Goto Chapter: Top 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 Bib Ind

72 Class Functions

72.8 Operations for Class Functions

72.8-1 IsCharacter

72.8-2 IsVirtualCharacter

72.8-3 IsIrreducibleCharacter

72.8-4 DegreeOfCharacter

72.8-5 ScalarProduct

72.8-6 MatScalarProducts

72.8-7 Norm

72.8-8 ConstituentsOfCharacter

72.8-9 KernelOfCharacter

72.8-10 ClassPositionsOfKernel

72.8-11 CentreOfCharacter

72.8-12 ClassPositionsOfCentre

72.8-13 InertiaSubgroup

72.8-14 CycleStructureClass

72.8-15 IsTransitive

72.8-16 Transitivity

72.8-17 CentralCharacter

72.8-18 DeterminantOfCharacter

72.8-19 EigenvaluesChar

72.8-20 Tensored

72.8-1 IsCharacter

72.8-2 IsVirtualCharacter

72.8-3 IsIrreducibleCharacter

72.8-4 DegreeOfCharacter

72.8-5 ScalarProduct

72.8-6 MatScalarProducts

72.8-7 Norm

72.8-8 ConstituentsOfCharacter

72.8-9 KernelOfCharacter

72.8-10 ClassPositionsOfKernel

72.8-11 CentreOfCharacter

72.8-12 ClassPositionsOfCentre

72.8-13 InertiaSubgroup

72.8-14 CycleStructureClass

72.8-15 IsTransitive

72.8-16 Transitivity

72.8-17 CentralCharacter

72.8-18 DeterminantOfCharacter

72.8-19 EigenvaluesChar

72.8-20 Tensored

This chapter describes operations for *class functions of finite groups*. For operations concerning *character tables*, see Chapter 71.

Several examples in this chapter require the **GAP** Character Table Library to be available. If it is not yet loaded then we load it now.

gap> LoadPackage( "ctbllib" ); true

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.

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.

Second, when one says that "\(\chi\) is a character of a group \(G\)" then this object \(\chi\) carries a lot of information. \(\chi\) has certain properties such as being irreducible or not. Several subgroups of \(G\) are related to \(\chi\), such as the kernel and the centre of \(\chi\). Other attributes of characters are the determinant and the central character. This knowledge cannot be stored in a plain list.

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.

The main idea behind class function objects is that a class function object is equal to its values list in the sense of `\=`

(31.11-1), 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. **GAP** library functions prefer to return class function objects rather than returning just values lists, for example `Irr`

(71.8-2) lists consist of class function objects, and `TrivialCharacter`

(72.7-1) returns a class function object.

Here is an *example* that shows both approaches. First we define some groups.

gap> S4:= SymmetricGroup( 4 );; SetName( S4, "S4" ); gap> D8:= SylowSubgroup( S4, 2 );; SetName( D8, "D8" );

We do some computations using the functions described later in this Chapter, first with class function objects.

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 ] )

Now we repeat these calculations with plain lists of character values. Here we need the character tables in some places.

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 ] )

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.)

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 ] )

Functions that compute *normal* subgroups related to characters have counterparts that return the list of class positions corresponding to these groups.

gap> ClassPositionsOfKernel( irrs5[2] ); [ 1, 2, 3, 4 ] gap> ClassPositionsOfCentre( irrs5[2] ); [ 1, 2, 3, 4, 5, 6, 7 ]

Non-normal subgroups cannot be described this way, so for example inertia subgroups (see `InertiaSubgroup`

(72.8-13)) can in general not be computed from character tables without access to their groups.

`‣ IsClassFunction` ( obj ) | ( category ) |

A *class function* (in characteristic \(p\)) of a finite group \(G\) is a map from the set of (\(p\)-regular) elements in \(G\) to the field of cyclotomics that is constant on conjugacy classes of \(G\).

Each class function in **GAP** is represented by an *immutable list*, where at the \(i\)-th position the value on the \(i\)-th conjugacy class of the character table of \(G\) 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 71.6); class functions always refer to the ordering of classes in the character table.

*Class function objects* in **GAP** are not just plain lists, they store the character table of the group \(G\) as value of the attribute `UnderlyingCharacterTable`

(72.2-1). The group \(G\) itself is accessible only via the character table and thus only if the character table stores its group, as value of the attribute `UnderlyingGroup`

(71.6-1). 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.

There are (at least) two reasons why class functions in **GAP** are *not* 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 \(G\) 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.

A further argument is that the typical operations for mappings such as `Image`

(32.4-6) and `PreImage`

(32.5-6) play no important role for class functions.

Basic operations for class functions are `UnderlyingCharacterTable`

(72.2-1), `ValuesOfClassFunction`

(72.2-2), and the basic operations for lists (see 21.2).

`‣ UnderlyingCharacterTable` ( psi ) | ( attribute ) |

For a class function `psi` of a group \(G\) the character table of \(G\) is stored as value of `UnderlyingCharacterTable`

. The ordering of entries in the list `psi` (see `ValuesOfClassFunction`

(72.2-2)) refers to the ordering of conjugacy classes in this character table.

If `psi` is an ordinary class function then the underlying character table is the ordinary character table of \(G\) (see `OrdinaryCharacterTable`

(71.8-4)), if `psi` is a class function in characteristic \(p \neq 0\) then the underlying character table is the \(p\)-modular Brauer table of \(G\) (see `BrauerTable`

(71.3-2)). So the underlying characteristic of `psi` can be read off from the underlying character table.

`‣ ValuesOfClassFunction` ( psi ) | ( attribute ) |

is the list of values of the class function `psi`, the \(i\)-th entry being the value on the \(i\)-th conjugacy class of the underlying character table (see `UnderlyingCharacterTable`

(72.2-1)).

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 ]

With respect to `\=`

(31.11-1) and `\<`

(31.11-1), class functions behave equally to their lists of values (see `ValuesOfClassFunction`

(72.2-2)). 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.

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> irr1:= Irr( grps[1] ); [ Character( t1, [ 1, 1, 1, 1, 1 ] ), Character( t1, [ 1, -1, -1, 1, 1 ] ), Character( t1, [ 1, -1, 1, 1, -1 ] ), Character( t1, [ 1, 1, -1, 1, -1 ] ), Character( t1, [ 2, 0, 0, -2, 0 ] ) ] gap> irr2:= Irr( grps[2] ); [ Character( t2, [ 1, 1, 1, 1, 1 ] ), Character( t2, [ 1, -1, -1, 1, 1 ] ), Character( t2, [ 1, -1, 1, 1, -1 ] ), Character( t2, [ 1, 1, -1, 1, -1 ] ), Character( t2, [ 2, 0, 0, -2, 0 ] ) ] gap> irr1 = irr2; true gap> IsSSortedList( irr1 ); false gap> irr1[1] < irr1[2]; false gap> irr1[2] < irr1[3]; true

Class functions are *row vectors* of cyclotomics. The *additive* 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 *multiplicative* 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 `ScalarProduct`

(72.8-5). In terms of filters, the arithmetic of class functions is based on the decision that they lie in `IsGeneralizedRowVector`

(21.12-1), with additive nesting depth \(1\), but they do *not* lie in `IsMultiplicativeGeneralizedRowVector`

(21.12-2).

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 `UnderlyingCharacterTable`

(72.2-1)) are again class functions of this table. The sum and the difference of a class function and a list that is *not* a class function are plain lists, as well as the sum and the difference of two class functions of different character tables.

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 ]

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 \(F\) of cyclotomics, the \(F\)-span of a given set of class functions forms an algebra.

The product of two class functions of *different* tables and the product of a class function and a list that is *not* 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.

gap> tens:= irr[3] * irr[4]; Character( S4, [ 6, 0, -2, 0, 0 ] ) gap> ValuesOfClassFunction( irr[3] ) * ValuesOfClassFunction( irr[4] ); 4

Class functions without zero values are invertible, the *inverse* is defined pointwise. As a consequence, for example groups of linear characters can be formed.

gap> tens / irr[1]; Character( S4, [ 6, 0, -2, 0, 0 ] )

Other (somewhat strange) implications of the definition of arithmetic operations for class functions, together with the general rules of list arithmetic (see 21.11), apply to the case of products involving *lists* 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.

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 ] ]

Also the product of a class function with a list of class functions is *not* a vector-matrix product but the list of pointwise products.

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 ] ) ]

And the product of two lists of class functions is *not* the matrix product but the sum of the pointwise products.

gap> irr * irr; Character( S4, [ 24, 4, 8, 3, 4 ] )

The *powering* operator `\^`

(31.12-1) 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 `\^`

(31.12-1); instead one can use `Permuted`

(21.20-17).) The power of a class function by a group or a character table is the induced class function (see `InducedClassFunction`

(72.9-3)). The power of a group element by a class function is the class function value at (the conjugacy class containing) this element.

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)

`‣ Characteristic` ( chi ) | ( attribute ) |

The *characteristic* of class functions is zero, as for all list of cyclotomics. For class functions of a \(p\)-modular character table, such as Brauer characters, the prime \(p\) is given by the `UnderlyingCharacteristic`

(71.9-5) value of the character table.

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

`‣ ComplexConjugate` ( chi ) | ( attribute ) |

`‣ GaloisCyc` ( chi, k ) | ( operation ) |

`‣ Permuted` ( chi, pi ) | ( method ) |

The operations `ComplexConjugate`

, `GaloisCyc`

, and `Permuted`

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.

