#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Thomas Breuer.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
## This file contains the definition of categories of class functions,
## and the corresponding properties, attributes, and operations.
##
## 1. Why Class Functions?
## 2. Basic Operations for Class Functions
## 3. Comparison of Class Functions
## 4. Arithmetic Operations for Class Functions
## 5. Printing Class Functions
## 6. Creating Class Functions from Values Lists
## 7. Creating Class Functions using Groups
## 8. Operations for Class Functions
## 9. Restricted and Induced Class Functions
## 10. Reducing Virtual Characters
## 11. Symmetrizations of Class Functions
## 12. Operations for Brauer Characters
## 13. Domains Generated by Class Functions
## 14. Auxiliary operations
##
#############################################################################
##
#C IsClassFunction( <obj> )
##
## <#GAPDoc Label="IsClassFunction">
## <ManSection>
## <Filt Name="IsClassFunction" Arg='obj' Type='Category'/>
##
## <Description>
## <Index>class function</Index><Index>class function objects</Index>
## A <E>class function</E> (in characteristic <M>p</M>) of a finite group
## <M>G</M> is a map from the set of (<M>p</M>-regular) elements in <M>G</M>
## to the field of cyclotomics
## that is constant on conjugacy classes of <M>G</M>.
## <P/>
## Each class function in &GAP; is represented by an <E>immutable list</E>,
## where at the <M>i</M>-th position the value on the <M>i</M>-th conjugacy
## class of the character table of <M>G</M> 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 <Ref Sect="The Interface between Character Tables and Groups"/>);
## class functions always refer to the ordering of classes in the character
## table.
## <P/>
## <E>Class function objects</E> in &GAP; are not just plain lists,
## they store the character table of the group <M>G</M> as value of the
## attribute <Ref Attr="UnderlyingCharacterTable"/>.
## The group <M>G</M> itself is accessible only via the character table
## and thus only if the character table stores its group, as value of the
## attribute <Ref Attr="UnderlyingGroup" Label="for character tables"/>.
## 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 &GAP;
## character table library do in general not have access to their
## underlying groups.
## <P/>
## There are (at least) two reasons why class functions in &GAP; are
## <E>not</E> 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 <M>G</M> 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.
## <P/>
## A further argument is that the typical operations for mappings such as
## <Ref Func="Image" Label="set of images of the source of a general mapping"/>
## and
## <Ref Func="PreImage" Label="set of preimages of the range of a general mapping"/>
## play no important role for class functions.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsClassFunction",
IsScalar and IsCommutativeElement and IsAssociativeElement
and IsHomogeneousList and IsScalarCollection and IsFinite
and IsGeneralizedRowVector );
#############################################################################
##
#F CharacterString( <char>, <str> )
##
## <ManSection>
## <Func Name="CharacterString" Arg='char, str'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "CharacterString" );
#############################################################################
##
## 1. Why Class Functions?
##
## <#GAPDoc Label="[1]{ctblfuns}">
## In 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.
## <P/>
## First, 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.
## <P/>
## Second, when one says that
## <Q><M>\chi</M> is a character of a group <M>G</M></Q>
## then this object <M>\chi</M> carries a lot of information.
## <M>\chi</M> has certain properties such as being irreducible or not.
## Several subgroups of <M>G</M> are related to <M>\chi</M>,
## such as the kernel and the centre of <M>\chi</M>.
## Other attributes of characters are the determinant and the central
## character.
## This knowledge cannot be stored in a plain list.
## <P/>
## For dealing with a group together with its characters, and maybe also
## subgroups and their characters, it is desirable that &GAP; 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 &GAP; 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.
