108X
        [33X[0;6Ythe list of found irreducible characters,[133X
  
  [8X[10Xremainders[110X[8X[108X
        [33X[0;6Ythe remaining reducible characters, and[133X
  
  [8X[10Xmatrix[110X[8X[108X
        [33X[0;6Ythe   decomposition   matrix  of  the  characters  in  the  [10Xremainders[110X
        component.[133X
  
  [33X[0;0YIn  the following example we temporarily decrease the line length limit from
  its default value [22X80[122X to [22X62[122X in order to get a nicer output format.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs4:= CharacterTable( "Symmetric", 4 );;[127X[104X
    [4X[25Xgap>[125X [27Xx:= Irr( s4 );;[127X[104X
    [4X[25Xgap>[125X [27Xred:= [ x[1]+x[2], -x[1]-x[3], -x[1]+x[3], -x[2]-x[4] ];;[127X[104X
    [4X[25Xgap>[125X [27Xmat:= MatScalarProducts( s4, red, red );[127X[104X
    [4X[28X[ [ 2, -1, -1, -1 ], [ -1, 2, 0, 0 ], [ -1, 0, 2, 0 ], [128X[104X
    [4X[28X  [ -1, 0, 0, 2 ] ][128X[104X
    [4X[25Xgap>[125X [27Xemb:= OrthogonalEmbeddings( mat );[127X[104X
    [4X[28Xrec( norms := [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ], [128X[104X
    [4X[28X  solutions := [ [ 1, 6, 7, 12 ], [ 2, 5, 8, 11 ], [ 3, 4, 9, 10 ] ], [128X[104X
    [4X[28X  vectors := [ [ -1, 1, 1, 0 ], [ -1, 1, 0, 1 ], [ 1, -1, 0, 0 ], [128X[104X
    [4X[28X      [ -1, 0, 1, 1 ], [ -1, 0, 1, 0 ], [ -1, 0, 0, 1 ], [128X[104X
    [4X[28X      [ 0, -1, 1, 0 ], [ 0, -1, 0, 1 ], [ 0, 1, 0, 0 ], [128X[104X
    [4X[28X      [ 0, 0, -1, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ] )[128X[104X
    [4X[25Xgap>[125X [27Xdec:= Decreased( s4, red, emb.vectors{ emb.solutions[1] } );[127X[104X
    [4X[28Xrec( [128X[104X
    [4X[28X  irreducibles := [128X[104X
    [4X[28X    [ Character( CharacterTable( "Sym(4)" ), [ 3, -1, -1, 0, 1 ] ), [128X[104X
    [4X[28X      Character( CharacterTable( "Sym(4)" ), [ 1, -1, 1, 1, -1 ] ), [128X[104X
    [4X[28X      Character( CharacterTable( "Sym(4)" ), [ 2, 0, 2, -1, 0 ] ), [128X[104X
    [4X[28X      Character( CharacterTable( "Sym(4)" ), [ 3, 1, -1, 0, -1 ] ) ], [128X[104X
    [4X[28X  matrix := [  ], remainders := [  ] )[128X[104X
    [4X[25Xgap>[125X [27XDecreased( s4, red, emb.vectors{ emb.solutions[2] } );[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27XDecreased( s4, red, emb.vectors{ emb.solutions[3] } );[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  [1X72.10-8 DnLattice[101X
  
  [29X[2XDnLattice[102X( [3Xtbl[103X, [3Xgrammat[103X, [3Xreducibles[103X ) [32X function
  
  [33X[0;0YLet  [3Xtbl[103X  be  an  ordinary character table, and [3Xreducibles[103X a list of virtual
  characters of [3Xtbl[103X.[133X
  
  [33X[0;0Y[2XDnLattice[102X  searches for sublattices isomorphic to root lattices of type [22XD_n[122X,
  for  [22Xn  ≥  4[122X, in the lattice that is generated by [3Xreducibles[103X; each vector in
  [3Xreducibles[103X   must   have   norm   [22X2[122X,  and  the  matrix  of  scalar  products
  (see [2XMatScalarProducts[102X  ([14X72.8-6[114X))  of [3Xreducibles[103X must be entered as argument
  [3Xgrammat[103X.[133X
  
  [33X[0;0Y[2XDnLattice[102X  is  able  to find irreducible characters if there is a lattice of
  type  [22XD_n[122X  with  [22Xn  >  4[122X. In the case [22Xn = 4[122X, [2XDnLattice[102X may fail to determine
  irreducibles.[133X
  
  [33X[0;0Y[2XDnLattice[102X returns a record with components[133X
  
  [8X[10Xirreducibles[110X[8X[108X
        [33X[0;6Ythe list of found irreducible characters,[133X
  
  [8X[10Xremainders[110X[8X[108X
        [33X[0;6Ythe list of remaining reducible virtual characters, and[133X
  
  [8X[10Xgram[110X[8X[108X
        [33X[0;6Ythe Gram matrix of the vectors in [10Xremainders[110X.[133X
  
  [33X[0;0YThe  [10Xremainders[110X  list is transformed in such a way that the [10Xgram[110X matrix is a
  block  diagonal  matrix that exhibits the structure of the lattice generated
  by  the vectors in [10Xremainders[110X. So [2XDnLattice[102X might be useful even if it fails
  to find irreducible characters.[133X
  
  [33X[0;0YIn  the following example we temporarily decrease the line length limit from
  its default value [22X80[122X to [22X62[122X in order to get a nicer output format.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs4:= CharacterTable( "Symmetric", 4 );;[127X[104X
    [4X[25Xgap>[125X [27Xred:= [ [ 2, 0, 2, 2, 0 ], [ 4, 0, 0, 1, 2 ],[127X[104X
    [4X[25X>[125X [27X           [ 5, -1, 1, -1, 1 ], [ -1, 1, 3, -1, -1 ] ];;[127X[104X
    [4X[25Xgap>[125X [27Xgram:= MatScalarProducts( s4, red, red );[127X[104X
    [4X[28X[ [ 2, 1, 0, 0 ], [ 1, 2, 1, -1 ], [ 0, 1, 2, 0 ], [ 0, -1, 0, 2 ] ][128X[104X
    [4X[25Xgap>[125X [27Xdn:= DnLattice( s4, gram, red );[127X[104X
    [4X[28Xrec( gram := [  ], [128X[104X
    [4X[28X  irreducibles := [128X[104X
    [4X[28X    [ Character( CharacterTable( "Sym(4)" ), [ 2, 0, 2, -1, 0 ] ), [128X[104X
    [4X[28X      Character( CharacterTable( "Sym(4)" ), [ 1, -1, 1, 1, -1 ] ), [128X[104X
    [4X[28X      Character( CharacterTable( "Sym(4)" ), [ 1, 1, 1, 1, 1 ] ), [128X[104X
    [4X[28X      Character( CharacterTable( "Sym(4)" ), [ 3, -1, -1, 0, 1 ] ) ], [128X[104X
    [4X[28X  remainders := [  ] )[128X[104X
  [4X[32X[104X
  
  [1X72.10-9 DnLatticeIterative[101X
  
  [29X[2XDnLatticeIterative[102X( [3Xtbl[103X, [3Xreducibles[103X ) [32X function
  
  [33X[0;0YLet  [3Xtbl[103X  be  an  ordinary  character table, and [3Xreducibles[103X either a list of
  virtual  characters of [3Xtbl[103X or a record with components [10Xremainders[110X and [10Xnorms[110X,
  for example a record returned by [2XLLL[102X ([14X72.10-4[114X).[133X
  
  [33X[0;0Y[2XDnLatticeIterative[102X  was  designed  for iterative use of [2XDnLattice[102X ([14X72.10-8[114X).
  [2XDnLatticeIterative[102X  selects  the  vectors  of norm [22X2[122X among the given virtual
  character,   calls   [2XDnLattice[102X  ([14X72.10-8[114X)  for  them,  reduces  the  virtual
  characters  with found irreducibles, calls [2XDnLattice[102X ([14X72.10-8[114X) again for the
  remaining  virtual  characters,  and  so  on,  until no new irreducibles are
  found.[133X
  
  [33X[0;0Y[2XDnLatticeIterative[102X  returns a record with the same components and meaning of
  components as [2XLLL[102X ([14X72.10-4[114X).[133X
  
  [33X[0;0YIn  the following example we temporarily decrease the line length limit from
  its default value [22X80[122X to [22X62[122X in order to get a nicer output format.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs4:= CharacterTable( "Symmetric", 4 );;[127X[104X
    [4X[25Xgap>[125X [27Xred:= [ [ 2, 0, 2, 2, 0 ], [ 4, 0, 0, 1, 2 ],[127X[104X
    [4X[25X>[125X [27X           [ 5, -1, 1, -1, 1 ], [ -1, 1, 3, -1, -1 ] ];;[127X[104X
    [4X[25Xgap>[125X [27Xdn:= DnLatticeIterative( s4, red );[127X[104X
    [4X[28Xrec( [128X[104X
    [4X[28X  irreducibles := [128X[104X
    [4X[28X    [ Character( CharacterTable( "Sym(4)" ), [ 2, 0, 2, -1, 0 ] ), [128X[104X
    [4X[28X      Character( CharacterTable( "Sym(4)" ), [ 1, -1, 1, 1, -1 ] ), [128X[104X
    [4X[28X      Character( CharacterTable( "Sym(4)" ), [ 1, 1, 1, 1, 1 ] ), [128X[104X
    [4X[28X      Character( CharacterTable( "Sym(4)" ), [ 3, -1, -1, 0, 1 ] ) ], [128X[104X
    [4X[28X  norms := [  ], remainders := [  ] )[128X[104X
  [4X[32X[104X
  