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) ] )

`‣ Order` ( chi ) | ( attribute ) |

By definition of `Order`

(31.10-10) for arbitrary monoid elements, the return value of `Order`

(31.10-10) for a character must be its multiplicative order. The *determinantal order* (see `DeterminantOfCharacter`

(72.8-18)) of a character `chi` can be computed as `Order( Determinant( `

.`chi` ) )

gap> det:= Determinant( irr[3] ); Character( S4, [ 1, -1, 1, 1, -1 ] ) gap> Order( det ); 2

`‣ ViewObj` ( chi ) | ( method ) |

The default `ViewObj`

(6.3-5) methods for class functions print one of the strings `"ClassFunction"`

, `"VirtualCharacter"`

, `"Character"`

(depending on whether the class function is known to be a character or virtual character, see `IsCharacter`

(72.8-1), `IsVirtualCharacter`

(72.8-2)), followed by the `ViewObj`

(6.3-5) output for the underlying character table (see 71.13), and the list of values. The table is chosen (and not the group) in order to distinguish class functions of different underlying characteristic (see `UnderlyingCharacteristic`

(71.9-5)).

`‣ PrintObj` ( chi ) | ( method ) |

The default `PrintObj`

(6.3-5) method for class functions does the same as `ViewObj`

(6.3-5), except that the character table is `Print`

(6.3-4)-ed instead of `View`

(6.3-3)-ed.

*Note* that if a class function is shown only with one of the strings `"ClassFunction"`

, `"VirtualCharacter"`

, 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.

In order to reduce the space that is needed to print a class function, it may be useful to give a name (see `Name`

(12.8-2)) to the underlying character table.

`‣ Display` ( chi ) | ( method ) |

The default `Display`

(6.3-6) method for a class function `chi` calls `Display`

(6.3-6) for its underlying character table (see 71.13), with `chi` as the only entry in the `chars`

list of the options record.

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

`‣ ClassFunction` ( tbl, values ) | ( operation ) |

`‣ ClassFunction` ( G, values ) | ( operation ) |

In the first form, `ClassFunction`

returns the class function of the character table `tbl` with values given by the list `values` of cyclotomics. In the second form, `G` must be a group, and the class function of its ordinary character table is returned.

Note that `tbl` determines the underlying characteristic of the returned class function (see `UnderlyingCharacteristic`

(71.9-5)).

`‣ VirtualCharacter` ( tbl, values ) | ( operation ) |

`‣ VirtualCharacter` ( G, values ) | ( operation ) |

`VirtualCharacter`

returns the virtual character (see `IsVirtualCharacter`

(72.8-2)) of the character table `tbl` or the group `G`, respectively, with values given by the list `values`.

It is *not* checked whether the given values really describe a virtual character.

`‣ Character` ( tbl, values ) | ( operation ) |

`‣ Character` ( G, values ) | ( operation ) |

`Character`

returns the character (see `IsCharacter`

(72.8-1)) of the character table `tbl` or the group `G`, respectively, with values given by the list `values`.

It is *not* checked whether the given values really describe a character.

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 ] )

`‣ ClassFunctionSameType` ( tbl, chi, values ) | ( function ) |

Let `tbl` be a character table, `chi` a class function object (*not* necessarily a class function of `tbl`), and `values` a list of cyclotomics. `ClassFunctionSameType`

returns the class function \(\psi\) of `tbl` with values list `values`, constructed with `ClassFunction`

(72.6-1).

If `chi` is known to be a (virtual) character then \(\psi\) is also known to be a (virtual) character.

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 ] )

`‣ TrivialCharacter` ( tbl ) | ( attribute ) |

`‣ TrivialCharacter` ( G ) | ( attribute ) |

is the *trivial character* of the group `G` or its character table `tbl`, respectively. This is the class function with value equal to \(1\) for each class.

gap> TrivialCharacter( CharacterTable( "A5" ) ); Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ) gap> TrivialCharacter( SymmetricGroup( 3 ) ); Character( CharacterTable( Sym( [ 1 .. 3 ] ) ), [ 1, 1, 1 ] )

`‣ NaturalCharacter` ( G ) | ( attribute ) |

`‣ NaturalCharacter` ( hom ) | ( attribute ) |

If the argument is a permutation group `G` then `NaturalCharacter`

returns the (ordinary) character of the natural permutation representation of `G` on the set of moved points (see `MovedPoints`

(42.3-3)), that is, the value on each class is the number of points among the moved points of `G` that are fixed by any permutation in that class.

If the argument is a matrix group `G` in characteristic zero then `NaturalCharacter`

returns the (ordinary) character of the natural matrix representation of `G`, that is, the value on each class is the trace of any matrix in that class.

If the argument is a group homomorphism `hom` whose image is a permutation group or a matrix group then `NaturalCharacter`

returns the restriction of the natural character of the image of `hom` to the preimage of `hom`.

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:= IsomorphismPermGroup( d8 );; gap> NaturalCharacter( hom ); Character( CharacterTable( <pc group of size 8 with 3 generators> ), [ 8, 0, 0, 0, 0 ] )

`‣ PermutationCharacter` ( G, D, opr ) | ( operation ) |

`‣ PermutationCharacter` ( G, U ) | ( operation ) |

Called with a group `G`, an action domain or proper set `D`, and an action function `opr` (see Chapter 41), `PermutationCharacter`

returns the *permutation character* of the action of `G` on `D` via `opr`, that is, the value on each class is the number of points in `D` that are fixed by an element in this class under the action `opr`.

If the arguments are a group `G` and a subgroup `U` of `G` then `PermutationCharacter`

returns the permutation character of the action of `G` on the right cosets of `U` via right multiplication.

To compute the permutation character of a *transitive permutation group* `G` on the cosets of a point stabilizer `U`, the attribute `NaturalCharacter`

(72.7-2) of `G` can be used instead of `PermutationCharacter( `

.`G`, `U` )

More facilities concerning permutation characters are the transitivity test (see Section 72.8) and several tools for computing possible permutation characters (see 72.13, 72.14).

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 ] )

In the description of the following operations, the optional first argument `tbl` is needed only if the argument `chi` is a plain list and not a class function object. In this case, `tbl` must always be the character table of which `chi` shall be regarded as a class function.

`‣ IsCharacter` ( [tbl, ]chi ) | ( property ) |

An *ordinary character* of a group \(G\) is a class function of \(G\) whose values are the traces of a complex matrix representation of \(G\).

A *Brauer character* of \(G\) in characteristic \(p\) is a class function of \(G\) whose values are the complex lifts of a matrix representation of \(G\) with image a finite field of characteristic \(p\).

`‣ IsVirtualCharacter` ( [tbl, ]chi ) | ( property ) |

A *virtual character* is a class function that can be written as the difference of two proper characters (see `IsCharacter`

(72.8-1)).

`‣ IsIrreducibleCharacter` ( [tbl, ]chi ) | ( property ) |

A character is *irreducible* 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 `ScalarProduct`

(72.8-5)). For Brauer characters there is no generic method for checking irreducibility.

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

`‣ DegreeOfCharacter` ( chi ) | ( attribute ) |

is the value of the character `chi` on the identity element. This can also be obtained as `chi``[1]`

.

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

`‣ ScalarProduct` ( [tbl, ]chi, psi ) | ( operation ) |

Returns: the scalar product of the class functions `chi` and `psi`, which belong to the same character table `tbl`.

If `chi` and `psi` are class function objects, the argument `tbl` is not needed, but `tbl` is necessary if at least one of `chi`, `psi` is just a plain list.

The scalar product of two *ordinary* class functions \(\chi\), \(\psi\) of a group \(G\) is defined as

\(( \sum_{{g \in G}} \chi(g) \psi(g^{{-1}}) ) / |G|\).

For two *\(p\)-modular* class functions, the scalar product is defined as \(( \sum_{{g \in S}} \chi(g) \psi(g^{{-1}}) ) / |G|\), where \(S\) is the set of \(p\)-regular elements in \(G\).

`‣ MatScalarProducts` ( [tbl, ]list[, list2] ) | ( operation ) |

Called with two lists `list`, `list2` of class functions of the same character table (which may be given as the argument `tbl`), `MatScalarProducts`

returns the matrix of scalar products (see `ScalarProduct`

(72.8-5)) More precisely, this matrix contains in the \(i\)-th row the list of scalar products of \(\textit{list2}[i]\) with the entries of `list`.

If only one list `list` of class functions is given then a lower triangular matrix of scalar products is returned, containing (for \(j \leq i\)) in the \(i\)-th row in column \(j\) the value `ScalarProduct`

\(( \textit{tbl}, \textit{list}[j], \textit{list}[i] )\).

`‣ Norm` ( [tbl, ]chi ) | ( attribute ) |

For an ordinary class function `chi` of a group \(G\) we have \(\textit{chi} = \sum_{{\chi \in Irr(G)}} a_{\chi} \chi\), with complex coefficients \(a_{\chi}\). The *norm* of `chi` is defined as \(\sum_{{\chi \in Irr(G)}} a_{\chi} \overline{{a_{\chi}}}\).

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 ]

`‣ ConstituentsOfCharacter` ( [tbl, ]chi ) | ( attribute ) |

Let `chi` be an ordinary or modular (virtual) character. If an ordinary or modular character table `tbl` is given then `chi` may also be a list of character values.

`ConstituentsOfCharacter`

returns the set of those irreducible characters that occur in the decomposition of `chi` with nonzero coefficient.