## <P/>
## The main idea behind class function objects is that a class function
## object is equal to its values list in the sense of <Ref Oper="\="/>,
## 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.
## <!-- Note that a class function object lies in the same family as its list of-->
## <!-- values.-->
## <!-- As a consequence, there is no filter <C>IsClassFunctionCollection</C>,-->
## <!-- so we have no special treatment of spaces and algebras without hacks.-->
## &GAP; library functions prefer to return class function objects
## rather than returning just values lists,
## for example <Ref Attr="Irr" Label="for a group"/> lists
## consist of class function objects,
## and <Ref Attr="TrivialCharacter" Label="for a group"/>
## returns a class function object.
## <P/>
## Here is an <E>example</E> that shows both approaches.
## First we define some groups.
## <P/>
## <Example><![CDATA[
## gap> S4:= SymmetricGroup( 4 );; SetName( S4, "S4" );
## gap> D8:= SylowSubgroup( S4, 2 );; SetName( D8, "D8" );
## ]]></Example>
## <P/>
## We do some computations using the functions described later in this
## Chapter, first with class function objects.
## <P/>
## <Example><![CDATA[
## gap> irrS4:= Irr( S4 );;
## gap> irrD8:= Irr( D8 );;
## gap> chi:= irrD8[4];
## Character( CharacterTable( D8 ), [ 1, -1, 1, -1, 1 ] )
## gap> chi * chi;
## Character( CharacterTable( D8 ), [ 1, 1, 1, 1, 1 ] )
## gap> ind:= chi ^ S4;
## Character( CharacterTable( S4 ), [ 3, -1, -1, 0, 1 ] )
## gap> List( irrS4, x -> ScalarProduct( x, ind ) );
## [ 0, 1, 0, 0, 0 ]
## gap> det:= Determinant( ind );
## Character( CharacterTable( S4 ), [ 1, 1, 1, 1, 1 ] )
## gap> cent:= CentralCharacter( ind );
## ClassFunction( CharacterTable( S4 ), [ 1, -2, -1, 0, 2 ] )
## gap> rest:= Restricted( cent, D8 );
## ClassFunction( CharacterTable( D8 ), [ 1, -2, -1, -1, 2 ] )
## ]]></Example>
## <P/>
## Now we repeat these calculations with plain lists of character values.
## Here we need the character tables in some places.
## <P/>
## <Example><![CDATA[
## gap> tS4:= CharacterTable( S4 );;
## gap> tD8:= CharacterTable( D8 );;
## gap> chi:= ValuesOfClassFunction( irrD8[4] );
## [ 1, -1, 1, -1, 1 ]
## gap> Tensored( [ chi ], [ chi ] )[1];
## [ 1, 1, 1, 1, 1 ]
## gap> ind:= InducedClassFunction( tD8, chi, tS4 );
## ClassFunction( CharacterTable( S4 ), [ 3, -1, -1, 0, 1 ] )
## gap> List( Irr( tS4 ), x -> ScalarProduct( tS4, x, ind ) );
## [ 0, 1, 0, 0, 0 ]
## gap> det:= DeterminantOfCharacter( tS4, ind );
## ClassFunction( CharacterTable( S4 ), [ 1, 1, 1, 1, 1 ] )
## gap> cent:= CentralCharacter( tS4, ind );
## ClassFunction( CharacterTable( S4 ), [ 1, -2, -1, 0, 2 ] )
## gap> rest:= Restricted( tS4, cent, tD8 );
## ClassFunction( CharacterTable( D8 ), [ 1, -2, -1, -1, 2 ] )
## ]]></Example>
## <P/>
## If one deals with character tables from the &GAP; 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.)
## <P/>
## <Example><![CDATA[
## gap> s5 := CharacterTable( "A5.2" );; irrs5 := Irr( s5 );;
## gap> m11:= CharacterTable( "M11" );; irrm11:= Irr( m11 );;
## gap> chi:= TrivialCharacter( s5 );
## Character( CharacterTable( "A5.2" ), [ 1, 1, 1, 1, 1, 1, 1 ] )
## gap> chi ^ m11;
## Character( CharacterTable( "M11" ), [ 66, 10, 3, 2, 1, 1, 0, 0, 0, 0
## ] )
## gap> Determinant( irrs5[4] );
## Character( CharacterTable( "A5.2" ), [ 1, 1, 1, 1, -1, -1, -1 ] )
## ]]></Example>
## <P/>
## Functions that compute <E>normal</E> subgroups related to characters
## have counterparts that return the list of class positions corresponding
## to these groups.
## <P/>
## <Example><![CDATA[
## gap> ClassPositionsOfKernel( irrs5[2] );
## [ 1, 2, 3, 4 ]
## gap> ClassPositionsOfCentre( irrs5[2] );
## [ 1, 2, 3, 4, 5, 6, 7 ]
## ]]></Example>
## <P/>
## Non-normal subgroups cannot be described this way,
## so for example inertia subgroups (see <Ref Oper="InertiaSubgroup"/>)
## can in general not be computed from character tables without access to
## their groups.
## <#/GAPDoc>
##
#############################################################################
##
## 2. Basic Operations for Class Functions
##
## <#GAPDoc Label="[2]{ctblfuns}">
## Basic operations for class functions are
## <Ref Attr="UnderlyingCharacterTable"/>,
## <Ref Attr="ValuesOfClassFunction"/>,
## and the basic operations for lists
## (see <Ref Sect="Basic Operations for Lists"/>).
## <#/GAPDoc>
##
#############################################################################
##
#A UnderlyingCharacterTable( <psi> )
##
## <#GAPDoc Label="UnderlyingCharacterTable">
## <ManSection>
## <Attr Name="UnderlyingCharacterTable" Arg='psi'/>
##
## <Description>
## For a class function <A>psi</A> of a group <M>G</M>
## the character table of <M>G</M> is stored as value of
## <Ref Attr="UnderlyingCharacterTable"/>.
## The ordering of entries in the list <A>psi</A>
## (see <Ref Attr="ValuesOfClassFunction"/>)
## refers to the ordering of conjugacy classes in this character table.
## <P/>
## If <A>psi</A> is an ordinary class function then the underlying character
## table is the ordinary character table of <M>G</M>
## (see <Ref Attr="OrdinaryCharacterTable" Label="for a group"/>),
## if <A>psi</A> is a class function in characteristic <M>p \neq 0</M> then
## the underlying character table is the <M>p</M>-modular Brauer table of
## <M>G</M>
## (see <Ref Oper="BrauerTable"
## Label="for a group, and a prime integer"/>).
## So the underlying characteristic of <A>psi</A> can be read off from the
## underlying character table.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "UnderlyingCharacterTable", IsClassFunction );
#############################################################################
##
#A ValuesOfClassFunction( <psi> ) . . . . . . . . . . . . . . list of values
##
## <#GAPDoc Label="ValuesOfClassFunction">
## <ManSection>
## <Attr Name="ValuesOfClassFunction" Arg='psi'/>
##
## <Description>
## is the list of values of the class function <A>psi</A>,
## the <M>i</M>-th entry being the value on the <M>i</M>-th conjugacy class
## of the underlying character table
## (see <Ref Attr="UnderlyingCharacterTable"/>).
## <P/>
## <Example><![CDATA[
## gap> g:= SymmetricGroup( 4 );
## Sym( [ 1 .. 4 ] )
## gap> psi:= TrivialCharacter( g );
## Character( CharacterTable( Sym( [ 1 .. 4 ] ) ), [ 1, 1, 1, 1, 1 ] )
## gap> UnderlyingCharacterTable( psi );
## CharacterTable( Sym( [ 1 .. 4 ] ) )
## gap> ValuesOfClassFunction( psi );
## [ 1, 1, 1, 1, 1 ]
## gap> IsList( psi );
## true
## gap> psi[1];
## 1
## gap> Length( psi );
## 5
## gap> IsBound( psi[6] );
## false
## gap> Concatenation( psi, [ 2, 3 ] );
## [ 1, 1, 1, 1, 1, 2, 3 ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "ValuesOfClassFunction", IsClassFunction );
#############################################################################
##
## 3. Comparison of Class Functions
##
## <#GAPDoc Label="[3]{ctblfuns}">
## With respect to <Ref Oper="\="/> and <Ref Oper="\<"/>,
## class functions behave equally to their lists of values
## (see <Ref Attr="ValuesOfClassFunction"/>).
## 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.
## <P/>
## <Example><![CDATA[
## gap> grps:= Filtered( AllSmallGroups( 8 ), g -> not IsAbelian( g ) );
## [ <pc group of size 8 with 3 generators>,
## <pc group of size 8 with 3 generators> ]
## gap> t1:= CharacterTable( grps[1] ); SetName( t1, "t1" );
## CharacterTable( <pc group of size 8 with 3 generators> )
## gap> t2:= CharacterTable( grps[2] ); SetName( t2, "t2" );
## CharacterTable( <pc group of size 8 with 3 generators> )
## gap> CharacterDegrees( t1 );
## [ [ 1, 4 ], [ 2, 1 ] ]
## gap> irr1:= Irr( grps[1] );;
## gap> irr2:= Irr( grps[2] );;
## gap> irr1 = irr2;
## true
## gap> irr1[1];
## Character( t1, [ 1, 1, 1, 1, 1 ] )
## gap> irr1[5];
## Character( t1, [ 2, 0, 0, -2, 0 ] )
## gap> irr2[1];
## Character( t2, [ 1, 1, 1, 1, 1 ] )
## gap> IsSSortedList( irr1 );
## false
## gap> irr1[1] < irr1[2]; # the triv. character has no '-1'
## false
## gap> irr1[2] < irr1[5];
## true
## ]]></Example>
## <#/GAPDoc>
##
#############################################################################
##
## 4. Arithmetic Operations for Class Functions
##
## <#GAPDoc Label="[4]{ctblfuns}">
## Class functions are <E>row vectors</E> of cyclotomics.
## The <E>additive</E> 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 <E>multiplicative</E> 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