  
  [1X72.11 [33X[0;0YSymmetrizations of Class Functions[133X[101X
  
  [1X72.11-1 Symmetrizations[101X
  
  [29X[2XSymmetrizations[102X( [[3Xtbl[103X, ][3Xcharacters[103X, [3Xn[103X ) [32X operation
  
  [33X[0;0Y[2XSymmetrizations[102X  returns  the  list  of  symmetrizations  of  the characters
  [3Xcharacters[103X of the ordinary character table [3Xtbl[103X with the ordinary irreducible
  characters of the symmetric group of degree [3Xn[103X; instead of the integer [3Xn[103X, the
  character table of the symmetric group can be entered.[133X
  
  [33X[0;0YThe  symmetrization  [22Xχ^[λ][122X of the character [22Xχ[122X of [3Xtbl[103X with the character [22Xλ[122X of
  the symmetric group [22XS_n[122X of degree [22Xn[122X is defined by[133X
  
  
  [24X[33X[0;6Yχ^[λ](g) = ( ∑_{ρ ∈ S_n} λ(ρ) ∏_{k=1}^n χ(g^k)^{a_k(ρ)} ) / n! ,[133X[124X
  
  [33X[0;0Ywhere [22Xa_k(ρ)[122X is the number of cycles of length [22Xk[122X in [22Xρ[122X.[133X
  
  [33X[0;0YFor   special   kinds   of  symmetrizations,  see [2XSymmetricParts[102X  ([14X72.11-2[114X),
  [2XAntiSymmetricParts[102X      ([14X72.11-3[114X),      [2XMinusCharacter[102X      ([14X73.6-5[114X)     and
  [2XOrthogonalComponents[102X ([14X72.11-4[114X), [2XSymplecticComponents[102X ([14X72.11-5[114X).[133X
  
  [33X[0;0Y[13XNote[113X that the returned list may contain zero class functions, and duplicates
  are not deleted.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtbl:= CharacterTable( "A5" );;[127X[104X
    [4X[25Xgap>[125X [27XSymmetrizations( Irr( tbl ){ [ 1 .. 3 ] }, 3 );[127X[104X
    [4X[28X[ VirtualCharacter( CharacterTable( "A5" ), [ 0, 0, 0, 0, 0 ] ), [128X[104X
    [4X[28X  VirtualCharacter( CharacterTable( "A5" ), [ 0, 0, 0, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [128X[104X
    [4X[28X    [ 8, 0, -1, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 10, -2, 1, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [128X[104X
    [4X[28X    [ 8, 0, -1, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 10, -2, 1, 0, 0 ] ) ][128X[104X
  [4X[32X[104X
  
  [1X72.11-2 SymmetricParts[101X
  
  [29X[2XSymmetricParts[102X( [3Xtbl[103X, [3Xcharacters[103X, [3Xn[103X ) [32X function
  
  [33X[0;0Yis the list of symmetrizations of the characters [3Xcharacters[103X of the character
  table  [3Xtbl[103X  with  the  trivial  character of the symmetric group of degree [3Xn[103X
  (see [2XSymmetrizations[102X ([14X72.11-1[114X)).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSymmetricParts( tbl, Irr( tbl ), 3 );[127X[104X
    [4X[28X[ Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 10, -2, 1, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 10, -2, 1, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 20, 0, 2, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 35, 3, 2, 0, 0 ] ) ][128X[104X
  [4X[32X[104X
  
  [1X72.11-3 AntiSymmetricParts[101X
  
  [29X[2XAntiSymmetricParts[102X( [3Xtbl[103X, [3Xcharacters[103X, [3Xn[103X ) [32X function
  
  [33X[0;0Yis the list of symmetrizations of the characters [3Xcharacters[103X of the character
  table  [3Xtbl[103X with the alternating character of the symmetric group of degree [3Xn[103X
  (see [2XSymmetrizations[102X ([14X72.11-1[114X)).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XAntiSymmetricParts( tbl, Irr( tbl ), 3 );[127X[104X
    [4X[28X[ VirtualCharacter( CharacterTable( "A5" ), [ 0, 0, 0, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 4, 0, 1, -1, -1 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 10, -2, 1, 0, 0 ] ) ][128X[104X
  [4X[32X[104X
  
  [1X72.11-4 OrthogonalComponents[101X
  
  [29X[2XOrthogonalComponents[102X( [3Xtbl[103X, [3Xchars[103X, [3Xm[103X ) [32X function
  
  [33X[0;0YIf  [22Xχ[122X  is a nonlinear character with indicator [22X+1[122X, a splitting of the tensor
  power [22Xχ^m[122X is given by the so-called Murnaghan functions (see [Mur58]). These
  components   in   general  have  fewer  irreducible  constituents  than  the
  symmetrizations  with  the  symmetric group of degree [3Xm[103X (see [2XSymmetrizations[102X
  ([14X72.11-1[114X)).[133X
  
  [33X[0;0Y[2XOrthogonalComponents[102X  returns  the  Murnaghan  components  of  the nonlinear
  characters  of  the character table [3Xtbl[103X in the list [3Xchars[103X up to the power [3Xm[103X,
  where [3Xm[103X is an integer between 2 and 6.[133X
  
  [33X[0;0YThe Murnaghan functions are implemented as in [Fra82].[133X
  
  [33X[0;0Y[13XNote[113X:  If  [3Xchars[103X  is  a list of character objects (see [2XIsCharacter[102X ([14X72.8-1[114X))
  then  also  the result consists of class function objects. It is not checked
  whether  all  characters  in [3Xchars[103X do really have indicator [22X+1[122X; if there are
  characters  with  indicator  [22X0[122X  or  [22X-1[122X,  the  result  might  contain virtual
  characters  (see also [2XSymplecticComponents[102X ([14X72.11-5[114X)), therefore the entries
  of the result do in general not know that they are characters.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtbl:= CharacterTable( "A8" );;  chi:= Irr( tbl )[2];[127X[104X
    [4X[28XCharacter( CharacterTable( "A8" ), [ 7, -1, 3, 4, 1, -1, 1, 2, 0, -1, [128X[104X
    [4X[28X  0, 0, -1, -1 ] )[128X[104X
    [4X[25Xgap>[125X [27XOrthogonalComponents( tbl, [ chi ], 3 );[127X[104X
    [4X[28X[ ClassFunction( CharacterTable( "A8" ), [128X[104X
    [4X[28X    [ 21, -3, 1, 6, 0, 1, -1, 1, -2, 0, 0, 0, 1, 1 ] ), [128X[104X
    [4X[28X  ClassFunction( CharacterTable( "A8" ), [128X[104X
    [4X[28X    [ 27, 3, 7, 9, 0, -1, 1, 2, 1, 0, -1, -1, -1, -1 ] ), [128X[104X
    [4X[28X  ClassFunction( CharacterTable( "A8" ), [128X[104X
    [4X[28X    [ 105, 1, 5, 15, -3, 1, -1, 0, -1, 1, 0, 0, 0, 0 ] ), [128X[104X
    [4X[28X  ClassFunction( CharacterTable( "A8" ), [128X[104X
    [4X[28X    [ 35, 3, -5, 5, 2, -1, -1, 0, 1, 0, 0, 0, 0, 0 ] ), [128X[104X
    [4X[28X  ClassFunction( CharacterTable( "A8" ), [128X[104X
    [4X[28X    [ 77, -3, 13, 17, 2, 1, 1, 2, 1, 0, 0, 0, 2, 2 ] ) ][128X[104X
  [4X[32X[104X
  
  [1X72.11-5 SymplecticComponents[101X
  
  [29X[2XSymplecticComponents[102X( [3Xtbl[103X, [3Xchars[103X, [3Xm[103X ) [32X function
  
  [33X[0;0YIf [22Xχ[122X is a (nonlinear) character with indicator [22X-1[122X, a splitting of the tensor
  power   [22Xχ^m[122X   is  given  in  terms  of  the  so-called  Murnaghan  functions
  (see [Mur58]).   These   components   in   general  have  fewer  irreducible
  constituents  than  the symmetrizations with the symmetric group of degree [3Xm[103X
  (see [2XSymmetrizations[102X ([14X72.11-1[114X)).[133X
  
  [33X[0;0Y[2XSymplecticComponents[102X returns the symplectic symmetrizations of the nonlinear
  characters  of  the character table [3Xtbl[103X in the list [3Xchars[103X up to the power [3Xm[103X,
  where [3Xm[103X is an integer between [22X2[122X and [22X5[122X.[133X
  