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 ] ) ]

`‣ KernelOfCharacter` ( [tbl, ]chi ) | ( attribute ) |

For a class function `chi` of a group \(G\), `KernelOfCharacter`

returns the normal subgroup of \(G\) that is formed by those conjugacy classes for which the value of `chi` equals the degree of `chi`. If the underlying character table of `chi` does not store the group \(G\) then an error is signalled. (See `ClassPositionsOfKernel`

(72.8-10) for a way to handle the kernel implicitly, by listing the positions of conjugacy classes in the kernel.)

The returned group is the kernel of any representation of \(G\) that affords `chi`.

gap> List( Irr( S4 ), KernelOfCharacter ); [ Alt( [ 1 .. 4 ] ), Group(()), Group([ (1,2)(3,4), (1,3)(2,4) ]), Group(()), Group([ (), (1,2), (1,2)(3,4), (1,2,3), (1,2,3,4) ]) ]

`‣ ClassPositionsOfKernel` ( chi ) | ( attribute ) |

is the list of positions of those conjugacy classes that form the kernel of the character `chi`, that is, those positions with character value equal to the character degree.

gap> List( Irr( S4 ), ClassPositionsOfKernel ); [ [ 1, 3, 4 ], [ 1 ], [ 1, 3 ], [ 1 ], [ 1, 2, 3, 4, 5 ] ]

`‣ CentreOfCharacter` ( [tbl, ]chi ) | ( attribute ) |

For a character `chi` of a group \(G\), `CentreOfCharacter`

returns the *centre* of `chi`, that is, the normal subgroup of all those elements of \(G\) for which the quotient of the value of `chi` by the degree of `chi` is a root of unity.

If the underlying character table of `psi` does not store the group \(G\) then an error is signalled. (See `ClassPositionsOfCentre`

(72.8-12) for a way to handle the centre implicitly, by listing the positions of conjugacy classes in the centre.)

gap> List( Irr( S4 ), CentreOfCharacter ); [ Group([ (), (1,2), (1,2)(3,4), (1,2,3), (1,2,3,4) ]), Group(()), Group([ (1,2)(3,4), (1,3)(2,4) ]), Group(()), Group([ (), (1,2), (1,2)(3,4), (1,2,3), (1,2,3,4) ]) ]

`‣ ClassPositionsOfCentre` ( chi ) | ( attribute ) |

is the list of positions of classes forming the centre of the character `chi` (see `CentreOfCharacter`

(72.8-11)).

gap> List( Irr( S4 ), ClassPositionsOfCentre ); [ [ 1, 2, 3, 4, 5 ], [ 1 ], [ 1, 3 ], [ 1 ], [ 1, 2, 3, 4, 5 ] ]

`‣ InertiaSubgroup` ( [tbl, ]G, chi ) | ( operation ) |

Let `chi` be a character of a group \(H\) and `tbl` the character table of \(H\); if the argument `tbl` is not given then the underlying character table of `chi` (see `UnderlyingCharacterTable`

(72.2-1)) is used instead. Furthermore, let `G` be a group that contains \(H\) as a normal subgroup.

`InertiaSubgroup`

returns the stabilizer in `G` of `chi`, w.r.t. the action of `G` on the classes of \(H\) via conjugation. In other words, `InertiaSubgroup`

returns the group of all those elements \(g \in \textit{G}\) that satisfy \(\textit{chi}^g = \textit{chi}\).

gap> der:= DerivedSubgroup( S4 ); Alt( [ 1 .. 4 ] ) gap> List( Irr( der ), chi -> InertiaSubgroup( S4, chi ) ); [ S4, Alt( [ 1 .. 4 ] ), Alt( [ 1 .. 4 ] ), S4 ]

`‣ CycleStructureClass` ( [tbl, ]chi, class ) | ( operation ) |

Let `permchar` be a permutation character, and `class` be the position of a conjugacy class of the character table of `permchar`. `CycleStructureClass`

returns a list describing the cycle structure of each element in class `class` in the underlying permutation representation, in the same format as the result of `CycleStructurePerm`

(42.4-2).

gap> nat:= NaturalCharacter( S4 ); Character( CharacterTable( S4 ), [ 4, 2, 0, 1, 0 ] ) gap> List( [ 1 .. 5 ], i -> CycleStructureClass( nat, i ) ); [ [ ], [ 1 ], [ 2 ], [ , 1 ], [ ,, 1 ] ]

`‣ IsTransitive` ( [tbl, ]chi ) | ( property ) |

For a permutation character `chi` of the group \(G\) that corresponds to an action on the \(G\)-set \(\Omega\) (see `PermutationCharacter`

(72.7-3)), `IsTransitive`

(41.10-1) returns `true`

if the action of \(G\) on \(\Omega\) is transitive, and `false`

otherwise.

`‣ Transitivity` ( [tbl, ]chi ) | ( attribute ) |

For a permutation character `chi` of the group \(G\) that corresponds to an action on the \(G\)-set \(\Omega\) (see `PermutationCharacter`

(72.7-3)), `Transitivity`

returns the maximal nonnegative integer \(k\) such that the action of \(G\) on \(\Omega\) is \(k\)-transitive.

gap> IsTransitive( nat ); Transitivity( nat ); true 4 gap> Transitivity( 2 * TrivialCharacter( S4 ) ); 0

`‣ CentralCharacter` ( [tbl, ]chi ) | ( attribute ) |

For a character `chi` of a group \(G\), `CentralCharacter`

returns the *central character* of `chi`.

The central character of \(\chi\) is the class function \(\omega_{\chi}\) defined by \(\omega_{\chi}(g) = |g^G| \cdot \chi(g)/\chi(1)\) for each \(g \in G\).

`‣ DeterminantOfCharacter` ( [tbl, ]chi ) | ( attribute ) |

`DeterminantOfCharacter`

returns the *determinant character* of the character `chi`. This is defined to be the character obtained by taking the determinant of representing matrices of any representation affording `chi`; the determinant can be computed using `EigenvaluesChar`

(72.8-19).

It is also possible to call `Determinant`

(24.4-4) instead of `DeterminantOfCharacter`

.

Note that the determinant character is well-defined for virtual characters.

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 ] )

`‣ EigenvaluesChar` ( [tbl, ]chi, class ) | ( operation ) |

Let `chi` be a character of a group \(G\). For an element \(g \in G\) in the `class`-th conjugacy class, of order \(n\), let \(M\) be a matrix of a representation affording `chi`.

`EigenvaluesChar`

returns the list of length \(n\) where at position \(k\) the multiplicity of `E`

\((n)^k = \exp(2 \pi i k / n)\) as an eigenvalue of \(M\) is stored.

We have

.`chi`[ `class` ] = List( [ 1 .. n ], k -> E(n)^k ) * EigenvaluesChar( `tbl`, `chi`, `class` )

It is also possible to call `Eigenvalues`

(24.8-3) instead of `EigenvaluesChar`

.

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 ] ]

`‣ Tensored` ( chars1, chars2 ) | ( operation ) |

Let `chars1` and `chars2` be lists of (values lists of) class functions of the same character table. `Tensored`

returns the list of tensor products of all entries in `chars1` with all entries in `chars2`.

gap> irra5:= Irr( CharacterTable( "A5" ) );; gap> chars1:= irra5{ [ 1 .. 3 ] };; chars2:= irra5{ [ 2, 3 ] };; gap> Tensored( chars1, chars2 ); [ Character( CharacterTable( "A5" ), [ 3, -1, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ] ), Character( CharacterTable( "A5" ), [ 3, -1, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ] ), Character( CharacterTable( "A5" ), [ 9, 1, 0, -2*E(5)-E(5)^2-E(5)^3-2*E(5)^4, -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4 ] ), Character( CharacterTable( "A5" ), [ 9, 1, 0, -1, -1 ] ), Character( CharacterTable( "A5" ), [ 9, 1, 0, -1, -1 ] ), Character( CharacterTable( "A5" ), [ 9, 1, 0, -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4, -2*E(5)-E(5)^2-E(5)^3-2*E(5)^4 ] ) ]

For restricting a class function of a group \(G\) to a subgroup \(H\) and for inducing a class function of \(H\) to \(G\), the *class fusion* from \(H\) to \(G\) must be known (see 73.3).

If \(F\) is the factor group of \(G\) by the normal subgroup \(N\) then each class function of \(F\) can be naturally regarded as a class function of \(G\), with \(N\) in its kernel. For a class function of \(F\), the corresponding class function of \(G\) is called the *inflated* 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 \(G\) and \(F\); either one needs the natural homomorphism, or the factor fusion to the character table of \(F\) must be stored on the table of \(G\). This explains the different syntax for computing restricted and inflated class functions.

In the following, the meaning of the optional first argument `tbl` is the same as in Section 72.8.

`‣ RestrictedClassFunction` ( [tbl, ]chi, target ) | ( operation ) |

Let `chi` be a class function of a group \(G\) and let `target` be either a subgroup \(H\) of \(G\) or an injective homomorphism from \(H\) to \(G\) or the character table of `H`. Then `RestrictedClassFunction`

returns the class function of \(H\) obtained by restricting `chi` to \(H\).

If `chi` is a class function of a *factor group* \(G\)of \(H\), where `target` is either the group \(H\) or a homomorphism from \(H\) to \(G\) or the character table of \(H\) 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 \(H\) to \(G\) is stored on the character table of \(H\).