## <Ref Oper="ScalarProduct" Label="for characters"/>.
## In terms of filters, the arithmetic of class functions is based on the
## decision that they lie in <Ref Filt="IsGeneralizedRowVector"/>,
## with additive nesting depth <M>1</M>, but they do <E>not</E> lie in
## <Ref Filt="IsMultiplicativeGeneralizedRowVector"/>.
## <P/>
## More 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 <Ref Attr="UnderlyingCharacterTable"/>)
## are again class functions of this table.
## The sum and the difference of a class function and a list that is
## <E>not</E> a class function are plain lists,
## as well as the sum and the difference of two class functions of different
## character tables.
## <P/>
## <Example><![CDATA[
## gap> g:= SymmetricGroup( 4 );; tbl:= CharacterTable( g );;
## gap> SetName( tbl, "S4" ); irr:= Irr( g );
## [ Character( S4, [ 1, -1, 1, 1, -1 ] ),
## Character( S4, [ 3, -1, -1, 0, 1 ] ),
## Character( S4, [ 2, 0, 2, -1, 0 ] ),
## Character( S4, [ 3, 1, -1, 0, -1 ] ),
## Character( S4, [ 1, 1, 1, 1, 1 ] ) ]
## gap> 2 * irr[5];
## Character( S4, [ 2, 2, 2, 2, 2 ] )
## gap> irr[1] / 7;
## ClassFunction( S4, [ 1/7, -1/7, 1/7, 1/7, -1/7 ] )
## gap> lincomb:= irr[3] + irr[1] - irr[5];
## VirtualCharacter( S4, [ 2, -2, 2, -1, -2 ] )
## gap> lincomb:= lincomb + 2 * irr[5];
## VirtualCharacter( S4, [ 4, 0, 4, 1, 0 ] )
## gap> IsCharacter( lincomb );
## true
## gap> lincomb;
## Character( S4, [ 4, 0, 4, 1, 0 ] )
## gap> irr[5] + 2;
## [ 3, 3, 3, 3, 3 ]
## gap> irr[5] + [ 1, 2, 3, 4, 5 ];
## [ 2, 3, 4, 5, 6 ]
## gap> zero:= 0 * irr[1];
## VirtualCharacter( S4, [ 0, 0, 0, 0, 0 ] )
## gap> zero + Z(3);
## [ Z(3), Z(3), Z(3), Z(3), Z(3) ]
## gap> irr[5] + TrivialCharacter( DihedralGroup( 8 ) );
## [ 2, 2, 2, 2, 2 ]
## ]]></Example>
## <P/>
## <Index Subkey="as ring elements">class functions</Index>
## The 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 <M>F</M> of cyclotomics, the <M>F</M>-span of a given
## set of class functions forms an algebra.
## <P/>
## The product of two class functions of <E>different</E> tables and the
## product of a class function and a list that is <E>not</E> 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.
## <P/>
## <Example><![CDATA[
## gap> tens:= irr[3] * irr[4];
## Character( S4, [ 6, 0, -2, 0, 0 ] )
## gap> ValuesOfClassFunction( irr[3] ) * ValuesOfClassFunction( irr[4] );
## 4
## ]]></Example>
## <P/>
## <Index Subkey="of class function">inverse</Index>
## Class functions without zero values are invertible,
## the <E>inverse</E> is defined pointwise.
## As a consequence, for example groups of linear characters can be formed.
## <P/>
## <Example><![CDATA[
## gap> tens / irr[1];
## Character( S4, [ 6, 0, -2, 0, 0 ] )
## ]]></Example>
## <P/>
## Other (somewhat strange) implications of the definition of arithmetic
## operations for class functions, together with the general rules of list
## arithmetic (see <Ref Sect="Arithmetic for Lists"/>),
## apply to the case of products involving <E>lists</E> 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.
## <P/>
## <Example><![CDATA[
## gap> Inverse( List( irr, ValuesOfClassFunction ) );
## [ [ 1/24, 1/8, 1/12, 1/8, 1/24 ], [ -1/4, -1/4, 0, 1/4, 1/4 ],
## [ 1/8, -1/8, 1/4, -1/8, 1/8 ], [ 1/3, 0, -1/3, 0, 1/3 ],
## [ -1/4, 1/4, 0, -1/4, 1/4 ] ]
## ]]></Example>
## <P/>
## Also the product of a class function with a list of class functions is
## <E>not</E> a vector-matrix product but the list of pointwise products.
## <P/>
## <Example><![CDATA[
## gap> irr[1] * irr{ [ 1 .. 3 ] };
## [ Character( S4, [ 1, 1, 1, 1, 1 ] ),
## Character( S4, [ 3, 1, -1, 0, -1 ] ),
## Character( S4, [ 2, 0, 2, -1, 0 ] ) ]
## ]]></Example>
## <P/>
## And the product of two lists of class functions is <E>not</E> the matrix
## product but the sum of the pointwise products.
## <P/>
## <Example><![CDATA[
## gap> irr * irr;
## Character( S4, [ 24, 4, 8, 3, 4 ] )
## ]]></Example>
## <P/>
## <Index Subkey="of group element using powering operator">character value</Index>
## <Index Subkey="meaning for class functions">power</Index>
## <Index Key="^" Subkey="for class functions"><C>^</C></Index>
## The <E>powering</E> operator <Ref Oper="\^"/> 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 <Ref Oper="\^"/>; instead one can use
## <Ref Oper="Permuted"/>.)
## The power of a class function by a group or a character table is the
## induced class function (see <Ref Oper="InducedClassFunction"
## Label="for the character table of a supergroup"/>).
## The power of a group element by a class function is the class function
## value at (the conjugacy class containing) this element.
## <P/>
## <Example><![CDATA[
## gap> irr[3] ^ 3;
## Character( S4, [ 8, 0, 8, -1, 0 ] )
## gap> lin:= LinearCharacters( DerivedSubgroup( g ) );
## [ Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, 1, 1, 1 ] ),
## Character( CharacterTable( Alt( [ 1 .. 4 ] ) ),
## [ 1, 1, E(3), E(3)^2 ] ),
## Character( CharacterTable( Alt( [ 1 .. 4 ] ) ),
## [ 1, 1, E(3)^2, E(3) ] ) ]
## gap> List( lin, chi -> chi ^ (1,2) );
## [ Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, 1, 1, 1 ] ),
## Character( CharacterTable( Alt( [ 1 .. 4 ] ) ),
## [ 1, 1, E(3)^2, E(3) ] ),
## Character( CharacterTable( Alt( [ 1 .. 4 ] ) ),
## [ 1, 1, E(3), E(3)^2 ] ) ]
## gap> Orbit( GaloisGroup( CF(3) ), lin[2] );
## [ Character( CharacterTable( Alt( [ 1 .. 4 ] ) ),
## [ 1, 1, E(3), E(3)^2 ] ),
## Character( CharacterTable( Alt( [ 1 .. 4 ] ) ),
## [ 1, 1, E(3)^2, E(3) ] ) ]
## gap> lin[1]^g;
## Character( S4, [ 2, 0, 2, 2, 0 ] )
## gap> (1,2,3)^lin[2];
## E(3)
## ]]></Example>
##
## <ManSection>
## <Attr Name="Characteristic" Arg='chi' Label="for a class function"/>
##
## <Description>
## The <E>characteristic</E> of class functions is zero,
## as for all list of cyclotomics.
## For class functions of a <M>p</M>-modular character table, such as Brauer
## characters, the prime <M>p</M> is given by the
## <Ref Attr="UnderlyingCharacteristic" Label="for a character table"/>
## value of the character table.
## <P/>
## <Example><![CDATA[
## gap> Characteristic( irr[1] );
## 0
## gap> irrmod2:= Irr( g, 2 );
## [ Character( BrauerTable( Sym( [ 1 .. 4 ] ), 2 ), [ 1, 1 ] ),
## Character( BrauerTable( Sym( [ 1 .. 4 ] ), 2 ), [ 2, -1 ] ) ]
## gap> Characteristic( irrmod2[1] );
## 0
## gap> UnderlyingCharacteristic( UnderlyingCharacterTable( irrmod2[1] ) );
## 2
## ]]></Example>
## </Description>
## </ManSection>
##
## <ManSection>
## <Attr Name="ComplexConjugate" Arg='chi' Label="for a class function"/>
## <Oper Name="GaloisCyc" Arg='chi, k' Label="for a class function"/>
## <Meth Name="Permuted" Arg='chi, pi' Label="for a class function"/>
##
## <Description>
## The operations
## <Ref Attr="ComplexConjugate" Label="for a class function"/>,
## <Ref Oper="GaloisCyc" Label="for a class function"/>,
## and <Ref Meth="Permuted" Label="for a class function"/> 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.
## <P/>
## <Example><![CDATA[
## gap> ComplexConjugate( lin[2] );
## Character( CharacterTable( Alt( [ 1 .. 4 ] ) ),
## [ 1, 1, E(3)^2, E(3) ] )
## gap> GaloisCyc( lin[2], 5 );
## Character( CharacterTable( Alt( [ 1 .. 4 ] ) ),
## [ 1, 1, E(3)^2, E(3) ] )
## gap> Permuted( lin[2], (2,3,4) );
## ClassFunction( CharacterTable( Alt( [ 1 .. 4 ] ) ),
## [ 1, E(3)^2, 1, E(3) ] )
## ]]></Example>
## </Description>
## </ManSection>
## <P/>
## <ManSection>
## <Attr Name="Order" Arg='chi' Label="for a class function"/>
##
## <Description>
## By definition of <Ref Attr="Order"/> for arbitrary monoid elements,
## the return value of <Ref Attr="Order"/> for a character must be its
## multiplicative order.
## The <E>determinantal order</E>
## (see <Ref Attr="DeterminantOfCharacter"/>) of a character <A>chi</A>
## can be computed as <C>Order( Determinant( <A>chi</A> ) )</C>.
## <P/>
## <Example><![CDATA[
## gap> det:= Determinant( irr[3] );
## Character( S4, [ 1, -1, 1, 1, -1 ] )
## gap> Order( det );
## 2
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
#############################################################################
##
#A GlobalPartitionOfClasses( <tbl> )