  [33X[0;0Y[13XNote[113X:  If  [3Xchars[103X  is  a list of character objects (see [2XIsCharacter[102X ([14X72.8-1[114X))
  then  also  the result consists of class function objects. It is not checked
  whether  all  characters  in [3Xchars[103X do really have indicator [22X-1[122X; if there are
  characters  with  indicator  [22X0[122X  or  [22X+1[122X,  the  result  might  contain virtual
  characters  (see also [2XOrthogonalComponents[102X ([14X72.11-4[114X)), therefore the entries
  of the result do in general not know that they are characters.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtbl:= CharacterTable( "U3(3)" );;  chi:= Irr( tbl )[2];[127X[104X
    [4X[28XCharacter( CharacterTable( "U3(3)" ), [128X[104X
    [4X[28X[ 6, -2, -3, 0, -2, -2, 2, 1, -1, -1, 0, 0, 1, 1 ] )[128X[104X
    [4X[25Xgap>[125X [27XSymplecticComponents( tbl, [ chi ], 3 );[127X[104X
    [4X[28X[ ClassFunction( CharacterTable( "U3(3)" ), [128X[104X
    [4X[28X    [ 14, -2, 5, -1, 2, 2, 2, 1, 0, 0, 0, 0, -1, -1 ] ), [128X[104X
    [4X[28X  ClassFunction( CharacterTable( "U3(3)" ), [128X[104X
    [4X[28X    [ 21, 5, 3, 0, 1, 1, 1, -1, 0, 0, -1, -1, 1, 1 ] ), [128X[104X
    [4X[28X  ClassFunction( CharacterTable( "U3(3)" ), [128X[104X
    [4X[28X    [ 64, 0, -8, -2, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0 ] ), [128X[104X
    [4X[28X  ClassFunction( CharacterTable( "U3(3)" ), [128X[104X
    [4X[28X    [ 14, 6, -4, 2, -2, -2, 2, 0, 0, 0, 0, 0, -2, -2 ] ), [128X[104X
    [4X[28X  ClassFunction( CharacterTable( "U3(3)" ), [128X[104X
    [4X[28X    [ 56, -8, 2, 2, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0 ] ) ][128X[104X
  [4X[32X[104X
  
  
  [1X72.12 [33X[0;0YMolien Series[133X[101X
  
  [1X72.12-1 MolienSeries[101X
  
  [29X[2XMolienSeries[102X( [[3Xtbl[103X, ][3Xpsi[103X[, [3Xchi[103X] ) [32X function
  
  [33X[0;0YThe  [13XMolien  series[113X  of the character [22Xψ[122X, relative to the character [22Xχ[122X, is the
  rational  function  given  by  the series [22XM_{ψ,χ}(z) = ∑_{d = 0}^∞ [χ,ψ^[d]]
  z^d[122X,  where [22Xψ^[d][122X denotes the symmetrization of [22Xψ[122X with the trivial character
  of the symmetric group [22XS_d[122X (see [2XSymmetricParts[102X ([14X72.11-2[114X)).[133X
  
  [33X[0;0Y[2XMolienSeries[102X  returns  the  Molien series of [3Xpsi[103X, relative to [3Xchi[103X, where [3Xpsi[103X
  and  [3Xchi[103X  must be characters of the same character table; this table must be
  entered  as  [3Xtbl[103X  if  [3Xchi[103X  and  [3Xpsi[103X  are only lists of character values. The
  default for [3Xchi[103X is the trivial character of [3Xtbl[103X.[133X
  
  [33X[0;0YThe   return  value  of  [2XMolienSeries[102X  stores  a  value  for  the  attribute
  [2XMolienSeriesInfo[102X  ([14X72.12-2[114X).  This admits the computation of coefficients of
  the  series  with  [2XValueMolienSeries[102X  ([14X72.12-3[114X). Furthermore, this attribute
  gives  access  to  numerator  and denominator of the Molien series viewed as
  rational  function, where the denominator is a product of polynomials of the
  form  [22X(1-z^r)^k[122X; the Molien series is also displayed in this form. Note that
  such     a     representation     is     not    unique,    one    can    use
  [2XMolienSeriesWithGivenDenominator[102X  ([14X72.12-4[114X)  to  obtain  the  series  with a
  prescribed denominator.[133X
  
  [33X[0;0YFor more information about Molien series, see [NPP84].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( AlternatingGroup( 5 ) );;[127X[104X
    [4X[25Xgap>[125X [27Xpsi:= First( Irr( t ), x -> Degree( x ) = 3 );;[127X[104X
    [4X[25Xgap>[125X [27Xmol:= MolienSeries( psi );[127X[104X
    [4X[28X( 1-z^2-z^3+z^6+z^7-z^9 ) / ( (1-z^5)*(1-z^3)*(1-z^2)^2 )[128X[104X
  [4X[32X[104X
  
  [1X72.12-2 MolienSeriesInfo[101X
  
  [29X[2XMolienSeriesInfo[102X( [3Xratfun[103X ) [32X attribute
  
  [33X[0;0YIf the rational function [3Xratfun[103X was constructed by [2XMolienSeries[102X ([14X72.12-1[114X), a
  representation as quotient of polynomials is known such that the denominator
  is  a product of terms of the form [22X(1-z^r)^k[122X. This information is encoded as
  value of [2XMolienSeriesInfo[102X. Additionally, there is a special [2XPrintObj[102X ([14X6.3-5[114X)
  method for Molien series based on this.[133X
  
  [33X[0;0Y[2XMolienSeriesInfo[102X  returns  a  record  that  describes  the rational function
  [3Xratfun[103X as a Molien series. The components of this record are[133X
  
  [8X[10Xnumer[110X[8X[108X
        [33X[0;6Ynumerator  of  [3Xratfun[103X (in general a multiple of the numerator one gets
        by [2XNumeratorOfRationalFunction[102X ([14X66.4-2[114X)),[133X
  
  [8X[10Xdenom[110X[8X[108X
        [33X[0;6Ydenominator  of  [3Xratfun[103X  (in general a multiple of the denominator one
        gets by [2XNumeratorOfRationalFunction[102X ([14X66.4-2[114X)),[133X
  
  [8X[10Xratfun[110X[8X[108X
        [33X[0;6Ythe rational function [3Xratfun[103X itself,[133X
  
  [8X[10Xnumerstring[110X[8X[108X
        [33X[0;6Ystring corresponding to the polynomial [10Xnumer[110X, expressed in terms of [10Xz[110X,[133X
  
  [8X[10Xdenomstring[110X[8X[108X
        [33X[0;6Ystring corresponding to the polynomial [10Xdenom[110X, expressed in terms of [10Xz[110X,[133X
  
  [8X[10Xdenominfo[110X[8X[108X
        [33X[0;6Ya list of the form [22X[ [ r_1, k_1 ], ..., [ r_n, k_n ] ][122X such that [10Xdenom[110X
        is [22X∏_{i = 1}^n (1-z^{r_i})^{k_i}[122X.[133X
  
  [8X[10Xsummands[110X[8X[108X
        [33X[0;6Ya  list  of  records,  each  with  the  components  [10Xnumer[110X,  [10Xr[110X,  and [10Xk[110X,
        describing the summand [10Xnumer[110X[22X/ (1-z^r)^k[122X,[133X
  
  [8X[10Xsize[110X[8X[108X
        [33X[0;6Ythe order of the underlying matrix group,[133X
  
  [8X[10Xdegree[110X[8X[108X
        [33X[0;6Ythe degree of the underlying matrix representation.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XHasMolienSeriesInfo( mol );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XMolienSeriesInfo( mol );[127X[104X
    [4X[28Xrec( degree := 3, [128X[104X
    [4X[28X  denom := x_1^12-2*x_1^10-x_1^9+x_1^8+x_1^7+x_1^5+x_1^4-x_1^3-2*x_1^2\[128X[104X
    [4X[28X+1, denominfo := [ 5, 1, 3, 1, 2, 2 ], [128X[104X
    [4X[28X  denomstring := "(1-z^5)*(1-z^3)*(1-z^2)^2", [128X[104X
    [4X[28X  numer := -x_1^9+x_1^7+x_1^6-x_1^3-x_1^2+1, [128X[104X
    [4X[28X  numerstring := "1-z^2-z^3+z^6+z^7-z^9", [128X[104X
    [4X[28X  ratfun := ( 1-z^2-z^3+z^6+z^7-z^9 ) / ( (1-z^5)*(1-z^3)*(1-z^2)^2 ),[128X[104X
    [4X[28X  size := 60, [128X[104X
    [4X[28X  summands := [ rec( k := 1, numer := [ -24, -12, -24 ], r := 5 ), [128X[104X
    [4X[28X      rec( k := 1, numer := [ -20 ], r := 3 ), [128X[104X
    [4X[28X      rec( k := 2, numer := [ -45/4, 75/4, -15/4, -15/4 ], r := 2 ), [128X[104X
    [4X[28X      rec( k := 3, numer := [ -1 ], r := 1 ), [128X[104X
    [4X[28X      rec( k := 1, numer := [ -15/4 ], r := 1 ) ] )[128X[104X
  [4X[32X[104X
  