`‣ RestrictedClassFunctions` ( [tbl, ]chars, target ) | ( operation ) |

`RestrictedClassFunctions`

is similar to `RestrictedClassFunction`

(72.9-1), the only difference is that it takes a list `chars` of class functions instead of one class function, and returns the list of restricted class functions.

gap> a5:= CharacterTable( "A5" );; s5:= CharacterTable( "S5" );; gap> RestrictedClassFunction( Irr( s5 )[2], a5 ); Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ) gap> RestrictedClassFunctions( Irr( s5 ), a5 ); [ Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ), Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ), Character( CharacterTable( "A5" ), [ 6, -2, 0, 1, 1 ] ), Character( CharacterTable( "A5" ), [ 4, 0, 1, -1, -1 ] ), Character( CharacterTable( "A5" ), [ 4, 0, 1, -1, -1 ] ), Character( CharacterTable( "A5" ), [ 5, 1, -1, 0, 0 ] ), Character( CharacterTable( "A5" ), [ 5, 1, -1, 0, 0 ] ) ] gap> hom:= NaturalHomomorphismByNormalSubgroup( S4, der );; gap> RestrictedClassFunctions( Irr( Image( hom ) ), hom ); [ Character( CharacterTable( S4 ), [ 1, 1, 1, 1, 1 ] ), Character( CharacterTable( S4 ), [ 1, -1, 1, 1, -1 ] ) ]

`‣ InducedClassFunction` ( [tbl, ]chi, H ) | ( operation ) |

`‣ InducedClassFunction` ( [tbl, ]chi, hom ) | ( operation ) |

`‣ InducedClassFunction` ( [tbl, ]chi, suptbl ) | ( operation ) |

Let `chi` be a class function of a group \(G\) and let `target` be either a supergroup \(H\) of \(G\) or an injective homomorphism from \(H\) to \(G\) or the character table of `H`. Then `InducedClassFunction`

returns the class function of \(H\) obtained by inducing `chi` to \(H\).

`‣ InducedClassFunctions` ( [tbl, ]chars, target ) | ( operation ) |

`InducedClassFunctions`

is similar to `InducedClassFunction`

(72.9-3), the only difference is that it takes a list `chars` of class functions instead of one class function, and returns the list of induced class functions.

gap> InducedClassFunctions( Irr( a5 ), s5 ); [ Character( CharacterTable( "A5.2" ), [ 2, 2, 2, 2, 0, 0, 0 ] ), Character( CharacterTable( "A5.2" ), [ 6, -2, 0, 1, 0, 0, 0 ] ), Character( CharacterTable( "A5.2" ), [ 6, -2, 0, 1, 0, 0, 0 ] ), Character( CharacterTable( "A5.2" ), [ 8, 0, 2, -2, 0, 0, 0 ] ), Character( CharacterTable( "A5.2" ), [ 10, 2, -2, 0, 0, 0, 0 ] ) ]

`‣ InducedClassFunctionsByFusionMap` ( subtbl, tbl, chars, fusionmap ) | ( function ) |

Let `subtbl` and `tbl` be two character tables of groups \(H\) and \(G\), such that \(H\) is a subgroup of \(G\), let `chars` be a list of class functions of `subtbl`, and let `fusionmap` be a fusion map from `subtbl` to `tbl`. The function returns the list of induced class functions of `tbl` that correspond to `chars`, w.r.t. the given fusion map.

`InducedClassFunctionsByFusionMap`

is the function that does the work for `InducedClassFunction`

(72.9-3) and `InducedClassFunctions`

(72.9-4).

gap> fus:= PossibleClassFusions( a5, s5 ); [ [ 1, 2, 3, 4, 4 ] ] gap> InducedClassFunctionsByFusionMap( a5, s5, Irr( a5 ), fus[1] ); [ Character( CharacterTable( "A5.2" ), [ 2, 2, 2, 2, 0, 0, 0 ] ), Character( CharacterTable( "A5.2" ), [ 6, -2, 0, 1, 0, 0, 0 ] ), Character( CharacterTable( "A5.2" ), [ 6, -2, 0, 1, 0, 0, 0 ] ), Character( CharacterTable( "A5.2" ), [ 8, 0, 2, -2, 0, 0, 0 ] ), Character( CharacterTable( "A5.2" ), [ 10, 2, -2, 0, 0, 0, 0 ] ) ]

`‣ InducedCyclic` ( tbl[, classes][, "all"] ) | ( operation ) |

`InducedCyclic`

calculates characters induced up from cyclic subgroups of the ordinary character table `tbl` to `tbl`, and returns the strictly sorted list of the induced characters.

If the string `"all"`

is specified then all irreducible characters of these subgroups are induced, otherwise only the permutation characters are calculated.

If a list `classes` is specified then only those cyclic subgroups generated by these classes are considered, otherwise all classes of `tbl` are considered.

gap> InducedCyclic( a5, "all" ); [ Character( CharacterTable( "A5" ), [ 12, 0, 0, 2, 2 ] ), Character( CharacterTable( "A5" ), [ 12, 0, 0, E(5)^2+E(5)^3, E(5)+E(5)^4 ] ), Character( CharacterTable( "A5" ), [ 12, 0, 0, E(5)+E(5)^4, E(5)^2+E(5)^3 ] ), Character( CharacterTable( "A5" ), [ 20, 0, -1, 0, 0 ] ), Character( CharacterTable( "A5" ), [ 20, 0, 2, 0, 0 ] ), Character( CharacterTable( "A5" ), [ 30, -2, 0, 0, 0 ] ), Character( CharacterTable( "A5" ), [ 30, 2, 0, 0, 0 ] ), Character( CharacterTable( "A5" ), [ 60, 0, 0, 0, 0 ] ) ]

The 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 \(1\).

`‣ ReducedClassFunctions` ( [tbl, ][constituents, ]reducibles ) | ( operation ) |

Let `reducibles` be a list of ordinary virtual characters of a group \(G\). If `constituents` is given then it must also be a list of ordinary virtual characters of \(G\), otherwise we have `constituents` equal to `reducibles` in the following.

`ReducedClassFunctions`

returns a record with the components `remainders`

and `irreducibles`

, both lists of virtual characters of \(G\). These virtual characters are computed as follows.

Let `rems`

be the set of nonzero class functions obtained by subtraction of

\[
\sum_{\chi} ( [`reducibles`[i], \chi] / [\chi, \chi] ) \cdot \chi
\]

from \(\textit{reducibles}[i]\), where the summation runs over `constituents` and \([\chi, \psi]\) denotes the scalar product of \(G\)-class functions. Let `irrs`

be the list of irreducible characters in `rems`

.

We project `rems`

into the orthogonal space of `irrs`

and all those irreducibles found this way until no new irreducibles arise. Then the `irreducibles`

list is the set of all found irreducible characters, and the `remainders`

list is the set of all nonzero remainders.

`‣ ReducedCharacters` ( [tbl, ]constituents, reducibles ) | ( operation ) |

`ReducedCharacters`

is similar to `ReducedClassFunctions`

(72.10-1), the only difference is that `constituents` and `reducibles` are assumed to be lists of characters. This means that only those scalar products must be formed where the degree of the character in `constituents` does not exceed the degree of the character in `reducibles`.

gap> tbl:= CharacterTable( "A5" );; gap> chars:= Irr( tbl ){ [ 2 .. 4 ] };; gap> chars:= Set( Tensored( chars, chars ) );; gap> red:= ReducedClassFunctions( chars ); rec( irreducibles := [ Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ), Character( CharacterTable( "A5" ), [ 3, -1, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ] ), Character( CharacterTable( "A5" ), [ 3, -1, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ] ), Character( CharacterTable( "A5" ), [ 4, 0, 1, -1, -1 ] ), Character( CharacterTable( "A5" ), [ 5, 1, -1, 0, 0 ] ) ], remainders := [ ] )

`‣ IrreducibleDifferences` ( tbl, reducibles, reducibles2[, scprmat] ) | ( function ) |

`IrreducibleDifferences`

returns the list of irreducible characters which occur as difference of an element of `reducibles` and an element of `reducibles2`, where these two arguments are lists of class functions of the character table `tbl`.

If `reducibles2` is the string `"triangle"`

then the differences of elements in `reducibles` are considered.

If `scprmat` is not specified then it will be calculated, otherwise we must have

or `scprmat` = MatScalarProducts( `tbl`, `reducibles`, `reducibles2` )

, respectively.`scprmat` = MatScalarProducts( `tbl`, `reducibles` )

gap> IrreducibleDifferences( a5, chars, "triangle" ); [ Character( CharacterTable( "A5" ), [ 3, -1, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ] ), Character( CharacterTable( "A5" ), [ 3, -1, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ] ) ]

`‣ LLL` ( tbl, characters[, y][, "sort"][, "linearcomb"] ) | ( function ) |

`LLL`

calls the LLL algorithm (see `LLLReducedBasis`

(25.5-1)) in the case of lattices spanned by the virtual characters `characters` of the ordinary character table `tbl` (see `ScalarProduct`

(72.8-5)). By finding shorter vectors in the lattice spanned by `characters`, i.e., virtual characters of smaller norm, in some cases `LLL`

is able to find irreducible characters.

`LLL`

returns a record with at least components `irreducibles`

(the list of found irreducible characters), `remainders`

(a list of reducible virtual characters), and `norms`

(the list of norms of the vectors in `remainders`

). `irreducibles`

together with `remainders`

form a basis of the \(ℤ\)-lattice spanned by `characters`.

Note that the vectors in the `remainders`

list are in general *not* orthogonal (see `ReducedClassFunctions`

(72.10-1)) to the irreducible characters in `irreducibles`

.