##
## <ManSection>
## <Attr Name="GlobalPartitionOfClasses" Arg='tbl'/>
##
## <Description>
## Let <M>n</M> be the number of conjugacy classes of the character table
## <A>tbl</A>.
## <Ref Func="GlobalPartitionOfClasses"/> returns a list of subsets of the
## range <M>[ 1 .. n ]</M> that forms a partition of <M>[ 1 .. n ]</M>.
## This partition is respected by each table automorphism of <A>tbl</A>
## (see <Ref Func="AutomorphismsOfTable"/>);
## <E>note</E> that also fixed points occur.
## <P/>
## This is useful for the computation of table automorphisms
## and of conjugate class functions.
## <P/>
## Since group automorphisms induce table automorphisms, the partition is
## also respected by the permutation group that occurs in the computation
## of inertia groups and conjugate class functions.
## <P/>
## If the group of table automorphisms is already known then its orbits
## form the finest possible global partition.
## <P/>
## Otherwise the subsets in the partition are the sets of classes with
## same centralizer order and same element order, and
## –if more about the character table is known–
## also with the same number of <M>p</M>-th root classes,
## for all <M>p</M> for which the power maps are stored.
## </Description>
## </ManSection>
##
DeclareAttribute( "GlobalPartitionOfClasses", IsNearlyCharacterTable );
#############################################################################
##
#O CorrespondingPermutations( <tbl>[, <chi>], <elms> )
##
## <ManSection>
## <Oper Name="CorrespondingPermutations" Arg='tbl[, chi], elms'/>
##
## <Description>
## Called with two arguments <A>tbl</A> and <A>elms</A>,
## <Ref Oper="CorrespondingPermutations"/> returns the list of those
## permutations of conjugacy classes of the character table <A>tbl</A>
## that are induced by the action of the group elements in the list
## <A>elms</A>.
## If an element of <A>elms</A> does <E>not</E> act on the classes of
## <A>tbl</A> then either <K>fail</K> or a (meaningless) permutation
## is returned.
## <P/>
## In the call with three arguments, the second argument <A>chi</A> must be
## (the values list of) a class function of <A>tbl</A>,
## and the returned permutations will at least yield the same
## conjugate class functions as the permutations of classes that are induced
## by <A>elms</A>,
## that is, the images are not necessarily the same for orbits on which
## <A>chi</A> is constant.
## <P/>
## This function is used for computing conjugate class functions.
## </Description>
## </ManSection>
##
DeclareOperation( "CorrespondingPermutations",
[ IsOrdinaryTable, IsHomogeneousList ] );
DeclareOperation( "CorrespondingPermutations",
[ IsOrdinaryTable, IsClassFunction, IsHomogeneousList ] );
#############################################################################
##
## 5. Printing Class Functions
##
## <#GAPDoc Label="[5]{ctblfuns}">
## <ManSection>
## <Meth Name="ViewObj" Arg='chi' Label="for class functions"/>
##
## <Description>
## The default <Ref Oper="ViewObj"/> methods for class functions
## print one of the strings <C>"ClassFunction"</C>,
## <C>"VirtualCharacter"</C>, <C>"Character"</C> (depending on whether the
## class function is known to be a character or virtual character,
## see <Ref Prop="IsCharacter"/>, <Ref Prop="IsVirtualCharacter"/>),
## followed by the <Ref Oper="ViewObj"/> output for the underlying character
## table (see <Ref Sect="Printing Character Tables"/>),
## and the list of values.
## The table is chosen (and not the group) in order to distinguish class
## functions of different underlying characteristic
## (see <Ref Attr="UnderlyingCharacteristic" Label="for a character"/>).
## </Description>
## </ManSection>
##
## <ManSection>
## <Meth Name="PrintObj" Arg='chi' Label="for class functions"/>
##
## <Description>
## The default <Ref Oper="PrintObj"/> method for class functions
## does the same as <Ref Oper="ViewObj"/>,
## except that the character table is <Ref Func="Print"/>-ed instead of
## <Ref Func="View"/>-ed.
## <P/>
## <E>Note</E> that if a class function is shown only with one of the
## strings <C>"ClassFunction"</C>, <C>"VirtualCharacter"</C>,
## 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.
## <P/>
## In order to reduce the space that is needed to print a class function,
## it may be useful to give a name (see <Ref Attr="Name"/>) to the
## underlying character table.
## </Description>
## </ManSection>
##
## <ManSection>
## <Meth Name="Display" Arg='chi' Label="for class functions"/>
##
## <Description>
## The default <Ref Oper="Display"/> method for a class function <A>chi</A>
## calls <Ref Oper="Display"/> for its underlying character table
## (see <Ref Sect="Printing Character Tables"/>),
## with <A>chi</A> as the only entry in the <C>chars</C> list of the options
## record.
## <P/>
## <Example><![CDATA[
## gap> chi:= TrivialCharacter( CharacterTable( "A5" ) );
## Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] )
## gap> Display( chi );
## A5
##
## 2 2 2 . . .
## 3 1 . 1 . .
## 5 1 . . 1 1
##
## 1a 2a 3a 5a 5b
## 2P 1a 1a 3a 5b 5a
## 3P 1a 2a 1a 5b 5a
## 5P 1a 2a 3a 1a 1a
##
## Y.1 1 1 1 1 1
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
#############################################################################
##
## 6. Creating Class Functions from Values Lists
##
#############################################################################
##
#O ClassFunction( <tbl>, <values> )
#O ClassFunction( <G>, <values> )
##
## <#GAPDoc Label="ClassFunction">
## <ManSection>
## <Oper Name="ClassFunction" Arg='tbl, values'
## Label="for a character table and a list"/>
## <Oper Name="ClassFunction" Arg='G, values'
## Label="for a group and a list"/>
##
## <Description>
## In the first form,
## <Ref Oper="ClassFunction" Label="for a character table and a list"/>
## returns the class function of the character table <A>tbl</A> with values
## given by the list <A>values</A> of cyclotomics.
## In the second form, <A>G</A> must be a group,
## and the class function of its ordinary character table is returned.
## <P/>
## Note that <A>tbl</A> determines the underlying characteristic of the
## returned class function
## (see <Ref Attr="UnderlyingCharacteristic" Label="for a character"/>).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "ClassFunction", [ IsNearlyCharacterTable, IsDenseList ] );
DeclareOperation( "ClassFunction", [ IsGroup, IsDenseList ] );
#############################################################################
##
#O VirtualCharacter( <tbl>, <values> )
#O VirtualCharacter( <G>, <values> )
##
## <#GAPDoc Label="VirtualCharacter">
## <ManSection>
## <Oper Name="VirtualCharacter" Arg='tbl, values'
## Label="for a character table and a list"/>
## <Oper Name="VirtualCharacter" Arg='G, values'
## Label="for a group and a list"/>
##
## <Description>
## <Ref Oper="VirtualCharacter" Label="for a character table and a list"/>
## returns the virtual character
## (see <Ref Prop="IsVirtualCharacter"/>)
## of the character table <A>tbl</A> or the group <A>G</A>,
## respectively, with values given by the list <A>values</A>.
## <P/>
## It is <E>not</E> checked whether the given values really describe a
## virtual character.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "VirtualCharacter",
[ IsNearlyCharacterTable, IsDenseList ] );
DeclareOperation( "VirtualCharacter", [ IsGroup, IsDenseList ] );
#############################################################################
##
#O Character( <tbl>, <values> )
##
## <#GAPDoc Label="Character">
## <ManSection>
## <Oper Name="Character" Arg='tbl, values'
## Label="for a character table and a list"/>
## <Oper Name="Character" Arg='G, values'
## Label="for a group and a list"/>
##
## <Description>
## <Ref Oper="Character" Label="for a character table and a list"/>
## returns the character (see <Ref Prop="IsCharacter"/>)
## of the character table <A>tbl</A> or the group <A>G</A>,
## respectively, with values given by the list <A>values</A>.
## <P/>
## It is <E>not</E> checked whether the given values really describe a
## character.
## <Example><![CDATA[
## gap> g:= DihedralGroup( 8 ); tbl:= CharacterTable( g );
## <pc group of size 8 with 3 generators>
## CharacterTable( <pc group of size 8 with 3 generators> )
## gap> SetName( tbl, "D8" );
## gap> phi:= ClassFunction( g, [ 1, -1, 0, 2, -2 ] );
## ClassFunction( D8, [ 1, -1, 0, 2, -2 ] )
## gap> psi:= ClassFunction( tbl,
## > List( Irr( g ), chi -> ScalarProduct( chi, phi ) ) );
## ClassFunction( D8, [ -3/8, 9/8, 5/8, 1/8, -1/4 ] )
## gap> chi:= VirtualCharacter( g, [ 0, 0, 8, 0, 0 ] );
## VirtualCharacter( D8, [ 0, 0, 8, 0, 0 ] )
## gap> reg:= Character( tbl, [ 8, 0, 0, 0, 0 ] );
## Character( D8, [ 8, 0, 0, 0, 0 ] )
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "Character", [ IsNearlyCharacterTable, IsDenseList ] );
DeclareOperation( "Character", [ IsGroup, IsDenseList ] );
#############################################################################
##
#F ClassFunctionSameType( <tbl>, <chi>, <values> )