  [1X72.12-3 ValueMolienSeries[101X
  
  [29X[2XValueMolienSeries[102X( [3Xmolser[103X, [3Xi[103X ) [32X function
  
  [33X[0;0Yis the [3Xi[103X-th coefficient of the Molien series [3Xseries[103X computed by [2XMolienSeries[102X
  ([14X72.12-1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XList( [ 0 .. 20 ], i -> ValueMolienSeries( mol, i ) );[127X[104X
    [4X[28X[ 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 3, 0, 4, 0, 4, 1, 5, 1, 6, 1, 7 ][128X[104X
  [4X[32X[104X
  
  [1X72.12-4 MolienSeriesWithGivenDenominator[101X
  
  [29X[2XMolienSeriesWithGivenDenominator[102X( [3Xmolser[103X, [3Xlist[103X ) [32X function
  
  [33X[0;0Yis  a  Molien  series  equal  to  [3Xmolser[103X as rational function, but viewed as
  quotient  with denominator [22X∏_{i = 1}^n (1-z^{r_i})[122X, where [22X[3Xlist[103X = [ r_1, r_2,
  ..., r_n ][122X. If [3Xmolser[103X cannot be represented this way, [9Xfail[109X is returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMolienSeriesWithGivenDenominator( mol, [ 2, 6, 10 ] );[127X[104X
    [4X[28X( 1+z^15 ) / ( (1-z^10)*(1-z^6)*(1-z^2) )[128X[104X
  [4X[32X[104X
  
  
  [1X72.13 [33X[0;0YPossible Permutation Characters[133X[101X
  
  [33X[0;0YFor  groups  [22XH[122X and [22XG[122X with [22XH ≤ G[122X, the induced character [22X(1_G)^H[122X is called the
  [13Xpermutation  character[113X  of  the  operation of [22XG[122X on the right cosets of [22XH[122X. If
  only  the  character table of [22XG[122X is available and not the group [22XG[122X itself, one
  can  try  to  get information about possible subgroups of [22XG[122X by inspection of
  those  [22XG[122X-class  functions  that  might be permutation characters, using that
  such  a  class  function [22Xπ[122X must have at least the following properties. (For
  details, see [Isa76, Theorem 5.18.]),[133X
  
  [8X(a)[108X
        [33X[0;6Y[22Xπ[122X is a character of [22XG[122X,[133X
  
  [8X(b)[108X
        [33X[0;6Y[22Xπ(g)[122X is a nonnegative integer for all [22Xg ∈ G[122X,[133X
  
  [8X(c)[108X
        [33X[0;6Y[22Xπ(1)[122X divides [22X|G|[122X,[133X
  
  [8X(d)[108X
        [33X[0;6Y[22Xπ(g^n) ≥ π(g)[122X for [22Xg ∈ G[122X and integers [22Xn[122X,[133X
  
  [8X(e)[108X
        [33X[0;6Y[22X[π, 1_G] = 1[122X,[133X
  
  [8X(f)[108X
        [33X[0;6Ythe  multiplicity  of  any  rational  irreducible  [22XG[122X-character  [22Xψ[122X as a
        constituent of [22Xπ[122X is at most [22Xψ(1)/[ψ, ψ][122X,[133X
  
  [8X(g)[108X
        [33X[0;6Y[22Xπ(g) = 0[122X if the order of [22Xg[122X does not divide [22X|G|/π(1)[122X,[133X
  
  [8X(h)[108X
        [33X[0;6Y[22Xπ(1) |N_G(g)|[122X divides [22Xπ(g) |G|[122X for all [22Xg ∈ G[122X,[133X
  
  [8X(i)[108X
        [33X[0;6Y[22Xπ(g)  ≤  (|G|  - π(1)) / (|g^G| |Gal_G(g)|)[122X for all nonidentity [22Xg ∈ G[122X,
        where  [22X|Gal_G(g)|[122X  denotes  the  number of conjugacy classes of [22XG[122X that
        contain generators of the group [22X⟨ g ⟩[122X,[133X
  
  [8X(j)[108X
        [33X[0;6Yif  [22Xp[122X  is a prime that divides [22X|G|/π(1)[122X only once then [22Xs/(p-1)[122X divides
        [22X|G|/π(1)[122X  and  is  congruent  to  [22X1[122X modulo [22Xp[122X, where [22Xs[122X is the number of
        elements  of  order  [22Xp[122X  in the (hypothetical) subgroup [22XH[122X for which [22Xπ =
        (1_H)^G[122X  holds.  (Note  that  [22Xs/(p-1)[122X  equals  the  number  of Sylow [22Xp[122X
        subgroups in [22XH[122X.)[133X
  
  [33X[0;0YAny  [22XG[122X-class function with these properties is called a [13Xpossible permutation
  character[113X in [5XGAP[105X.[133X
  
  [33X[0;0Y(Condition  (d)  is checked only for those power maps that are stored in the
  character table of [22XG[122X; clearly (d) holds for all integers if it holds for all
  prime divisors of the group order [22X|G|[122X.)[133X
  
  [33X[0;0Y[5XGAP[105X  provides  some  algorithms  to  compute possible permutation characters
  (see [2XPermChars[102X  ([14X72.14-1[114X)),  and also provides functions to check a few more
  criteria whether a given character can be a transitive permutation character
  (see [2XTestPerm1[102X ([14X72.14-2[114X)).[133X
  
  [33X[0;0YSome  information  about the subgroup [22XU[122X can be computed from the permutation
  character [22X(1_U)^G[122X using [2XPermCharInfo[102X ([14X72.13-1[114X).[133X
  
  [1X72.13-1 PermCharInfo[101X
  
  [29X[2XPermCharInfo[102X( [3Xtbl[103X, [3Xpermchars[103X[, [3Xformat[103X] ) [32X function
  
  [33X[0;0YLet [3Xtbl[103X be the ordinary character table of the group [22XG[122X, and [3Xpermchars[103X either
  the  permutation character [22X(1_U)^G[122X, for a subgroup [22XU[122X of [22XG[122X, or a list of such
  permutation  characters.  [2XPermCharInfo[102X  returns  a record with the following
  components.[133X
  
  [8X[10Xcontained[110X[8X:[108X
        [33X[0;6Ya list containing, for each character [22Xψ = (1_U)^G[122X in [3Xpermchars[103X, a list
        containing  at position [22Xi[122X the number [22Xψ[i] |U| /[122X [10XSizesCentralizers( [110X[3Xtbl[103X[10X
        )[110X[22X[i][122X,  which  equals  the  number  of  those  elements  of  [22XU[122X that are
        contained in class [22Xi[122X of [3Xtbl[103X,[133X
  
  [8X[10Xbound[110X[8X:[108X
        [33X[0;6Ya list containing, for each character [22Xψ = (1_U)^G[122X in [3Xpermchars[103X, a list
        containing at position [22Xi[122X the number [22X|U| / gcd( |U|,[122X [10XSizesCentralizers(
        [3Xtbl[103X[10X )[110X[22X[i] )[122X, which divides the class length in [22XU[122X of an element in class
        [22Xi[122X of [3Xtbl[103X,[133X
  
  [8X[10Xdisplay[110X[8X:[108X
        [33X[0;6Ya  record  that  can  be used as second argument of [2XDisplay[102X ([14X6.3-6[114X) to
        display  each permutation character in [3Xpermchars[103X and the corresponding
        components  [10Xcontained[110X  and [10Xbound[110X, for those classes where at least one
        character of [3Xpermchars[103X is nonzero,[133X
  