Optional arguments of `LLL`

are

`y`controls the sensitivity of the algorithm, see

`LLLReducedBasis`

(25.5-1),`"sort"``LLL`

sorts`characters`and the`remainders`

component of the result according to the degrees,`"linearcomb"`the returned record contains components

`irreddecomp`

and`reddecomp`

, which are decomposition matrices of`irreducibles`

and`remainders`

, with respect to`characters`.

gap> s4:= CharacterTable( "Symmetric", 4 );; gap> chars:= [ [ 8, 0, 0, -1, 0 ], [ 6, 0, 2, 0, 2 ], > [ 12, 0, -4, 0, 0 ], [ 6, 0, -2, 0, 0 ], [ 24, 0, 0, 0, 0 ], > [ 12, 0, 4, 0, 0 ], [ 6, 0, 2, 0, -2 ], [ 12, -2, 0, 0, 0 ], > [ 8, 0, 0, 2, 0 ], [ 12, 2, 0, 0, 0 ], [ 1, 1, 1, 1, 1 ] ];; gap> LLL( s4, chars ); rec( irreducibles := [ Character( CharacterTable( "Sym(4)" ), [ 2, 0, 2, -1, 0 ] ), Character( CharacterTable( "Sym(4)" ), [ 1, 1, 1, 1, 1 ] ), Character( CharacterTable( "Sym(4)" ), [ 3, 1, -1, 0, -1 ] ), Character( CharacterTable( "Sym(4)" ), [ 3, -1, -1, 0, 1 ] ), Character( CharacterTable( "Sym(4)" ), [ 1, -1, 1, 1, -1 ] ) ], norms := [ ], remainders := [ ] )

`‣ Extract` ( tbl, reducibles, grammat[, missing] ) | ( function ) |

Let `tbl` be an ordinary character table, `reducibles` a list of characters of `tbl`, and `grammat` the matrix of scalar products of `reducibles` (see `MatScalarProducts`

(72.8-6)). `Extract`

tries to find irreducible characters by drawing conclusions out of the scalar products, using combinatorial and backtrack means.

The optional argument `missing` is the maximal number of irreducible characters that occur as constituents of `reducibles`. Specification of `missing` may accelerate `Extract`

.

`Extract`

returns a record `ext` with the components `solution`

and `choice`

, where the value of `solution`

is a list \([ M_1, \ldots, M_n ]\) of decomposition matrices \(M_i\) (up to permutations of rows) with the property that \(M_i^{tr} \cdot X\) is equal to the sublist at the positions `ext``.choice[i]`

of `reducibles`, for a matrix \(X\) of irreducible characters; the value of `choice`

is a list of length \(n\) whose entries are lists of indices.

So the \(j\)-th column in each matrix \(M_i\) corresponds to \(\textit{reducibles}[j]\), and each row in \(M_i\) corresponds to an irreducible character. `Decreased`

(72.10-7) can be used to examine the solution for computable irreducibles.

gap> s4:= CharacterTable( "Symmetric", 4 );; gap> red:= [ [ 5, 1, 5, 2, 1 ], [ 2, 0, 2, 2, 0 ], [ 3, -1, 3, 0, -1 ], > [ 6, 0, -2, 0, 0 ], [ 4, 0, 0, 1, 2 ] ];; gap> gram:= MatScalarProducts( s4, red, red ); [ [ 6, 3, 2, 0, 2 ], [ 3, 2, 1, 0, 1 ], [ 2, 1, 2, 0, 0 ], [ 0, 0, 0, 2, 1 ], [ 2, 1, 0, 1, 2 ] ] gap> ext:= Extract( s4, red, gram, 5 ); rec( choice := [ [ 2, 5, 3, 4, 1 ] ], solution := [ [ [ 1, 1, 0, 0, 2 ], [ 1, 0, 1, 0, 1 ], [ 0, 1, 0, 1, 0 ], [ 0, 0, 1, 0, 1 ], [ 0, 0, 0, 1, 0 ] ] ] ) gap> dec:= Decreased( s4, red, ext.solution[1], ext.choice[1] ); rec( irreducibles := [ Character( CharacterTable( "Sym(4)" ), [ 1, 1, 1, 1, 1 ] ), Character( CharacterTable( "Sym(4)" ), [ 3, -1, -1, 0, 1 ] ), Character( CharacterTable( "Sym(4)" ), [ 1, -1, 1, 1, -1 ] ), Character( CharacterTable( "Sym(4)" ), [ 3, 1, -1, 0, -1 ] ), Character( CharacterTable( "Sym(4)" ), [ 2, 0, 2, -1, 0 ] ) ], matrix := [ ], remainders := [ ] )

`‣ OrthogonalEmbeddingsSpecialDimension` ( tbl, reducibles, grammat[, "positive"], dim ) | ( function ) |

`OrthogonalEmbeddingsSpecialDimension`

is a variant of `OrthogonalEmbeddings`

(25.6-1) for the situation that `tbl` is an ordinary character table, `reducibles` is a list of virtual characters of `tbl`, `grammat` is the matrix of scalar products (see `MatScalarProducts`

(72.8-6)), and `dim` is an upper bound for the number of irreducible characters of `tbl` that occur as constituents of `reducibles`; if the vectors in `reducibles` are known to be proper characters then the string `"positive"`

may be entered as fourth argument. (See `OrthogonalEmbeddings`

(25.6-1) for information why this may help.)

`OrthogonalEmbeddingsSpecialDimension`

first uses `OrthogonalEmbeddings`

(25.6-1) to compute all orthogonal embeddings of `grammat` into a standard lattice of dimension up to `dim`, and then calls `Decreased`

(72.10-7) in order to find irreducible characters of `tbl`.

`OrthogonalEmbeddingsSpecialDimension`

returns a record with the following components.

`irreducibles`

a list of found irreducibles, the intersection of all lists of irreducibles found by

`Decreased`

(72.10-7), for all possible embeddings, and`remainders`

a list of remaining reducible virtual characters.

gap> s6:= CharacterTable( "S6" );; gap> red:= InducedCyclic( s6, "all" );; gap> Add( red, TrivialCharacter( s6 ) ); gap> lll:= LLL( s6, red );; gap> irred:= lll.irreducibles; [ Character( CharacterTable( "A6.2_1" ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ), Character( CharacterTable( "A6.2_1" ), [ 9, 1, 0, 0, 1, -1, -3, -3, 1, 0, 0 ] ), Character( CharacterTable( "A6.2_1" ), [ 16, 0, -2, -2, 0, 1, 0, 0, 0, 0, 0 ] ) ] gap> Set( Flat( MatScalarProducts( s6, irred, lll.remainders ) ) ); [ 0 ] gap> dim:= NrConjugacyClasses( s6 ) - Length( lll.irreducibles ); 8 gap> rem:= lll.remainders;; Length( rem ); 8 gap> gram:= MatScalarProducts( s6, rem, rem );; RankMat( gram ); 8 gap> emb1:= OrthogonalEmbeddings( gram, 8 ); rec( norms := [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ], solutions := [ [ 1, 2, 3, 7, 11, 12, 13, 15 ], [ 1, 2, 4, 8, 10, 12, 13, 14 ], [ 1, 2, 5, 6, 9, 12, 13, 16 ] ], vectors := [ [ -1, 0, 1, 0, 1, 0, 1, 0 ], [ 1, 0, 0, 1, 0, 1, 0, 0 ], [ 0, 1, 1, 0, 0, 0, 1, 1 ], [ 0, 1, 1, 0, 0, 0, 1, 0 ], [ 0, 1, 1, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 1, 0 ], [ 0, -1, 0, 0, 0, 0, 0, 1 ], [ 0, 1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0, 1, 1 ], [ 0, 0, 1, 0, 0, 0, 0, 1 ], [ 0, 0, 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, -1, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 1, 1 ], [ 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 1 ] ] )

In the following example we temporarily decrease the line length limit from its default value \(80\) to \(62\) in order to get a nicer output format.

gap> emb2:= OrthogonalEmbeddingsSpecialDimension( s6, rem, gram, 8 ); rec( irreducibles := [ Character( CharacterTable( "A6.2_1" ), [ 5, 1, -1, 2, -1, 0, 1, -3, -1, 1, 0 ] ), Character( CharacterTable( "A6.2_1" ), [ 5, 1, 2, -1, -1, 0, -3, 1, -1, 0, 1 ] ), Character( CharacterTable( "A6.2_1" ), [ 10, -2, 1, 1, 0, 0, -2, 2, 0, 1, -1 ] ), Character( CharacterTable( "A6.2_1" ), [ 10, -2, 1, 1, 0, 0, 2, -2, 0, -1, 1 ] ) ], remainders := [ VirtualCharacter( CharacterTable( "A6.2_1" ), [ 0, 0, 3, -3, 0, 0, 4, -4, 0, 1, -1 ] ), VirtualCharacter( CharacterTable( "A6.2_1" ), [ 6, 2, 3, 0, 0, 1, 2, -2, 0, -1, -2 ] ), VirtualCharacter( CharacterTable( "A6.2_1" ), [ 10, 2, 1, 1, 2, 0, 2, 2, -2, -1, -1 ] ), VirtualCharacter( CharacterTable( "A6.2_1" ), [ 14, 2, 2, -1, 0, -1, 6, 2, 0, 0, -1 ] ) ] )

`‣ Decreased` ( tbl, chars, decompmat[, choice] ) | ( function ) |

Let `tbl` be an ordinary character table, `chars` a list of virtual characters of `tbl`, and `decompmat` a decomposition matrix, that is, a matrix \(M\) with the property that \(M^{tr} \cdot X = \textit{chars}\) holds, where \(X\) is a list of irreducible characters of `tbl`. `Decreased`

tries to compute the irreducibles in \(X\) or at least some of them.