##
## <#GAPDoc Label="ClassFunctionSameType">
## <ManSection>
## <Func Name="ClassFunctionSameType" Arg='tbl, chi, values'/>
##
## <Description>
## Let <A>tbl</A> be a character table, <A>chi</A> a class function object
## (<E>not</E> necessarily a class function of <A>tbl</A>),
## and <A>values</A> a list of cyclotomics.
## <Ref Func="ClassFunctionSameType"/> returns the class function
## <M>\psi</M> of <A>tbl</A> with values list <A>values</A>,
## constructed with
## <Ref Oper="ClassFunction" Label="for a character table and a list"/>.
## <P/>
## If <A>chi</A> is known to be a (virtual) character then <M>\psi</M>
## is also known to be a (virtual) character.
## <P/>
## <Example><![CDATA[
## gap> h:= Centre( g );;
## gap> centbl:= CharacterTable( h );; SetName( centbl, "C2" );
## gap> ClassFunctionSameType( centbl, phi, [ 1, 1 ] );
## ClassFunction( C2, [ 1, 1 ] )
## gap> ClassFunctionSameType( centbl, chi, [ 1, 1 ] );
## VirtualCharacter( C2, [ 1, 1 ] )
## gap> ClassFunctionSameType( centbl, reg, [ 1, 1 ] );
## Character( C2, [ 1, 1 ] )
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "ClassFunctionSameType" );
#############################################################################
##
## 7. Creating Class Functions using Groups
##
#############################################################################
##
#A TrivialCharacter( <tbl> )
#A TrivialCharacter( <G> )
##
## <#GAPDoc Label="TrivialCharacter">
## <ManSection>
## <Heading>TrivialCharacter</Heading>
## <Attr Name="TrivialCharacter" Arg='tbl' Label="for a character table"/>
## <Attr Name="TrivialCharacter" Arg='G' Label="for a group"/>
##
## <Description>
## is the <E>trivial character</E> of the group <A>G</A>
## or its character table <A>tbl</A>, respectively.
## This is the class function with value equal to <M>1</M> for each class.
## <P/>
## <Example><![CDATA[
## gap> TrivialCharacter( CharacterTable( "A5" ) );
## Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] )
## gap> TrivialCharacter( SymmetricGroup( 3 ) );
## Character( CharacterTable( Sym( [ 1 .. 3 ] ) ), [ 1, 1, 1 ] )
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "TrivialCharacter", IsNearlyCharacterTable );
DeclareAttribute( "TrivialCharacter", IsGroup );
#############################################################################
##
#A NaturalCharacter( <G> )
#A NaturalCharacter( <hom> )
##
## <#GAPDoc Label="NaturalCharacter">
## <ManSection>
## <Attr Name="NaturalCharacter" Arg='G' Label="for a group"/>
## <Attr Name="NaturalCharacter" Arg='hom' Label="for a homomorphism"/>
##
## <Description>
## If the argument is a permutation group <A>G</A> then
## <Ref Attr="NaturalCharacter" Label="for a group"/>
## returns the (ordinary) character of the natural permutation
## representation of <A>G</A> on the set of moved points (see
## <Ref Attr="MovedPoints" Label="for a list or collection of permutations"/>),
## that is, the value on each class is the number of points among the moved
## points of <A>G</A> that are fixed by any permutation in that class.
## <P/>
## If the argument is a matrix group <A>G</A> in characteristic zero then
## <Ref Attr="NaturalCharacter" Label="for a group"/> returns the
## (ordinary) character of the natural matrix representation of <A>G</A>,
## that is, the value on each class is the trace of any matrix in that class.
## <P/>
## If the argument is a group homomorphism <A>hom</A> whose image is a
## permutation group or a matrix group then
## <Ref Attr="NaturalCharacter" Label="for a homomorphism"/> returns the
## restriction of the natural character of the image of <A>hom</A> to the
## preimage of <A>hom</A>.
## <P/>
## <Example><![CDATA[
## gap> NaturalCharacter( SymmetricGroup( 3 ) );
## Character( CharacterTable( Sym( [ 1 .. 3 ] ) ), [ 3, 1, 0 ] )
## gap> NaturalCharacter( Group( [ [ 0, -1 ], [ 1, -1 ] ] ) );
## Character( CharacterTable( Group([ [ [ 0, -1 ], [ 1, -1 ] ] ]) ),
## [ 2, -1, -1 ] )
## gap> d8:= DihedralGroup( 8 );; hom:= RegularActionHomomorphism( d8 );;
## gap> NaturalCharacter( hom );
## Character( CharacterTable( <pc group of size 8 with 3 generators> ),
## [ 8, 0, 0, 0, 0 ] )
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "NaturalCharacter", IsGroup );
DeclareAttribute( "NaturalCharacter", IsGeneralMapping );
#############################################################################
##
#O PermutationCharacter( <G>, <D>, <opr> )
#O PermutationCharacter( <G>, <U> )
##
## <#GAPDoc Label="PermutationCharacter">
## <ManSection>
## <Heading>PermutationCharacter</Heading>
## <Oper Name="PermutationCharacter" Arg='G, D, opr'
## Label="for a group, an action domain, and a function"/>
## <Oper Name="PermutationCharacter" Arg='G, U' Label="for two groups"/>
##
## <Description>
## Called with a group <A>G</A>, an action domain or proper set <A>D</A>,
## and an action function <A>opr</A>
## (see Chapter <Ref Chap="Group Actions"/>),
## <Ref Oper="PermutationCharacter"
## Label="for a group, an action domain, and a function"/>
## returns the <E>permutation character</E> of the action
## of <A>G</A> on <A>D</A> via <A>opr</A>,
## that is, the value on each class is the number of points in <A>D</A>
## that are fixed by an element in this class under the action <A>opr</A>.
## <P/>
## If the arguments are a group <A>G</A> and a subgroup <A>U</A> of <A>G</A>
## then <Ref Oper="PermutationCharacter" Label="for two groups"/> returns
## the permutation character of the action of <A>G</A> on the right cosets
## of <A>U</A> via right multiplication.
## <P/>
## To compute the permutation character of a
## <E>transitive permutation group</E>
## <A>G</A> on the cosets of a point stabilizer <A>U</A>,
## the attribute <Ref Attr="NaturalCharacter" Label="for a group"/>
## of <A>G</A> can be used instead of
## <C>PermutationCharacter( <A>G</A>, <A>U</A> )</C>.
## <P/>
## More facilities concerning permutation characters are the transitivity
## test (see Section <Ref Sect="Operations for Class Functions"/>)
## and several tools for computing possible permutation characters
## (see <Ref Sect="Possible Permutation Characters"/>,
## <Ref Sect="Computing Possible Permutation Characters"/>).
## <P/>
## <Example><![CDATA[
## gap> PermutationCharacter( GL(2,2), AsSSortedList( GF(2)^2 ), OnRight );
## Character( CharacterTable( SL(2,2) ), [ 4, 2, 1 ] )
## gap> s3:= SymmetricGroup( 3 );; a3:= DerivedSubgroup( s3 );;
## gap> PermutationCharacter( s3, a3 );
## Character( CharacterTable( Sym( [ 1 .. 3 ] ) ), [ 2, 0, 2 ] )
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "PermutationCharacter",
[ IsGroup, IsCollection, IsFunction ] );
DeclareOperation( "PermutationCharacter", [ IsGroup, IsGroup ] );