  [8X[10XATLAS[110X[8X:[108X
        [33X[0;6Ya  list  of  strings  describing  the decomposition of the permutation
        characters  in [3Xpermchars[103X into the irreducible characters of [3Xtbl[103X, given
        in   an   [5XAtlas[105X-like   notation.   This  means  that  the  irreducible
        constituents  are  indicated  by  their degrees followed by lower case
        letters  [10Xa[110X,  [10Xb[110X,  [10Xc[110X,  [22X...[122X,  which  indicate  the successive irreducible
        characters of [3Xtbl[103X of that degree, in the order in which they appear in
        [10XIrr(  [110X[3Xtbl[103X[10X  )[110X.  A  sequence of small letters (not necessarily distinct)
        after  a single number indicates a sum of irreducible constituents all
        of  the same degree, an exponent [3Xn[103X for the letter [3Xlett[103X means that [3Xlett[103X
        is  repeated  [3Xn[103X  times.  The  default  notation  for exponentiation is
        [10X[3Xlett[103X[10X^{[3Xn[103X[10X}[110X, this is also chosen if the optional third argument [3Xformat[103X is
        the  string  [10X"LaTeX"[110X;  if the third argument is the string [10X"HTML"[110X then
        exponentiation is denoted by [10X[3Xlett[103X[10X<sup>[3Xn[103X[10X</sup>[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "A6" );;[127X[104X
    [4X[25Xgap>[125X [27Xpsi:= Sum( Irr( t ){ [ 1, 3, 6 ] } );[127X[104X
    [4X[28XCharacter( CharacterTable( "A6" ), [ 15, 3, 0, 3, 1, 0, 0 ] )[128X[104X
    [4X[25Xgap>[125X [27Xinfo:= PermCharInfo( t, psi );[127X[104X
    [4X[28Xrec( ATLAS := [ "1a+5b+9a" ], bound := [ [ 1, 3, 8, 8, 6, 24, 24 ] ], [128X[104X
    [4X[28X  contained := [ [ 1, 9, 0, 8, 6, 0, 0 ] ], [128X[104X
    [4X[28X  display := [128X[104X
    [4X[28X    rec( [128X[104X
    [4X[28X      chars := [ [ 15, 3, 0, 3, 1, 0, 0 ], [ 1, 9, 0, 8, 6, 0, 0 ], [128X[104X
    [4X[28X          [ 1, 3, 8, 8, 6, 24, 24 ] ], classes := [ 1, 2, 4, 5 ], [128X[104X
    [4X[28X      letter := "I" ) )[128X[104X
    [4X[25Xgap>[125X [27XDisplay( t, info.display );[127X[104X
    [4X[28XA6[128X[104X
    [4X[28X[128X[104X
    [4X[28X     2  3  3  .  2[128X[104X
    [4X[28X     3  2  .  2  .[128X[104X
    [4X[28X     5  1  .  .  .[128X[104X
    [4X[28X[128X[104X
    [4X[28X       1a 2a 3b 4a[128X[104X
    [4X[28X    2P 1a 1a 3b 2a[128X[104X
    [4X[28X    3P 1a 2a 1a 4a[128X[104X
    [4X[28X    5P 1a 2a 3b 4a[128X[104X
    [4X[28X[128X[104X
    [4X[28XI.1    15  3  3  1[128X[104X
    [4X[28XI.2     1  9  8  6[128X[104X
    [4X[28XI.3     1  3  8  6[128X[104X
    [4X[25Xgap>[125X [27Xj1:= CharacterTable( "J1" );;[127X[104X
    [4X[25Xgap>[125X [27Xpsi:= TrivialCharacter( CharacterTable( "7:6" ) )^j1;[127X[104X
    [4X[28XCharacter( CharacterTable( "J1" ), [ 4180, 20, 10, 0, 0, 2, 1, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0 ] )[128X[104X
    [4X[25Xgap>[125X [27XPermCharInfo( j1, psi ).ATLAS;[127X[104X
    [4X[28X[ "1a+56aabb+76aaab+77aabbcc+120aaabbbccc+133a^{4}bbcc+209a^{5}" ][128X[104X
  [4X[32X[104X
  
  [1X72.13-2 PermCharInfoRelative[101X
  
  [29X[2XPermCharInfoRelative[102X( [3Xtbl[103X, [3Xtbl2[103X, [3Xpermchars[103X ) [32X function
  
  [33X[0;0YLet  [3Xtbl[103X  and  [3Xtbl2[103X  be the ordinary character tables of two groups [22XH[122X and [22XG[122X,
  respectively,  where  [22XH[122X  is  of  index  two  in  [22XG[122X, and [3Xpermchars[103X either the
  permutation  character  [22X(1_U)^G[122X,  for  a  subgroup [22XU[122X of [22XG[122X, or a list of such
  permutation  characters. [2XPermCharInfoRelative[102X returns a record with the same
  components as [2XPermCharInfo[102X ([14X72.13-1[114X), the only exception is that the entries
  of the [10XATLAS[110X component are names relative to [3Xtbl[103X.[133X
  
  [33X[0;0YMore precisely, the [22Xi[122X-th entry of the [10XATLAS[110X component is a string describing
  the   decomposition  of  the  [22Xi[122X-th  entry  in  [3Xpermchars[103X.  The  degrees  and
  distinguishing letters of the constituents refer to the irreducibles of [3Xtbl[103X,
  as  follows.  The  two irreducible characters of [3Xtbl2[103X of degree [22XN[122X, say, that
  extend  the  irreducible character [22XN[122X [10Xa[110X of [3Xtbl[103X are denoted by [22XN[122X [10Xa[110X[22X^+[122X and [22XN[122X[10Xa[110X[22X^-[122X.
  The  irreducible  character  of [3Xtbl2[103X of degree [22X2N[122X, say, whose restriction to
  [3Xtbl[103X is the sum of the irreducible characters [22XN[122X [10Xa[110X and [22XN[122X [10Xb[110X is denoted as [22XN[122X [10Xab[110X.
  Multiplicities larger than [22X1[122X of constituents are denoted by exponents.[133X
  
  [33X[0;0Y(This format is useful mainly for multiplicity free permutation characters.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "A5" );;[127X[104X
    [4X[25Xgap>[125X [27Xt2:= CharacterTable( "A5.2" );;[127X[104X
    [4X[25Xgap>[125X [27XList( Irr( t2 ), x -> x[1] );[127X[104X
    [4X[28X[ 1, 1, 6, 4, 4, 5, 5 ][128X[104X
    [4X[25Xgap>[125X [27XList( Irr( t ), x -> x[1] );[127X[104X
    [4X[28X[ 1, 3, 3, 4, 5 ][128X[104X
    [4X[25Xgap>[125X [27Xpermchars:= List( [ [1], [1,2], [1,7], [1,3,4,4,6,6,7] ],[127X[104X
    [4X[25X>[125X [27X                     l -> Sum( Irr( t2 ){ l } ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "A5.2" ), [ 1, 1, 1, 1, 1, 1, 1 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5.2" ), [ 2, 2, 2, 2, 0, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5.2" ), [ 6, 2, 0, 1, 0, 2, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5.2" ), [ 30, 2, 0, 0, 6, 0, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27Xinfo:= PermCharInfoRelative( t, t2, permchars );;[127X[104X
    [4X[25Xgap>[125X [27Xinfo.ATLAS;[127X[104X
    [4X[28X[ "1a^+", "1a^{\\pm}", "1a^++5a^-", [128X[104X
    [4X[28X  "1a^++3ab+4(a^+)^{2}+5a^+a^{\\pm}" ][128X[104X
  [4X[32X[104X
  
  
  [1X72.14 [33X[0;0YComputing Possible Permutation Characters[133X[101X
  
  [1X72.14-1 PermChars[101X
  
  [29X[2XPermChars[102X( [3Xtbl[103X[, [3Xcond[103X] ) [32X function
  
  [33X[0;0Y[5XGAP[105X provides several algorithms to determine possible permutation characters
  from  a  given  character table. They are described in detail in [BP98]. The
  algorithm  is  selected  from  the choice of the optional argument [3Xcond[103X. The
  user  is  encouraged  to  try different approaches, especially if one choice
  fails to come to an end.[133X
  
  [33X[0;0YRegardless  of  the  algorithm  used in a specific case, [2XPermChars[102X returns a
  list of [13Xall[113X possible permutation characters with the properties described by
  [3Xcond[103X.  There  is  no  guarantee  that  a character of this list is in fact a
  permutation   character.  But  an  empty  list  always  means  there  is  no
  permutation character with these properties (e.g., of a certain degree).[133X
  
  [33X[0;0YCalled  with only one argument, a character table [3Xtbl[103X, [2XPermChars[102X returns the
  list of all possible permutation characters of the group with this character
  table.  This  list  might be rather long for big groups, and its computation
  might  take much time. The algorithm is described in [BP98, Section 3.2]; it
  depends  on  a  preprocessing  step, where the inequalities arising from the
  condition [22Xπ(g) ≥ 0[122X are transformed into a system of inequalities that guides
  the  search  (see [2XInequalities[102X ([14X72.14-5[114X)). So the following commands compute
  the list of 39 possible permutation characters of the Mathieu group [22XM_11[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm11:= CharacterTable( "M11" );;[127X[104X
    [4X[25Xgap>[125X [27XSetName( m11, "m11" );[127X[104X
    [4X[25Xgap>[125X [27Xperms:= PermChars( m11 );;[127X[104X
    [4X[25Xgap>[125X [27XLength( perms );[127X[104X
    [4X[28X39[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThere  are  two  different search strategies for this algorithm. The default
  strategy  simply  constructs all characters with nonnegative values and then
  tests  for  each such character whether its degree is a divisor of the order
  of  the group. The other strategy uses the inequalities to predict whether a
  character  of a certain degree can lie in the currently searched part of the
  search  tree. To choose this strategy, enter a record as the second argument
  of  [2XPermChars[102X,  and  set its component [10Xdegree[110X to the range of degrees (which
  might  also  be a range containing all divisors of the group order) you want
  to   look  for;  additionally,  the  record  component  [10Xineq[110X  can  take  the
  inequalities computed by [2XInequalities[102X ([14X72.14-5[114X) if they are needed more than
  once.[133X
  