Usually `Decreased`

is applied to the output of `Extract`

(72.10-5) or `OrthogonalEmbeddings`

(25.6-1) or `OrthogonalEmbeddingsSpecialDimension`

(72.10-6). In the case of `Extract`

(72.10-5), the choice component corresponding to the decomposition matrix must be entered as argument `choice` of `Decreased`

.

`Decreased`

returns `fail`

if it can prove that no list \(X\) of irreducible characters corresponding to the arguments exists; otherwise `Decreased`

returns a record with the following components.

`irreducibles`

the list of found irreducible characters,

`remainders`

the remaining reducible characters, and

`matrix`

the decomposition matrix of the characters in the

`remainders`

component.

In the following example we temporarily decrease the line length limit from its default value \(80\) to \(62\) in order to get a nicer output format.

gap> s4:= CharacterTable( "Symmetric", 4 );; gap> x:= Irr( s4 );; gap> red:= [ x[1]+x[2], -x[1]-x[3], -x[1]+x[3], -x[2]-x[4] ];; gap> mat:= MatScalarProducts( s4, red, red ); [ [ 2, -1, -1, -1 ], [ -1, 2, 0, 0 ], [ -1, 0, 2, 0 ], [ -1, 0, 0, 2 ] ] gap> emb:= OrthogonalEmbeddings( mat ); rec( norms := [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ], solutions := [ [ 1, 6, 7, 12 ], [ 2, 5, 8, 11 ], [ 3, 4, 9, 10 ] ], vectors := [ [ -1, 1, 1, 0 ], [ -1, 1, 0, 1 ], [ 1, -1, 0, 0 ], [ -1, 0, 1, 1 ], [ -1, 0, 1, 0 ], [ -1, 0, 0, 1 ], [ 0, -1, 1, 0 ], [ 0, -1, 0, 1 ], [ 0, 1, 0, 0 ], [ 0, 0, -1, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ] ) gap> dec:= Decreased( s4, red, emb.vectors{ emb.solutions[1] } ); rec( irreducibles := [ Character( CharacterTable( "Sym(4)" ), [ 3, -1, -1, 0, 1 ] ), Character( CharacterTable( "Sym(4)" ), [ 1, -1, 1, 1, -1 ] ), Character( CharacterTable( "Sym(4)" ), [ 2, 0, 2, -1, 0 ] ), Character( CharacterTable( "Sym(4)" ), [ 3, 1, -1, 0, -1 ] ) ], matrix := [ ], remainders := [ ] ) gap> Decreased( s4, red, emb.vectors{ emb.solutions[2] } ); fail gap> Decreased( s4, red, emb.vectors{ emb.solutions[3] } ); fail

`‣ DnLattice` ( tbl, grammat, reducibles ) | ( function ) |

Let `tbl` be an ordinary character table, and `reducibles` a list of virtual characters of `tbl`.

`DnLattice`

searches for sublattices isomorphic to root lattices of type \(D_n\), for \(n \geq 4\), in the lattice that is generated by `reducibles`; each vector in `reducibles` must have norm \(2\), and the matrix of scalar products (see `MatScalarProducts`

(72.8-6)) of `reducibles` must be entered as argument `grammat`.

`DnLattice`

is able to find irreducible characters if there is a lattice of type \(D_n\) with \(n > 4\). In the case \(n = 4\), `DnLattice`

may fail to determine irreducibles.

`DnLattice`

returns a record with components

`irreducibles`

the list of found irreducible characters,

`remainders`

the list of remaining reducible virtual characters, and

`gram`

the Gram matrix of the vectors in

`remainders`

.

The `remainders`

list is transformed in such a way that the `gram`

matrix is a block diagonal matrix that exhibits the structure of the lattice generated by the vectors in `remainders`

. So `DnLattice`

might be useful even if it fails to find irreducible characters.

In the following example we temporarily decrease the line length limit from its default value \(80\) to \(62\) in order to get a nicer output format.

gap> s4:= CharacterTable( "Symmetric", 4 );; gap> red:= [ [ 2, 0, 2, 2, 0 ], [ 4, 0, 0, 1, 2 ], > [ 5, -1, 1, -1, 1 ], [ -1, 1, 3, -1, -1 ] ];; gap> gram:= MatScalarProducts( s4, red, red ); [ [ 2, 1, 0, 0 ], [ 1, 2, 1, -1 ], [ 0, 1, 2, 0 ], [ 0, -1, 0, 2 ] ] gap> dn:= DnLattice( s4, gram, red ); rec( gram := [ ], irreducibles := [ Character( CharacterTable( "Sym(4)" ), [ 2, 0, 2, -1, 0 ] ), Character( CharacterTable( "Sym(4)" ), [ 1, -1, 1, 1, -1 ] ), Character( CharacterTable( "Sym(4)" ), [ 1, 1, 1, 1, 1 ] ), Character( CharacterTable( "Sym(4)" ), [ 3, -1, -1, 0, 1 ] ) ], remainders := [ ] )

`‣ DnLatticeIterative` ( tbl, reducibles ) | ( function ) |

Let `tbl` be an ordinary character table, and `reducibles` either a list of virtual characters of `tbl` or a record with components `remainders`

and `norms`

, for example a record returned by `LLL`

(72.10-4).

`DnLatticeIterative`

was designed for iterative use of `DnLattice`

(72.10-8). `DnLatticeIterative`

selects the vectors of norm \(2\) among the given virtual character, calls `DnLattice`

(72.10-8) for them, reduces the virtual characters with found irreducibles, calls `DnLattice`

(72.10-8) again for the remaining virtual characters, and so on, until no new irreducibles are found.

`DnLatticeIterative`

returns a record with the same components and meaning of components as `LLL`

(72.10-4).

gap> s4:= CharacterTable( "Symmetric", 4 );; gap> red:= [ [ 2, 0, 2, 2, 0 ], [ 4, 0, 0, 1, 2 ], > [ 5, -1, 1, -1, 1 ], [ -1, 1, 3, -1, -1 ] ];; gap> dn:= DnLatticeIterative( s4, red ); rec( irreducibles := [ Character( CharacterTable( "Sym(4)" ), [ 2, 0, 2, -1, 0 ] ), Character( CharacterTable( "Sym(4)" ), [ 1, -1, 1, 1, -1 ] ), Character( CharacterTable( "Sym(4)" ), [ 1, 1, 1, 1, 1 ] ), Character( CharacterTable( "Sym(4)" ), [ 3, -1, -1, 0, 1 ] ) ], norms := [ ], remainders := [ ] )

`‣ Symmetrizations` ( [tbl, ]characters, n ) | ( operation ) |

`Symmetrizations`

returns the list of symmetrizations of the characters `characters` of the ordinary character table `tbl` with the ordinary irreducible characters of the symmetric group of degree `n`; instead of the integer `n`, the character table of the symmetric group can be entered.

The symmetrization \(\chi^{[\lambda]}\) of the character \(\chi\) of `tbl` with the character \(\lambda\) of the symmetric group \(S_n\) of degree \(n\) is defined by

\[ \chi^{[\lambda]}(g) = \left( \sum_{{\rho \in S_n}} \lambda(\rho) \prod_{{k=1}}^n \chi(g^k)^{{a_k(\rho)}} \right) / n! , \]

where \(a_k(\rho)\) is the number of cycles of length \(k\) in \(\rho\).

For special kinds of symmetrizations, see `SymmetricParts`

(72.11-2), `AntiSymmetricParts`

(72.11-3), `MinusCharacter`

(73.6-5) and `OrthogonalComponents`

(72.11-4), `SymplecticComponents`

(72.11-5).

*Note* that the returned list may contain zero class functions, and duplicates are not deleted.

gap> tbl:= CharacterTable( "A5" );; gap> Symmetrizations( Irr( tbl ){ [ 1 .. 3 ] }, 3 ); [ VirtualCharacter( CharacterTable( "A5" ), [ 0, 0, 0, 0, 0 ] ), VirtualCharacter( CharacterTable( "A5" ), [ 0, 0, 0, 0, 0 ] ), Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ), Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ), Character( CharacterTable( "A5" ), [ 8, 0, -1, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ] ), Character( CharacterTable( "A5" ), [ 10, -2, 1, 0, 0 ] ), Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ), Character( CharacterTable( "A5" ), [ 8, 0, -1, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ] ), Character( CharacterTable( "A5" ), [ 10, -2, 1, 0, 0 ] ) ]

`‣ SymmetricParts` ( tbl, characters, n ) | ( function ) |

is the list of symmetrizations of the characters `characters` of the character table `tbl` with the trivial character of the symmetric group of degree `n` (see `Symmetrizations`

(72.11-1)).

gap> SymmetricParts( tbl, Irr( tbl ), 3 ); [ Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ), Character( CharacterTable( "A5" ), [ 10, -2, 1, 0, 0 ] ), Character( CharacterTable( "A5" ), [ 10, -2, 1, 0, 0 ] ), Character( CharacterTable( "A5" ), [ 20, 0, 2, 0, 0 ] ), Character( CharacterTable( "A5" ), [ 35, 3, 2, 0, 0 ] ) ]

`‣ AntiSymmetricParts` ( tbl, characters, n ) | ( function ) |

is the list of symmetrizations of the characters `characters` of the character table `tbl` with the alternating character of the symmetric group of degree `n` (see `Symmetrizations`

(72.11-1)).

gap> AntiSymmetricParts( tbl, Irr( tbl ), 3 ); [ VirtualCharacter( CharacterTable( "A5" ), [ 0, 0, 0, 0, 0 ] ), Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ), Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ), Character( CharacterTable( "A5" ), [ 4, 0, 1, -1, -1 ] ), Character( CharacterTable( "A5" ), [ 10, -2, 1, 0, 0 ] ) ]

`‣ OrthogonalComponents` ( tbl, chars, m ) | ( function ) |

If \(\chi\) is a nonlinear character with indicator \(+1\), a splitting of the tensor power \(\chi^m\) 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 `m` (see `Symmetrizations`

(72.11-1)).