#############################################################################
##
## 8. Operations for Class Functions
##
## <#GAPDoc Label="[6]{ctblfuns}">
## In the description of the following operations,
## the optional first argument <A>tbl</A> is needed only if the argument
## <A>chi</A> is a plain list and not a class function object.
## In this case, <A>tbl</A> must always be the character table of which
## <A>chi</A> shall be regarded as a class function.
## <#/GAPDoc>
##
#############################################################################
##
#P IsCharacter( [<tbl>, ]<chi> )
##
## <#GAPDoc Label="IsCharacter">
## <ManSection>
## <Prop Name="IsCharacter" Arg='[tbl, ]chi'/>
##
## <Description>
## <Index>ordinary character</Index>
## An <E>ordinary character</E> of a group <M>G</M> is a class function of
## <M>G</M> whose values are the traces of a complex matrix representation
## of <M>G</M>.
## <P/>
## <Index>Brauer character</Index>
## A <E>Brauer character</E> of <M>G</M> in characteristic <M>p</M> is
## a class function of <M>G</M> whose values are the complex lifts of a
## matrix representation of <M>G</M> with image a finite field of
## characteristic <M>p</M>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsCharacter", IsClassFunction );
DeclareOperation( "IsCharacter", [ IsCharacterTable, IsHomogeneousList ] );
#############################################################################
##
#P IsVirtualCharacter( [<tbl>, ]<chi> )
##
## <#GAPDoc Label="IsVirtualCharacter">
## <ManSection>
## <Prop Name="IsVirtualCharacter" Arg='[tbl, ]chi'/>
##
## <Description>
## <Index>virtual character</Index>
## A <E>virtual character</E> is a class function that can be written as the
## difference of two proper characters (see <Ref Prop="IsCharacter"/>).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsVirtualCharacter", IsClassFunction );
InstallTrueMethod( IsClassFunction, IsVirtualCharacter );
DeclareOperation( "IsVirtualCharacter",
[ IsCharacterTable, IsHomogeneousList ] );
#############################################################################
##
#M IsVirtualCharacter( <chi> ) . . . . . . . . . . . . . . . for a character
##
## Each character is of course a virtual character.
##
InstallTrueMethod( IsVirtualCharacter, IsCharacter );
#############################################################################
##
#P IsIrreducibleCharacter( [<tbl>, ]<chi> )
##
## <#GAPDoc Label="IsIrreducibleCharacter">
## <ManSection>
## <Prop Name="IsIrreducibleCharacter" Arg='[tbl, ]chi'/>
##
## <Description>
## <Index>irreducible character</Index>
## A character is <E>irreducible</E> 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 <Ref Oper="ScalarProduct" Label="for characters"/>).
## For Brauer characters there is no generic method for checking
## irreducibility.
## <P/>
## <Example><![CDATA[
## gap> S4:= SymmetricGroup( 4 );; SetName( S4, "S4" );
## gap> psi:= ClassFunction( S4, [ 1, 1, 1, -2, 1 ] );
## ClassFunction( CharacterTable( S4 ), [ 1, 1, 1, -2, 1 ] )
## gap> IsVirtualCharacter( psi );
## true
## gap> IsCharacter( psi );
## false
## gap> chi:= ClassFunction( S4, SizesCentralizers( CharacterTable( S4 ) ) );
## ClassFunction( CharacterTable( S4 ), [ 24, 4, 8, 3, 4 ] )
## gap> IsCharacter( chi );
## true
## gap> IsIrreducibleCharacter( chi );
## false
## gap> IsIrreducibleCharacter( TrivialCharacter( S4 ) );
## true
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsIrreducibleCharacter", IsClassFunction );
DeclareOperation( "IsIrreducibleCharacter",
[ IsCharacterTable, IsHomogeneousList ] );
#############################################################################
##
#O ScalarProduct( [<tbl>, ]<chi>, <psi> )
##
## <#GAPDoc Label="ScalarProduct:ctblfuns">
## <ManSection>
## <Oper Name="ScalarProduct" Arg='[tbl, ]chi, psi' Label="for characters"/>
##
## <Returns>
## the scalar product of the class functions <A>chi</A> and <A>psi</A>,
## which belong to the same character table <A>tbl</A>.
## </Returns>
## <Description>
## <Index Subkey="of a group character">constituent</Index>
## <Index Subkey="a group character">decompose</Index>
## <Index Subkey="of constituents of a group character">multiplicity</Index>
## <Index Subkey="of group characters">inner product</Index>
## If <A>chi</A> and <A>psi</A> are class function objects,
## the argument <A>tbl</A> is not needed,
## but <A>tbl</A> is necessary if at least one of <A>chi</A>, <A>psi</A>
## is just a plain list.
## <P/>
## The scalar product of two <E>ordinary</E> class functions <M>\chi</M>,
## <M>\psi</M> of a group <M>G</M> is defined as
## <P/>
## <M>( \sum_{{g \in G}} \chi(g) \psi(g^{{-1}}) ) / |G|</M>.
## <P/>
## For two <E><M>p</M>-modular</E> class functions,
## the scalar product is defined as
## <M>( \sum_{{g \in S}} \chi(g) \psi(g^{{-1}}) ) / |G|</M>,
## where <M>S</M> is the set of <M>p</M>-regular elements in <M>G</M>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "ScalarProduct",
[ IsCharacterTable, IsRowVector, IsRowVector ] );
#############################################################################
##
#O MatScalarProducts( [<tbl>, ]<list>[, <list2>] )
##
## <#GAPDoc Label="MatScalarProducts">
## <ManSection>
## <Oper Name="MatScalarProducts" Arg='[tbl, ]list[, list2]'/>
##
## <Description>
## Called with two lists <A>list</A>, <A>list2</A> of class functions of the
## same character table (which may be given as the argument <A>tbl</A>),
## <Ref Oper="MatScalarProducts"/> returns the matrix of scalar products
## (see <Ref Oper="ScalarProduct" Label="for characters"/>)
## More precisely, this matrix contains in the <M>i</M>-th row the list of
## scalar products of <M><A>list2</A>[i]</M>
## with the entries of <A>list</A>.
## <P/>
## If only one list <A>list</A> of class functions is given then
## a lower triangular matrix of scalar products is returned,
## containing (for <M>j \leq i</M>) in the <M>i</M>-th row in column
## <M>j</M> the value
## <C>ScalarProduct</C><M>( <A>tbl</A>, <A>list</A>[j], <A>list</A>[i] )</M>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "MatScalarProducts",
[ IsHomogeneousList, IsHomogeneousList ] );
DeclareOperation( "MatScalarProducts",
[ IsOrdinaryTable, IsHomogeneousList, IsHomogeneousList ] );
DeclareOperation( "MatScalarProducts", [ IsHomogeneousList ] );
DeclareOperation( "MatScalarProducts",
[ IsOrdinaryTable, IsHomogeneousList ] );
#############################################################################
##
#A Norm( [<tbl>, ]<chi> )