  [33X[0;0YIf  a  positive  integer  is  given  as  the second argument [3Xcond[103X, [2XPermChars[102X
  returns  the  list  of  all possible permutation characters of [3Xtbl[103X that have
  degree  [3Xcond[103X.  For  that  purpose,  a  preprocessing step is performed where
  essentially  the  rational character table is inverted in order to determine
  boundary points for the simplex in which the possible permutation characters
  of  the  given  degree must lie (see [2XPermBounds[102X ([14X72.14-3[114X)). The algorithm is
  described at the end of [BP98, Section 3.2]. Note that inverting big integer
  matrices  needs a lot of time and space. So this preprocessing is restricted
  to groups with less than 100 classes, say.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xdeg220:= PermChars( m11, 220 );[127X[104X
    [4X[28X[ Character( m11, [ 220, 4, 4, 0, 0, 4, 0, 0, 0, 0 ] ), [128X[104X
    [4X[28X  Character( m11, [ 220, 12, 4, 4, 0, 0, 0, 0, 0, 0 ] ), [128X[104X
    [4X[28X  Character( m11, [ 220, 20, 4, 0, 0, 2, 0, 0, 0, 0 ] ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YIf a record is given as the second argument [3Xcond[103X, [2XPermChars[102X returns the list
  of all possible permutation characters that have the properties described by
  the  components of this record. One such situation has been mentioned above.
  If  [3Xcond[103X  contains  a  degree  as  value of the record component [10Xdegree[110X then
  [2XPermChars[102X will behave exactly as if this degree was entered as [3Xcond[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xdeg220 = PermChars( m11, rec( degree:= 220 ) );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor   the   meaning   of  additional  components  of  [3Xcond[103X  besides  [10Xdegree[110X,
  see [2XPermComb[102X ([14X72.14-4[114X).[133X
  
  [33X[0;0YInstead  of  [10Xdegree[110X,  [3Xcond[103X may have the component [10Xtorso[110X bound to a list that
  contains  some  known  values  of  the  required  characters  at  the  right
  positions;  at  least  the  degree [3Xcond[103X[10X.torso[1][110X must be an integer. In this
  case,  the  algorithm  described  in  [BP98,  Section  3.3]  is  chosen. The
  component  [10Xchars[110X, if present, holds a list of all those [13Xrational[113X irreducible
  characters of [3Xtbl[103X that might be constituents of the required characters.[133X
  
  [33X[0;0Y([13XNote[113X:  If [3Xcond[103X[10X.chars[110X is bound and does not contain [13Xall[113X rational irreducible
  characters  of  [3Xtbl[103X,  [5XGAP[105X  checks  whether  the scalar products of all class
  functions   in  the  result  list  with  the  omitted  rational  irreducible
  characters of [3Xtbl[103X are nonnegative; so there should be nontrivial reasons for
  excluding  a  character that is known to be not a constituent of the desired
  possible permutation characters.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPermChars( m11, rec( torso:= [ 220 ] ) );[127X[104X
    [4X[28X[ Character( m11, [ 220, 4, 4, 0, 0, 4, 0, 0, 0, 0 ] ), [128X[104X
    [4X[28X  Character( m11, [ 220, 20, 4, 0, 0, 2, 0, 0, 0, 0 ] ), [128X[104X
    [4X[28X  Character( m11, [ 220, 12, 4, 4, 0, 0, 0, 0, 0, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XPermChars( m11, rec( torso:= [ 220,,,,, 2 ] ) );[127X[104X
    [4X[28X[ Character( m11, [ 220, 20, 4, 0, 0, 2, 0, 0, 0, 0 ] ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YAn  additional  restriction  on the possible permutation characters computed
  can  be  forced  if  [3Xcon[103X  contains,  in  addition  to  [10Xtorso[110X, the components
  [10Xnormalsubgroup[110X  and  [10Xnonfaithful[110X, with values a list of class positions of a
  normal subgroup [22XN[122X of the group [22XG[122X of [3Xtbl[103X and a possible permutation character
  [22Xπ[122X  of  [22XG[122X, respectively, such that [22XN[122X is contained in the kernel of [22Xπ[122X. In this
  case,  [2XPermChars[102X returns the list of those possible permutation characters [22Xψ[122X
  of  [3Xtbl[103X  coinciding  with [10Xtorso[110X wherever its values are bound and having the
  property  that  no  irreducible constituent of [22Xψ - π[122X has [22XN[122X in its kernel. If
  the  component  [10Xchars[110X  is  bound in [3Xcond[103X then the above statements apply. An
  interpretation  of  the  computed characters is the following. Suppose there
  exists  a  subgroup  [22XV[122X  of  [22XG[122X  such  that  [22Xπ = (1_V)^G[122X; Then [22XN ≤ V[122X, and if a
  computed  character  is of the form [22X(1_U)^G[122X, for a subgroup [22XU[122X of [22XG[122X, then [22XV =
  UN[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs4:= CharacterTable( "Symmetric", 4 );;[127X[104X
    [4X[25Xgap>[125X [27Xnsg:= ClassPositionsOfDerivedSubgroup( s4 );;[127X[104X
    [4X[25Xgap>[125X [27Xpi:= TrivialCharacter( s4 );;[127X[104X
    [4X[25Xgap>[125X [27XPermChars( s4, rec( torso:= [ 12 ], normalsubgroup:= nsg,[127X[104X
    [4X[25X>[125X [27X                       nonfaithful:= pi ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "Sym(4)" ), [ 12, 2, 0, 0, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27Xpi:= Sum( Filtered( Irr( s4 ),[127X[104X
    [4X[25X>[125X [27X             chi -> IsSubset( ClassPositionsOfKernel( chi ), nsg ) ) );[127X[104X
    [4X[28XCharacter( CharacterTable( "Sym(4)" ), [ 2, 0, 2, 2, 0 ] )[128X[104X
    [4X[25Xgap>[125X [27XPermChars( s4, rec( torso:= [ 12 ], normalsubgroup:= nsg,[127X[104X
    [4X[25X>[125X [27X                       nonfaithful:= pi ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "Sym(4)" ), [ 12, 0, 4, 0, 0 ] ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  class  functions  returned  by  [2XPermChars[102X have the properties tested by
  [2XTestPerm1[102X  ([14X72.14-2[114X),  [2XTestPerm2[102X ([14X72.14-2[114X), and [2XTestPerm3[102X ([14X72.14-2[114X). So they
  are  possible  permutation  characters. See [2XTestPerm1[102X ([14X72.14-2[114X) for criteria
  whether  a  possible  permutation  character  can  in  fact be a permutation
  character.[133X
  
  
  [1X72.14-2 [33X[0;0YTestPerm1, ..., TestPerm5[133X[101X
  
  [29X[2XTestPerm1[102X( [3Xtbl[103X, [3Xchar[103X ) [32X function
  [29X[2XTestPerm2[102X( [3Xtbl[103X, [3Xchar[103X ) [32X function
  [29X[2XTestPerm3[102X( [3Xtbl[103X, [3Xchars[103X ) [32X function
  [29X[2XTestPerm4[102X( [3Xtbl[103X, [3Xchars[103X ) [32X function
  [29X[2XTestPerm5[102X( [3Xtbl[103X, [3Xchars[103X, [3Xmodtbl[103X ) [32X function
  
  [33X[0;0YThe  first  three  of  these  functions implement tests of the properties of
  possible  permutation  characters  listed  in  Section [14X72.13[114X,  The other two
  implement  test  of additional properties. Let [3Xtbl[103X be the ordinary character
  table  of a group [22XG[122X, say, [3Xchar[103X a rational character of [3Xtbl[103X, and [3Xchars[103X a list
  of  rational  characters  of [3Xtbl[103X. For applying [2XTestPerm5[102X, the knowledge of a
  [22Xp[122X-modular  Brauer  table  [3Xmodtbl[103X  of  [22XG[122X is required. [2XTestPerm4[102X and [2XTestPerm5[102X
  expect  the  characters  in  [3Xchars[103X  to  satisfy  the  conditions  checked by
  [2XTestPerm1[102X and [2XTestPerm2[102X (see below).[133X
  
  [33X[0;0YThe  return values of the functions were chosen parallel to the tests listed
  in [NPP84].[133X
  
  [33X[0;0Y[2XTestPerm1[102X return [10X1[110X or [10X2[110X if [3Xchar[103X fails because of (T1) or (T2), respectively;
  this  corresponds  to  the  criteria (b) and (d). Note that only those power
  maps  are  considered  that  are  stored  on  [3Xtbl[103X.  If  [3Xchar[103X  satisfies  the
  conditions, [10X0[110X is returned.[133X
  
  [33X[0;0Y[2XTestPerm2[102X  returns  [10X1[110X if [3Xchar[103X fails because of the criterion (c), it returns
  [10X3[110X, [10X4[110X, or [10X5[110X if [3Xchar[103X fails because of (T3), (T4), or (T5), respectively; these
  tests  correspond  to  (g), a weaker form of (h), and (j). If [3Xchar[103X satisfies
  the conditions, [10X0[110X is returned.[133X
  
  [33X[0;0Y[2XTestPerm3[102X  returns  the  list of all those class functions in the list [3Xchars[103X
  that satisfy criterion (h); this is a stronger version of (T6).[133X
  