`OrthogonalComponents`

returns the Murnaghan components of the nonlinear characters of the character table `tbl` in the list `chars` up to the power `m`, where `m` is an integer between 2 and 6.

The Murnaghan functions are implemented as in [Fra82].

*Note*: If `chars` is a list of character objects (see `IsCharacter`

(72.8-1)) then also the result consists of class function objects. It is not checked whether all characters in `chars` do really have indicator \(+1\); if there are characters with indicator \(0\) or \(-1\), the result might contain virtual characters (see also `SymplecticComponents`

(72.11-5)), therefore the entries of the result do in general not know that they are characters.

gap> tbl:= CharacterTable( "A8" );; chi:= Irr( tbl )[2]; Character( CharacterTable( "A8" ), [ 7, -1, 3, 4, 1, -1, 1, 2, 0, -1, 0, 0, -1, -1 ] ) gap> OrthogonalComponents( tbl, [ chi ], 3 ); [ ClassFunction( CharacterTable( "A8" ), [ 21, -3, 1, 6, 0, 1, -1, 1, -2, 0, 0, 0, 1, 1 ] ), ClassFunction( CharacterTable( "A8" ), [ 27, 3, 7, 9, 0, -1, 1, 2, 1, 0, -1, -1, -1, -1 ] ), ClassFunction( CharacterTable( "A8" ), [ 105, 1, 5, 15, -3, 1, -1, 0, -1, 1, 0, 0, 0, 0 ] ), ClassFunction( CharacterTable( "A8" ), [ 35, 3, -5, 5, 2, -1, -1, 0, 1, 0, 0, 0, 0, 0 ] ), ClassFunction( CharacterTable( "A8" ), [ 77, -3, 13, 17, 2, 1, 1, 2, 1, 0, 0, 0, 2, 2 ] ) ]

`‣ SymplecticComponents` ( tbl, chars, m ) | ( function ) |

If \(\chi\) is a (nonlinear) character with indicator \(-1\), a splitting of the tensor power \(\chi^m\) 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 `m` (see `Symmetrizations`

(72.11-1)).

`SymplecticComponents`

returns the symplectic symmetrizations of the nonlinear characters of the character table `tbl` in the list `chars` up to the power `m`, where `m` is an integer between \(2\) and \(5\).

*Note*: If `chars` is a list of character objects (see `IsCharacter`

(72.8-1)) then also the result consists of class function objects. It is not checked whether all characters in `chars` do really have indicator \(-1\); if there are characters with indicator \(0\) or \(+1\), the result might contain virtual characters (see also `OrthogonalComponents`

(72.11-4)), therefore the entries of the result do in general not know that they are characters.

gap> tbl:= CharacterTable( "U3(3)" );; chi:= Irr( tbl )[2]; Character( CharacterTable( "U3(3)" ), [ 6, -2, -3, 0, -2, -2, 2, 1, -1, -1, 0, 0, 1, 1 ] ) gap> SymplecticComponents( tbl, [ chi ], 3 ); [ ClassFunction( CharacterTable( "U3(3)" ), [ 14, -2, 5, -1, 2, 2, 2, 1, 0, 0, 0, 0, -1, -1 ] ), ClassFunction( CharacterTable( "U3(3)" ), [ 21, 5, 3, 0, 1, 1, 1, -1, 0, 0, -1, -1, 1, 1 ] ), ClassFunction( CharacterTable( "U3(3)" ), [ 64, 0, -8, -2, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0 ] ), ClassFunction( CharacterTable( "U3(3)" ), [ 14, 6, -4, 2, -2, -2, 2, 0, 0, 0, 0, 0, -2, -2 ] ), ClassFunction( CharacterTable( "U3(3)" ), [ 56, -8, 2, 2, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0 ] ) ]

`‣ MolienSeries` ( [tbl, ]psi[, chi] ) | ( function ) |

The *Molien series* of the character \(\psi\), relative to the character \(\chi\), is the rational function given by the series \(M_{{\psi,\chi}}(z) = \sum_{{d = 0}}^{\infty} [\chi,\psi^{[d]}] z^d\), where \(\psi^{[d]}\) denotes the symmetrization of \(\psi\) with the trivial character of the symmetric group \(S_d\) (see `SymmetricParts`

(72.11-2)).

`MolienSeries`

returns the Molien series of `psi`, relative to `chi`, where `psi` and `chi` must be characters of the same character table; this table must be entered as `tbl` if `chi` and `psi` are only lists of character values. The default for `chi` is the trivial character of `tbl`.

The return value of `MolienSeries`

stores a value for the attribute `MolienSeriesInfo`

(72.12-2). This admits the computation of coefficients of the series with `ValueMolienSeries`

(72.12-3). 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 \((1-z^r)^k\); the Molien series is also displayed in this form. Note that such a representation is not unique, one can use `MolienSeriesWithGivenDenominator`

(72.12-4) to obtain the series with a prescribed denominator.

For more information about Molien series, see [NPP84].

gap> t:= CharacterTable( AlternatingGroup( 5 ) );; gap> psi:= First( Irr( t ), x -> Degree( x ) = 3 );; gap> mol:= MolienSeries( psi ); ( 1-z^2-z^3+z^6+z^7-z^9 ) / ( (1-z^5)*(1-z^3)*(1-z^2)^2 )

`‣ MolienSeriesInfo` ( ratfun ) | ( attribute ) |

If the rational function `ratfun` was constructed by `MolienSeries`

(72.12-1), a representation as quotient of polynomials is known such that the denominator is a product of terms of the form \((1-z^r)^k\). This information is encoded as value of `MolienSeriesInfo`

. Additionally, there is a special `PrintObj`

(6.3-5) method for Molien series based on this.

`MolienSeriesInfo`

returns a record that describes the rational function `ratfun` as a Molien series. The components of this record are

`numer`

numerator of

`ratfun`(in general a multiple of the numerator one gets by`NumeratorOfRationalFunction`

(66.4-2)),`denom`

denominator of

`ratfun`(in general a multiple of the denominator one gets by`NumeratorOfRationalFunction`

(66.4-2)),`ratfun`

the rational function

`ratfun`itself,`numerstring`

string corresponding to the polynomial

`numer`

, expressed in terms of`z`

,`denomstring`

string corresponding to the polynomial

`denom`

, expressed in terms of`z`

,`denominfo`

a list of the form \([ [ r_1, k_1 ], \ldots, [ r_n, k_n ] ]\) such that

`denom`

is \(\prod_{{i = 1}}^n (1-z^{{r_i}})^{{k_i}}\).`summands`

a list of records, each with the components

`numer`

,`r`

, and`k`

, describing the summand`numer`

\( / (1-z^r)^k\),`pol`

a list of coefficients, describing a final polynomial which is added to those described by

`summands`

,`size`

the order of the underlying matrix group,

`degree`

the degree of the underlying matrix representation.

gap> HasMolienSeriesInfo( mol ); true gap> MolienSeriesInfo( mol ); rec( degree := 3, 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\ +1, denominfo := [ 5, 1, 3, 1, 2, 2 ], denomstring := "(1-z^5)*(1-z^3)*(1-z^2)^2", numer := -x_1^9+x_1^7+x_1^6-x_1^3-x_1^2+1, numerstring := "1-z^2-z^3+z^6+z^7-z^9", pol := [ ], ratfun := ( 1-z^2-z^3+z^6+z^7-z^9 ) / ( (1-z^5)*(1-z^3)*(1-z^2)^2 ), size := 60, summands := [ rec( k := 1, numer := [ -24, -12, -24 ], r := 5 ), rec( k := 1, numer := [ -20 ], r := 3 ), rec( k := 2, numer := [ -45/4, 75/4, -15/4, -15/4 ], r := 2 ), rec( k := 3, numer := [ -1 ], r := 1 ), rec( k := 1, numer := [ -15/4 ], r := 1 ) ] )

`‣ ValueMolienSeries` ( molser, i ) | ( function ) |

is the `i`-th coefficient of the Molien series `series` computed by `MolienSeries`

(72.12-1).

gap> List( [ 0 .. 20 ], i -> ValueMolienSeries( mol, i ) ); [ 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 3, 0, 4, 0, 4, 1, 5, 1, 6, 1, 7 ]

`‣ MolienSeriesWithGivenDenominator` ( molser, list ) | ( function ) |

is a Molien series equal to `molser` as rational function, but viewed as quotient with denominator \(\prod_{{i = 1}}^n (1-z^{{r_i}})\), where \(\textit{list} = [ r_1, r_2, \ldots, r_n ]\). If `molser` cannot be represented this way, `fail`

is returned.

gap> MolienSeriesWithGivenDenominator( mol, [ 2, 6, 10 ] ); ( 1+z^15 ) / ( (1-z^10)*(1-z^6)*(1-z^2) )