##
## <#GAPDoc Label="Norm:ctblfuns">
## <ManSection>
## <Attr Name="Norm" Arg='[tbl, ]chi' Label="for a class function"/>
##
## <Description>
## <Index Subkey="of character" Key="Norm"><C>Norm</C></Index>
## For an ordinary class function <A>chi</A> of a group <M>G</M>
## we have <M><A>chi</A> = \sum_{{\chi \in Irr(G)}} a_{\chi} \chi</M>,
## with complex coefficients <M>a_{\chi}</M>.
## The <E>norm</E> of <A>chi</A> is defined as
## <M>\sum_{{\chi \in Irr(G)}} a_{\chi} \overline{{a_{\chi}}}</M>.
## <P/>
## <Example><![CDATA[
## gap> tbl:= CharacterTable( "A5" );;
## gap> ScalarProduct( TrivialCharacter( tbl ), Sum( Irr( tbl ) ) );
## 1
## gap> ScalarProduct( tbl, [ 1, 1, 1, 1, 1 ], Sum( Irr( tbl ) ) );
## 1
## gap> tbl2:= tbl mod 2;
## BrauerTable( "A5", 2 )
## gap> chi:= Irr( tbl2 )[1];
## Character( BrauerTable( "A5", 2 ), [ 1, 1, 1, 1 ] )
## gap> ScalarProduct( chi, chi );
## 3/4
## gap> ScalarProduct( tbl2, [ 1, 1, 1, 1 ], [ 1, 1, 1, 1 ] );
## 3/4
## gap> chars:= Irr( tbl ){ [ 2 .. 4 ] };;
## gap> chars:= Set( Tensored( chars, chars ) );;
## gap> MatScalarProducts( Irr( tbl ), chars );
## [ [ 0, 0, 0, 1, 1 ], [ 1, 1, 0, 0, 1 ], [ 1, 0, 1, 0, 1 ],
## [ 0, 1, 0, 1, 1 ], [ 0, 0, 1, 1, 1 ], [ 1, 1, 1, 1, 1 ] ]
## gap> MatScalarProducts( tbl, chars );
## [ [ 2 ], [ 1, 3 ], [ 1, 2, 3 ], [ 2, 2, 1, 3 ], [ 2, 1, 2, 2, 3 ],
## [ 2, 3, 3, 3, 3, 5 ] ]
## gap> List( chars, Norm );
## [ 2, 3, 3, 3, 3, 5 ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "Norm", IsClassFunction );
DeclareOperation( "Norm", [ IsOrdinaryTable, IsHomogeneousList ] );
#############################################################################
##
#A CentreOfCharacter( [<tbl>, ]<chi> )
##
## <#GAPDoc Label="CentreOfCharacter">
## <ManSection>
## <Attr Name="CentreOfCharacter" Arg='[tbl, ]chi'/>
##
## <Description>
## <Index Subkey="of a character">centre</Index>
## For a character <A>chi</A> of a group <M>G</M>,
## <Ref Attr="CentreOfCharacter"/> returns the <E>centre</E> of <A>chi</A>,
## that is, the normal subgroup of all those elements of <M>G</M> for which
## the quotient of the value of <A>chi</A> by the degree of <A>chi</A> is
## a root of unity.
## <P/>
## If the underlying character table of <A>psi</A> does not store the group
## <M>G</M> then an error is signalled.
## (See <Ref Attr="ClassPositionsOfCentre" Label="for a character"/>
## for a way to handle the centre implicitly,
## by listing the positions of conjugacy classes in the centre.)
## <P/>
## <Example><![CDATA[
## gap> List( Irr( S4 ), chi -> StructureDescription(CentreOfCharacter(chi)) );
## [ "S4", "1", "C2 x C2", "1", "S4" ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "CentreOfCharacter", IsClassFunction );
DeclareOperation( "CentreOfCharacter",
[ IsOrdinaryTable, IsHomogeneousList ] );
DeclareSynonym( "CenterOfCharacter", CentreOfCharacter );
#############################################################################
##
#A ClassPositionsOfCentre( <chi> )
##
## <#GAPDoc Label="ClassPositionsOfCentre:ctblfuns">
## <ManSection>
## <Attr Name="ClassPositionsOfCentre" Arg='chi' Label="for a character"/>
##
## <Description>
## is the list of positions of classes forming the centre of the character
## <A>chi</A> (see <Ref Attr="CentreOfCharacter"/>).
## <P/>
## <Example><![CDATA[
## gap> List( Irr( S4 ), ClassPositionsOfCentre );
## [ [ 1, 2, 3, 4, 5 ], [ 1 ], [ 1, 3 ], [ 1 ], [ 1, 2, 3, 4, 5 ] ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "ClassPositionsOfCentre", IsHomogeneousList );
#############################################################################
##
#A ConstituentsOfCharacter( [<tbl>, ]<chi> )
##
## <#GAPDoc Label="ConstituentsOfCharacter">
## <ManSection>
## <Attr Name="ConstituentsOfCharacter" Arg='[tbl, ]chi'/>
##
## <Description>
## Let <A>chi</A> be an ordinary or modular (virtual) character.
## If an ordinary or modular character table <A>tbl</A> is given then
## <A>chi</A> may also be a list of character values.
## <P/>
## <Ref Attr="ConstituentsOfCharacter"/> returns
## the set of those irreducible characters that occur in the decomposition
## of <A>chi</A> with nonzero coefficient.
## <P/>
## <Example><![CDATA[
## gap> nat:= NaturalCharacter( S4 );
## Character( CharacterTable( S4 ), [ 4, 2, 0, 1, 0 ] )
## gap> ConstituentsOfCharacter( nat );
## [ Character( CharacterTable( S4 ), [ 1, 1, 1, 1, 1 ] ),
## Character( CharacterTable( S4 ), [ 3, 1, -1, 0, -1 ] ) ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "ConstituentsOfCharacter", IsClassFunction );
DeclareOperation( "ConstituentsOfCharacter",
[ IsCharacterTable, IsHomogeneousList ] );
#############################################################################
##
#A DegreeOfCharacter( <chi> )
##
## <#GAPDoc Label="DegreeOfCharacter">
## <ManSection>
## <Attr Name="DegreeOfCharacter" Arg='chi'/>
##
## <Description>
## is the value of the character <A>chi</A> on the identity element.
## This can also be obtained as <A>chi</A><C>[1]</C>.
## <P/>
## <Example><![CDATA[
## gap> List( Irr( S4 ), DegreeOfCharacter );
## [ 1, 3, 2, 3, 1 ]
## gap> nat:= NaturalCharacter( S4 );
## Character( CharacterTable( S4 ), [ 4, 2, 0, 1, 0 ] )
## gap> nat[1];
## 4
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "DegreeOfCharacter", IsClassFunction );
#############################################################################
##
#O InertiaSubgroup( [<tbl>, ]<G>, <chi> )
##
## <#GAPDoc Label="InertiaSubgroup">
## <ManSection>
## <Oper Name="InertiaSubgroup" Arg='[tbl, ]G, chi'/>
##
## <Description>
## Let <A>chi</A> be a character of a group <M>H</M>
## and <A>tbl</A> the character table of <M>H</M>;
## if the argument <A>tbl</A> is not given then the underlying character
## table of <A>chi</A> (see <Ref Attr="UnderlyingCharacterTable"/>) is
## used instead.
## Furthermore, let <A>G</A> be a group that contains <M>H</M> as a normal
## subgroup.
## <P/>
## <Ref Oper="InertiaSubgroup"/> returns the stabilizer in <A>G</A> of
## <A>chi</A>, w.r.t. the action of <A>G</A> on the classes of <M>H</M>
## via conjugation.
## In other words, <Ref Oper="InertiaSubgroup"/> returns the group of all
## those elements <M>g \in <A>G</A></M> that satisfy
## <M><A>chi</A>^g = <A>chi</A></M>.
## <P/>
## <Example><![CDATA[
## gap> der:= DerivedSubgroup( S4 );
## Alt( [ 1 .. 4 ] )
## gap> List( Irr( der ), chi -> InertiaSubgroup( S4, chi ) );
## [ S4, S4, Alt( [ 1 .. 4 ] ), Alt( [ 1 .. 4 ] ) ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "InertiaSubgroup", [ IsGroup, IsClassFunction ] );
DeclareOperation( "InertiaSubgroup",
[ IsOrdinaryTable, IsGroup, IsHomogeneousList ] );
#############################################################################
##
#A KernelOfCharacter( [<tbl>, ]<chi> )