  [33X[0;0Y[2XTestPerm4[102X  returns  the  list of all those class functions in the list [3Xchars[103X
  that satisfy (T8) and (T9) for each prime divisor [22Xp[122X of the order of [22XG[122X; these
  tests  use modular representation theory but do not require the knowledge of
  decomposition matrices (cf. [2XTestPerm5[102X below).[133X
  
  [33X[0;0Y(T8) implements the test of the fact that in the case that [22Xp[122X divides [22X|G|[122X and
  the  degree  of  a  transitive  permutation  character  [22Xπ[122X  exactly once, the
  projective  cover  of the trivial character is a summand of [22Xπ[122X. (This test is
  omitted if the projective cover cannot be identified.)[133X
  
  [33X[0;0YGiven  a  permutation  character  [22Xπ[122X  of a group [22XG[122X and a prime integer [22Xp[122X, the
  restriction  [22Xπ_B[122X  to  a  [22Xp[122X-block [22XB[122X of [22XG[122X has the following property, which is
  checked  by (T9). For each [22Xg ∈ G[122X such that [22Xg^n[122X is a [22Xp[122X-element of [22XG[122X, [22Xπ_B(g^n)[122X
  is  a nonnegative integer that satisfies [22X|π_B(g)| ≤ π_B(g^n) ≤ π(g^n)[122X. (This
  is [Sco73, Corollary A on p. 113].)[133X
  
  [33X[0;0Y[2XTestPerm5[102X  requires the [22Xp[122X-modular Brauer table [3Xmodtbl[103X of [22XG[122X, for some prime [22Xp[122X
  dividing  the  order  of  [22XG[122X, and checks whether those characters in the list
  [3Xchars[103X  whose  degree  is  divisible  by  the [22Xp[122X-part of the order of [22XG[122X can be
  decomposed  into projective indecomposable characters; [2XTestPerm5[102X returns the
  sublist  of all those characters in [3Xchars[103X that either satisfy this condition
  or to which the test does not apply.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtbl:= CharacterTable( "A5" );;[127X[104X
    [4X[25Xgap>[125X [27Xrat:= RationalizedMat( Irr( tbl ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 6, -2, 0, 1, 1 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 4, 0, 1, -1, -1 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 5, 1, -1, 0, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27Xtup:= Filtered( Tuples( [ 0, 1 ], 4 ), x -> not IsZero( x ) );[127X[104X
    [4X[28X[ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 1 ], [ 0, 1, 0, 0 ], [128X[104X
    [4X[28X  [ 0, 1, 0, 1 ], [ 0, 1, 1, 0 ], [ 0, 1, 1, 1 ], [ 1, 0, 0, 0 ], [128X[104X
    [4X[28X  [ 1, 0, 0, 1 ], [ 1, 0, 1, 0 ], [ 1, 0, 1, 1 ], [ 1, 1, 0, 0 ], [128X[104X
    [4X[28X  [ 1, 1, 0, 1 ], [ 1, 1, 1, 0 ], [ 1, 1, 1, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27Xlincomb:= List( tup, coeff -> coeff * rat );;[127X[104X
    [4X[25Xgap>[125X [27XList( lincomb, psi -> TestPerm1( tbl, psi ) );[127X[104X
    [4X[28X[ 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0 ][128X[104X
    [4X[25Xgap>[125X [27XList( lincomb, psi -> TestPerm2( tbl, psi ) );[127X[104X
    [4X[28X[ 0, 5, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1 ][128X[104X
    [4X[25Xgap>[125X [27XSet( List( TestPerm3(tbl, lincomb), x -> Position(lincomb, x) ) );[127X[104X
    [4X[28X[ 1, 4, 6, 7, 8, 9, 10, 11, 13 ][128X[104X
    [4X[25Xgap>[125X [27Xtbl:= CharacterTable( "A7" );[127X[104X
    [4X[28XCharacterTable( "A7" )[128X[104X
    [4X[25Xgap>[125X [27Xperms:= PermChars( tbl, rec( degree:= 315 ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "A7" ), [ 315, 3, 0, 0, 3, 0, 0, 0, 0 ] )[128X[104X
    [4X[28X    , Character( CharacterTable( "A7" ), [128X[104X
    [4X[28X    [ 315, 15, 0, 0, 1, 0, 0, 0, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XTestPerm4( tbl, perms );[127X[104X
    [4X[28X[ Character( CharacterTable( "A7" ), [ 315, 15, 0, 0, 1, 0, 0, 0, 0 [128X[104X
    [4X[28X     ] ) ][128X[104X
    [4X[25Xgap>[125X [27Xperms:= PermChars( tbl, rec( degree:= 15 ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "A7" ), [ 15, 3, 0, 3, 1, 0, 0, 1, 1 ] ),[128X[104X
    [4X[28X  Character( CharacterTable( "A7" ), [ 15, 3, 3, 0, 1, 0, 3, 1, 1 ] ) [128X[104X
    [4X[28X ][128X[104X
    [4X[25Xgap>[125X [27XTestPerm5( tbl, perms, tbl mod 5 );[127X[104X
    [4X[28X[ Character( CharacterTable( "A7" ), [ 15, 3, 0, 3, 1, 0, 0, 1, 1 ] ) [128X[104X
    [4X[28X ][128X[104X
  [4X[32X[104X
  
  [1X72.14-3 PermBounds[101X
  
  [29X[2XPermBounds[102X( [3Xtbl[103X, [3Xd[103X ) [32X function
  
  [33X[0;0YLet  [3Xtbl[103X  be the ordinary character table of the group [22XG[122X. All [22XG[122X-characters [22Xπ[122X
  satisfying  [22Xπ(g)  >  0[122X  and [22Xπ(1) = [3Xd[103X[122X, for a given degree [3Xd[103X, lie in a simplex
  described  by  these  conditions. [2XPermBounds[102X computes the boundary points of
  this  simplex  for [22Xd = 0[122X, from which the boundary points for any other [3Xd[103X are
  easily  derived.  (Some  conditions  from  the  power  maps  of [3Xtbl[103X are also
  involved.)  For  this  purpose,  a  matrix similar to the rational character
  table  of  [22XG[122X has to be inverted. These boundary points are used by [2XPermChars[102X
  ([14X72.14-1[114X)  to construct all possible permutation characters (see [14X72.13[114X) of a
  given  degree.  [2XPermChars[102X  ([14X72.14-1[114X)  either  calls [2XPermBounds[102X or takes this
  information from the [10Xbounds[110X component of its argument record.[133X
  
  [1X72.14-4 PermComb[101X
  
  [29X[2XPermComb[102X( [3Xtbl[103X, [3Xarec[103X ) [32X function
  
  [33X[0;0Y[2XPermComb[102X computes possible permutation characters of the character table [3Xtbl[103X
  by  the  improved  combinatorial  approach  described  at  the end of [BP98,
  Section 3.2].[133X
  
  [33X[0;0YFor  computing  the  possible linear combinations [13Xwithout[113X prescribing better
  bounds (i.e., when the computation of bounds shall be suppressed), enter[133X
  
  [33X[0;0Y[10X[3Xarec[103X[10X:= rec( degree := [3Xdegree[103X[10X, bounds := false )[110X,[133X
  
  [33X[0;0Ywhere  [3Xdegree[103X  is the character degree; this is useful if the multiplicities
  are expected to be small, and if this is forced by high irreducible degrees.[133X
  
  [33X[0;0YA  list  of  upper bounds on the multiplicities of the rational irreducibles
  characters can be explicitly prescribed as a [10Xmaxmult[110X component in [3Xarec[103X.[133X
  
  [1X72.14-5 Inequalities[101X
  
  [29X[2XInequalities[102X( [3Xtbl[103X, [3Xchars[103X[, [3Xoption[103X] ) [32X operation
  
  [33X[0;0YLet [3Xtbl[103X be the ordinary character table of a group [22XG[122X. The condition [22Xπ(g) ≥ 0[122X
  for  every  possible permutation character [22Xπ[122X of [22XG[122X places restrictions on the
  multiplicities  [22Xa_i[122X  of  the irreducible constituents [22Xχ_i[122X of [22Xπ = ∑_{i = 1}^r
  a_i  χ_i[122X.  For  every element [22Xg ∈ G[122X, we have [22X∑_{i = 1}^r a_i χ_i(g) ≥ 0[122X. The
  power maps provide even stronger conditions.[133X
  
  [33X[0;0YThis  system  of inequalities is kind of diagonalized, resulting in a system
  of  inequalities  restricting [22Xa_i[122X in terms of [22Xa_j[122X, [22Xj < i[122X. These inequalities
  are  used  to  construct  characters  with nonnegative values (see [2XPermChars[102X
  ([14X72.14-1[114X)).  [2XPermChars[102X  ([14X72.14-1[114X)  either  calls  [2XInequalities[102X or takes this
  information from the [10Xineq[110X component of its argument record.[133X
  
  [33X[0;0YThe  number  of  inequalities  arising in the process of diagonalization may
  grow very strongly.[133X
  