##
## <#GAPDoc Label="KernelOfCharacter">
## <ManSection>
## <Attr Name="KernelOfCharacter" Arg='[tbl, ]chi'/>
##
## <Description>
## For a class function <A>chi</A> of a group <M>G</M>,
## <Ref Attr="KernelOfCharacter"/> returns the normal subgroup of <M>G</M>
## that is formed by those conjugacy classes for which the value of
## <A>chi</A> equals the degree of <A>chi</A>.
## If the underlying character table of <A>chi</A> does not store the group
## <M>G</M> then an error is signalled.
## (See <Ref Attr="ClassPositionsOfKernel"/> for a way to handle the
## kernel implicitly,
## by listing the positions of conjugacy classes in the kernel.)
## <P/>
## The returned group is the kernel of any representation of <M>G</M> that
## affords <A>chi</A>.
## <P/>
## <Example><![CDATA[
## gap> List( Irr( S4 ), chi -> StructureDescription(KernelOfCharacter(chi)) );
## [ "A4", "1", "C2 x C2", "1", "S4" ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "KernelOfCharacter", IsClassFunction );
DeclareOperation( "KernelOfCharacter",
[ IsOrdinaryTable, IsHomogeneousList ] );
#############################################################################
##
#A ClassPositionsOfKernel( <chi> )
##
## <#GAPDoc Label="ClassPositionsOfKernel">
## <ManSection>
## <Attr Name="ClassPositionsOfKernel" Arg='chi'/>
##
## <Description>
## is the list of positions of those conjugacy classes that form the kernel
## of the character <A>chi</A>, that is, those positions with character
## value equal to the character degree.
## <P/>
## <Example><![CDATA[
## gap> List( Irr( S4 ), ClassPositionsOfKernel );
## [ [ 1, 3, 4 ], [ 1 ], [ 1, 3 ], [ 1 ], [ 1, 2, 3, 4, 5 ] ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "ClassPositionsOfKernel", IsHomogeneousList );
#############################################################################
##
#O CycleStructureClass( [<tbl>, ]<chi>, <class> )
##
## <#GAPDoc Label="CycleStructureClass">
## <ManSection>
## <Oper Name="CycleStructureClass" Arg='[tbl, ]chi, class'/>
##
## <Description>
## Let <A>permchar</A> be a permutation character, and <A>class</A> be the
## position of a conjugacy class of the character table of <A>permchar</A>.
## <Ref Oper="CycleStructureClass"/> returns a list describing
## the cycle structure of each element in class <A>class</A> in the
## underlying permutation representation, in the same format as the result
## of <Ref Attr="CycleStructurePerm"/>.
## <P/>
## <Example><![CDATA[
## gap> nat:= NaturalCharacter( S4 );
## Character( CharacterTable( S4 ), [ 4, 2, 0, 1, 0 ] )
## gap> List( [ 1 .. 5 ], i -> CycleStructureClass( nat, i ) );
## [ [ ], [ 1 ], [ 2 ], [ , 1 ], [ ,, 1 ] ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "CycleStructureClass",
[ IsOrdinaryTable, IsHomogeneousList, IsPosInt ] );
DeclareOperation( "CycleStructureClass", [ IsClassFunction, IsPosInt ] );
#############################################################################
##
#P IsTransitive( [<tbl>, ]<chi> )
##
## <#GAPDoc Label="IsTransitive:ctblfuns">
## <ManSection>
## <Prop Name="IsTransitive" Arg='[tbl, ]chi' Label="for a character"/>
##
## <Description>
## For a permutation character <A>chi</A> of the group <M>G</M> that
## corresponds to an action on the <M>G</M>-set <M>\Omega</M>
## (see <Ref Oper="PermutationCharacter"
## Label="for a group, an action domain, and a function"/>),
## <Ref Oper="IsTransitive" Label="for a group, an action domain, etc."/>
## returns <K>true</K> if the action of <M>G</M> on <M>\Omega</M> is
## transitive, and <K>false</K> otherwise.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsTransitive", IsClassFunction );
DeclareOperation( "IsTransitive", [ IsCharacterTable, IsHomogeneousList ] );
#############################################################################
##
#A Transitivity( [<tbl>, ]<chi> )
##
## <#GAPDoc Label="Transitivity:ctblfuns">
## <ManSection>
## <Attr Name="Transitivity" Arg='[tbl, ]chi' Label="for a character"/>
##
## <Description>
## For a permutation character <A>chi</A> of the group <M>G</M>
## that corresponds to an action on the <M>G</M>-set <M>\Omega</M>
## (see <Ref Oper="PermutationCharacter"
## Label="for a group, an action domain, and a function"/>),
## <Ref Attr="Transitivity" Label="for a character"/> returns the maximal
## nonnegative integer <M>k</M> such that the action of <M>G</M> on
## <M>\Omega</M> is <M>k</M>-transitive.
## <P/>
## <Example><![CDATA[
## gap> IsTransitive( nat ); Transitivity( nat );
## true
## 4
## gap> Transitivity( 2 * TrivialCharacter( S4 ) );
## 0
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "Transitivity", IsClassFunction );
DeclareOperation( "Transitivity", [ IsOrdinaryTable, IsHomogeneousList ] );
#############################################################################
##
#A CentralCharacter( [<tbl>, ]<chi> )
##
## <#GAPDoc Label="CentralCharacter">
## <ManSection>
## <Attr Name="CentralCharacter" Arg='[tbl, ]chi'/>
##
## <Description>
## <Index>central character</Index>
## For a character <A>chi</A> of a group <M>G</M>,
## <Ref Attr="CentralCharacter"/> returns
## the <E>central character</E> of <A>chi</A>.
## <P/>
## The central character of <M>\chi</M> is the class function
## <M>\omega_{\chi}</M> defined by
## <M>\omega_{\chi}(g) = |g^G| \cdot \chi(g)/\chi(1)</M> for each
## <M>g \in G</M>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "CentralCharacter", IsClassFunction );
DeclareOperation( "CentralCharacter",
[ IsCharacterTable, IsHomogeneousList ] );
#############################################################################
##
#A DeterminantOfCharacter( [<tbl>, ]<chi> )
##
## <#GAPDoc Label="DeterminantOfCharacter">
## <ManSection>
## <Attr Name="DeterminantOfCharacter" Arg='[tbl, ]chi'/>
##
## <Description>
## <Index>determinant character</Index>
## <Ref Attr="DeterminantOfCharacter"/> returns the
## <E>determinant character</E> of the character <A>chi</A>.
## This is defined to be the character obtained by taking the determinant of
## representing matrices of any representation affording <A>chi</A>;
## the determinant can be computed using <Ref Oper="EigenvaluesChar"/>.
## <P/>
## It is also possible to call <Ref Oper="Determinant"/> instead of
## <Ref Attr="DeterminantOfCharacter"/>.
## <P/>
## Note that the determinant character is well-defined for virtual
## characters.
## <P/>
## <Example><![CDATA[
## gap> CentralCharacter( TrivialCharacter( S4 ) );
## ClassFunction( CharacterTable( S4 ), [ 1, 6, 3, 8, 6 ] )
## gap> DeterminantOfCharacter( Irr( S4 )[3] );
## Character( CharacterTable( S4 ), [ 1, -1, 1, 1, -1 ] )
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "DeterminantOfCharacter", IsClassFunction );
DeclareOperation( "DeterminantOfCharacter",
[ IsCharacterTable, IsHomogeneousList ] );
#############################################################################
##
#O EigenvaluesChar( [<tbl>, ]<chi>, <class> )
##
## <#GAPDoc Label="EigenvaluesChar">
## <ManSection>
## <Oper Name="EigenvaluesChar" Arg='[tbl, ]chi, class'/>
##
## <Description>
## Let <A>chi</A> be a character of a group <M>G</M>.
## For an element <M>g \in G</M> in the <A>class</A>-th conjugacy class,
## of order <M>n</M>, let <M>M</M> be a matrix of a representation affording
## <A>chi</A>.
## <P/>
## <Ref Oper="EigenvaluesChar"/> returns the list of length <M>n</M>
## where at position <M>k</M> the multiplicity
## of <C>E</C><M>(n)^k = \exp(2 \pi i k / n)</M>
## as an eigenvalue of <M>M</M> is stored.
## <P/>
## We have
## <C><A>chi</A>[ <A>class</A> ] = List( [ 1 .. n ], k -> E(n)^k )
## * EigenvaluesChar( <A>tbl</A>, <A>chi</A>, <A>class</A> )</C>.
## <P/>
## It is also possible to call <Ref Oper="Eigenvalues"/> instead of
## <Ref Oper="EigenvaluesChar"/>.
## <P/>
## <Example><![CDATA[
## gap> chi:= Irr( CharacterTable( "A5" ) )[2];
## Character( CharacterTable( "A5" ),
## [ 3, -1, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ] )
## gap> List( [ 1 .. 5 ], i -> Eigenvalues( chi, i ) );
## [ [ 3 ], [ 2, 1 ], [ 1, 1, 1 ], [ 0, 1, 1, 0, 1 ], [ 1, 0, 0, 1, 1 ] ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "EigenvaluesChar", [ IsClassFunction, IsPosInt ] );
DeclareOperation( "EigenvaluesChar",
[ IsCharacterTable, IsHomogeneousList, IsPosInt ] );
--> --------------------
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--> --------------------