  [33X[0;0YThere are two ways to organize the projection. The first, which is chosen if
  no  [3Xoption[103X  argument  is  present,  is the straight approach which takes the
  rational  irreducible  characters  in  their  original  order  and  by  this
  guarantees  the  character  with the smallest degree to be considered first.
  The  other  way,  which  is chosen if the string [10X"small"[110X is entered as third
  argument [3Xoption[103X, tries to keep the number of intermediate inequalities small
  by eventually changing the order of characters.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtbl:= CharacterTable( "M11" );;[127X[104X
    [4X[25Xgap>[125X [27XPermComb( tbl, rec( degree:= 110 ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "M11" ), [128X[104X
    [4X[28X    [ 110, 6, 2, 2, 0, 0, 2, 2, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "M11" ), [128X[104X
    [4X[28X    [ 110, 6, 2, 6, 0, 0, 0, 0, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "M11" ), [ 110, 14, 2, 2, 0, 2, 0, 0, 0, [128X[104X
    [4X[28X      0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27X# Now compute only multiplicity free permutation characters.[127X[104X
    [4X[25Xgap>[125X [27Xbounds:= List( RationalizedMat( Irr( tbl ) ), x -> 1 );;[127X[104X
    [4X[25Xgap>[125X [27XPermComb( tbl, rec( degree:= 110, maxmult:= bounds ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "M11" ), [128X[104X
    [4X[28X    [ 110, 6, 2, 2, 0, 0, 2, 2, 0, 0 ] ) ][128X[104X
  [4X[32X[104X
  
  
  [1X72.15 [33X[0;0YOperations for Brauer Characters[133X[101X
  
  [1X72.15-1 FrobeniusCharacterValue[101X
  
  [29X[2XFrobeniusCharacterValue[102X( [3Xvalue[103X, [3Xp[103X ) [32X function
  
  [33X[0;0YLet  [3Xvalue[103X  be a cyclotomic whose coefficients over the rationals are in the
  ring  [22Xℤ_[3Xp[103X[122X  of [3Xp[103X-local numbers, where [3Xp[103X is a prime integer. Assume that [3Xvalue[103X
  lies in [22Xℤ_[3Xp[103X[ζ][122X for [22Xζ = exp([3Xp[103X^n-1)[122X, for some positive integer [22Xn[122X.[133X
  
  [33X[0;0Y[2XFrobeniusCharacterValue[102X   returns   the   image  of  [3Xvalue[103X  under  the  ring
  homomorphism from [22Xℤ_[3Xp[103X[ζ][122X to the field with [22X[3Xp[103X^n[122X elements that is defined with
  the   help  of  Conway  polynomials  (see [2XConwayPolynomial[102X  ([14X59.5-1[114X)),  more
  information can be found in [JLPW95, Sections 2-5].[133X
  
  [33X[0;0YIf [3Xvalue[103X is a Brauer character value in characteristic [3Xp[103X then the result can
  be described as the corresponding value of the Frobenius character, that is,
  as the trace of a representing matrix with the given Brauer character value.[133X
  
  [33X[0;0YIf  the  result of [2XFrobeniusCharacterValue[102X cannot be expressed as an element
  of  a  finite  field  in  [5XGAP[105X  (see Chapter [14X59[114X) then [2XFrobeniusCharacterValue[102X
  returns [9Xfail[109X.[133X
  
  [33X[0;0YIf the Conway polynomial of degree [22Xn[122X is required for the computation then it
  is computed only if [2XIsCheapConwayPolynomial[102X ([14X59.5-2[114X) returns [9Xtrue[109X when it is
  called with [3Xp[103X and [22Xn[122X, otherwise [9Xfail[109X is returned.[133X
  
  [1X72.15-2 BrauerCharacterValue[101X
  
  [29X[2XBrauerCharacterValue[102X( [3Xmat[103X ) [32X attribute
  
  [33X[0;0YFor  an  invertible  matrix  [3Xmat[103X over a finite field [22XF[122X, [2XBrauerCharacterValue[102X
  returns  the Brauer character value of [3Xmat[103X if the order of [3Xmat[103X is coprime to
  the characteristic of [22XF[122X, and [9Xfail[109X otherwise.[133X
  
  [33X[0;0YThe  [13XBrauer  character  value[113X of a matrix is the sum of complex lifts of its
  eigenvalues.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= SL(2,4);;           # 2-dim. irreducible representation of A5[127X[104X
    [4X[25Xgap>[125X [27Xccl:= ConjugacyClasses( g );;[127X[104X
    [4X[25Xgap>[125X [27Xrep:= List( ccl, Representative );;[127X[104X
    [4X[25Xgap>[125X [27XList( rep, Order );[127X[104X
    [4X[28X[ 1, 2, 5, 5, 3 ][128X[104X
    [4X[25Xgap>[125X [27Xphi:= List( rep, BrauerCharacterValue );[127X[104X
    [4X[28X[ 2, fail, E(5)^2+E(5)^3, E(5)+E(5)^4, -1 ][128X[104X
    [4X[25Xgap>[125X [27XList( phi{ [ 1, 3, 4, 5 ] }, x -> FrobeniusCharacterValue( x, 2 ) );[127X[104X
    [4X[28X[ 0*Z(2), Z(2^2), Z(2^2)^2, Z(2)^0 ][128X[104X
    [4X[25Xgap>[125X [27XList( rep{ [ 1, 3, 4, 5 ] }, TraceMat );[127X[104X
    [4X[28X[ 0*Z(2), Z(2^2), Z(2^2)^2, Z(2)^0 ][128X[104X
  [4X[32X[104X
  
  [1X72.15-3 SizeOfFieldOfDefinition[101X
  
  [29X[2XSizeOfFieldOfDefinition[102X( [3Xval[103X, [3Xp[103X ) [32X function
  
  [33X[0;0YFor  a  cyclotomic  or  a  list  of  cyclotomics [3Xval[103X, and a prime integer [3Xp[103X,
  [2XSizeOfFieldOfDefinition[102X  returns  the  size  of the smallest finite field in
  characteristic [3Xp[103X that contains the [3Xp[103X-modular reduction of [3Xval[103X.[133X
  
  [33X[0;0YThe reduction map is defined as in [JLPW95], that is, the complex [22X([3Xp[103X^d-1)[122X-th
  root   of   unity   [22Xexp([3Xp[103X^d-1)[122X  is  mapped  to  the  residue  class  of  the
  indeterminate,   modulo   the   ideal   spanned  by  the  Conway  polynomial
  (see [2XConwayPolynomial[102X ([14X59.5-1[114X)) of degree [22Xd[122X over the field with [22Xp[122X elements.[133X
  
  [33X[0;0YIf  [3Xval[103X  is  a  Brauer  character then the value returned is the size of the
  smallest  finite  field  in  characteristic  [3Xp[103X  over which the corresponding
  representation lives.[133X
  
  [1X72.15-4 RealizableBrauerCharacters[101X
  
  [29X[2XRealizableBrauerCharacters[102X( [3Xmatrix[103X, [3Xq[103X ) [32X function
  
  [33X[0;0YFor   a   list   [3Xmatrix[103X  of  absolutely  irreducible  Brauer  characters  in
  characteristic  [22Xp[122X,  and a power [3Xq[103X of [22Xp[122X, [2XRealizableBrauerCharacters[102X returns a
  duplicate-free  list  of sums of Frobenius conjugates of the rows of [3Xmatrix[103X,
  each irreducible over the field with [3Xq[103X elements.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xirr:= Irr( CharacterTable( "A5" ) mod 2 );[127X[104X
    [4X[28X[ Character( BrauerTable( "A5", 2 ), [ 1, 1, 1, 1 ] ), [128X[104X
    [4X[28X  Character( BrauerTable( "A5", 2 ), [128X[104X
    [4X[28X    [ 2, -1, E(5)+E(5)^4, E(5)^2+E(5)^3 ] ), [128X[104X
    [4X[28X  Character( BrauerTable( "A5", 2 ), [128X[104X
    [4X[28X    [ 2, -1, E(5)^2+E(5)^3, E(5)+E(5)^4 ] ), [128X[104X
    [4X[28X  Character( BrauerTable( "A5", 2 ), [ 4, 1, -1, -1 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XList( irr, phi -> SizeOfFieldOfDefinition( phi, 2 ) );[127X[104X
    [4X[28X[ 2, 4, 4, 2 ][128X[104X
    [4X[25Xgap>[125X [27XRealizableBrauerCharacters( irr, 2 );[127X[104X
    [4X[28X[ Character( BrauerTable( "A5", 2 ), [ 1, 1, 1, 1 ] ), [128X[104X
    [4X[28X  ClassFunction( BrauerTable( "A5", 2 ), [ 4, -2, -1, -1 ] ), [128X[104X
    [4X[28X  Character( BrauerTable( "A5", 2 ), [ 4, 1, -1, -1 ] ) ][128X[104X
  [4X[32X[104X
  
  
  [1X72.16 [33X[0;0YDomains Generated by Class Functions[133X[101X
  
  [33X[0;0Y[5XGAP[105X  supports  groups,  vector  spaces,  and  algebras  generated  by  class
  functions.[133X
  