For groups \(H\) and \(G\) with \(H \leq G\), the induced character \((1_G)^H\) is called the *permutation character* of the operation of \(G\) on the right cosets of \(H\). If only the character table of \(G\) is available and not the group \(G\) itself, one can try to get information about possible subgroups of \(G\) by inspection of those \(G\)-class functions that might be permutation characters, using that such a class function \(\pi\) must have at least the following properties. (For details, see [Isa76, Theorem 5.18.]),

**(a)**\(\pi\) is a character of \(G\),

**(b)**\(\pi(g)\) is a nonnegative integer for all \(g \in G\),

**(c)**\(\pi(1)\) divides \(|G|\),

**(d)**\(\pi(g^n) \geq \pi(g)\) for \(g \in G\) and integers \(n\),

**(e)**\([\pi, 1_G] = 1\),

**(f)**the multiplicity of any rational irreducible \(G\)-character \(\psi\) as a constituent of \(\pi\) is at most \(\psi(1)/[\psi, \psi]\),

**(g)**\(\pi(g) = 0\) if the order of \(g\) does not divide \(|G|/\pi(1)\),

**(h)**\(\pi(1) |N_G(g)|\) divides \(\pi(g) |G|\) for all \(g \in G\),

**(i)**\(\pi(g) \leq (|G| - \pi(1)) / (|g^G| |Gal_G(g)|)\) for all nonidentity \(g \in G\), where \(|Gal_G(g)|\) denotes the number of conjugacy classes of \(G\) that contain generators of the group \(\langle g \rangle\),

**(j)**if \(p\) is a prime that divides \(|G|/\pi(1)\) only once then \(s/(p-1)\) divides \(|G|/\pi(1)\) and is congruent to \(1\) modulo \(p\), where \(s\) is the number of elements of order \(p\) in the (hypothetical) subgroup \(H\) for which \(\pi = (1_H)^G\) holds. (Note that \(s/(p-1)\) equals the number of Sylow \(p\) subgroups in \(H\).)

Any \(G\)-class function with these properties is called a *possible permutation character* in **GAP**.

(Condition (d) is checked only for those power maps that are stored in the character table of \(G\); clearly (d) holds for all integers if it holds for all prime divisors of the group order \(|G|\).)

**GAP** provides some algorithms to compute possible permutation characters (see `PermChars`

(72.14-1)), and also provides functions to check a few more criteria whether a given character can be a transitive permutation character (see `TestPerm1`

(72.14-2)).

Some information about the subgroup \(U\) can be computed from the permutation character \((1_U)^G\) using `PermCharInfo`

(72.13-1).

`‣ PermCharInfo` ( tbl, permchars[, format] ) | ( function ) |

Let `tbl` be the ordinary character table of the group \(G\), and `permchars` either the permutation character \((1_U)^G\), for a subgroup \(U\) of \(G\), or a list of such permutation characters. `PermCharInfo`

returns a record with the following components.

`contained`

:a list containing, for each character \(\psi = (1_U)^G\) in

`permchars`, a list containing at position \(i\) the number \(\psi[i] |U| /\)`SizesCentralizers(`

`tbl``)`

\([i]\), which equals the number of those elements of \(U\) that are contained in class \(i\) of`tbl`,`bound`

:a list containing, for each character \(\psi = (1_U)^G\) in

`permchars`, a list containing at position \(i\) the number \(|U| / \gcd( |U|,\)`SizesCentralizers(`

\([i] )\), which divides the class length in \(U\) of an element in class \(i\) of`tbl`)`tbl`,`display`

:a record that can be used as second argument of

`Display`

(6.3-6) to display each permutation character in`permchars`and the corresponding components`contained`

and`bound`

, for those classes where at least one character of`permchars`is nonzero,`ATLAS`

:a list of strings describing the decomposition of the permutation characters in

`permchars`into the irreducible characters of`tbl`, given in an**Atlas**-like notation. This means that the irreducible constituents are indicated by their degrees followed by lower case letters`a`

,`b`

,`c`

, \(\ldots\), which indicate the successive irreducible characters of`tbl`of that degree, in the order in which they appear in`Irr(`

`tbl``)`

. 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`n`for the letter`lett`means that`lett`is repeated`n`times. The default notation for exponentiation is

, this is also chosen if the optional third argument`lett`^{`n`}`format`is the string`"LaTeX"`

; if the third argument is the string`"HTML"`

then exponentiation is denoted by

.`lett`<sup>`n`</sup>

gap> t:= CharacterTable( "A6" );; gap> psi:= Sum( Irr( t ){ [ 1, 3, 6 ] } ); Character( CharacterTable( "A6" ), [ 15, 3, 0, 3, 1, 0, 0 ] ) gap> info:= PermCharInfo( t, psi ); rec( ATLAS := [ "1a+5b+9a" ], bound := [ [ 1, 3, 8, 8, 6, 24, 24 ] ], contained := [ [ 1, 9, 0, 8, 6, 0, 0 ] ], display := rec( chars := [ [ 15, 3, 0, 3, 1, 0, 0 ], [ 1, 9, 0, 8, 6, 0, 0 ], [ 1, 3, 8, 8, 6, 24, 24 ] ], classes := [ 1, 2, 4, 5 ], letter := "I" ) ) gap> Display( t, info.display ); A6 2 3 3 . 2 3 2 . 2 . 5 1 . . . 1a 2a 3b 4a 2P 1a 1a 3b 2a 3P 1a 2a 1a 4a 5P 1a 2a 3b 4a I.1 15 3 3 1 I.2 1 9 8 6 I.3 1 3 8 6 gap> j1:= CharacterTable( "J1" );; gap> psi:= TrivialCharacter( CharacterTable( "7:6" ) )^j1; Character( CharacterTable( "J1" ), [ 4180, 20, 10, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0 ] ) gap> PermCharInfo( j1, psi ).ATLAS; [ "1a+56aabb+76aaab+77aabbcc+120aaabbbccc+133a^{4}bbcc+209a^{5}" ]

`‣ PermCharInfoRelative` ( tbl, tbl2, permchars ) | ( function ) |

Let `tbl` and `tbl2` be the ordinary character tables of two groups \(H\) and \(G\), respectively, where \(H\) is of index two in \(G\), and `permchars` either the permutation character \((1_U)^G\), for a subgroup \(U\) of \(G\), or a list of such permutation characters. `PermCharInfoRelative`

returns a record with the same components as `PermCharInfo`

(72.13-1), the only exception is that the entries of the `ATLAS`

component are names relative to `tbl`.

More precisely, the \(i\)-th entry of the `ATLAS`

component is a string describing the decomposition of the \(i\)-th entry in `permchars`. The degrees and distinguishing letters of the constituents refer to the irreducibles of `tbl`, as follows. The two irreducible characters of `tbl2` of degree \(N\) that extend the irreducible character \(N\) `a`

of `tbl` are denoted by \(N\) `a`

\(^+\) and \(N \)`a`

\(^-\). The irreducible character of `tbl2` of degree \(2N\) whose restriction to `tbl` is the sum of the irreducible characters \(N\) `a`

and \(N\) `b`

is denoted as \(N\) `ab`

. Multiplicities larger than \(1\) of constituents are denoted by exponents.

(This format is useful mainly for multiplicity free permutation characters.)

gap> t:= CharacterTable( "A5" );; gap> t2:= CharacterTable( "A5.2" );; gap> List( Irr( t2 ), x -> x[1] ); [ 1, 1, 6, 4, 4, 5, 5 ] gap> List( Irr( t ), x -> x[1] ); [ 1, 3, 3, 4, 5 ] gap> permchars:= List( [ [1], [1,2], [1,7], [1,3,4,4,6,6,7] ], > l -> Sum( Irr( t2 ){ l } ) ); [ Character( CharacterTable( "A5.2" ), [ 1, 1, 1, 1, 1, 1, 1 ] ), Character( CharacterTable( "A5.2" ), [ 2, 2, 2, 2, 0, 0, 0 ] ), Character( CharacterTable( "A5.2" ), [ 6, 2, 0, 1, 0, 2, 0 ] ), Character( CharacterTable( "A5.2" ), [ 30, 2, 0, 0, 6, 0, 0 ] ) ] gap> info:= PermCharInfoRelative( t, t2, permchars );; gap> info.ATLAS; [ "1a^+", "1a^{\\pm}", "1a^++5a^-", "1a^++3ab+4(a^+)^{2}+5a^+a^{\\pm}" ]

`‣ PermChars` ( tbl[, cond] ) | ( function ) |

**GAP** 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 `cond`. The user is encouraged to try different approaches, especially if one choice fails to come to an end.

Regardless of the algorithm used in a specific case, `PermChars`

returns a list of *all* possible permutation characters with the properties described by `cond`. 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).

Called with only one argument, a character table `tbl`, `PermChars`

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 \(\pi(g) \geq 0\) are transformed into a system of inequalities that guides the search (see `Inequalities`

(72.14-5)). So the following commands compute the list of 39 possible permutation characters of the Mathieu group \(M_{11}\).

gap> m11:= CharacterTable( "M11" );; gap> SetName( m11, "m11" ); gap> perms:= PermChars( m11 );; gap> Length( perms ); 39

There 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 `PermChars`

, and set its component `degree`

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 `ineq`

can take the inequalities computed by `Inequalities`

(72.14-5) if they are needed more than once.

If a positive integer is given as the second argument `cond`, `PermChars`

returns the list of all possible permutation characters of `tbl` that have degree `cond`. 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 `PermBounds`

(72.14-3)). 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.

gap> deg220:= PermChars( m11, 220 ); [ Character( m11, [ 220, 4, 4, 0, 0, 4, 0, 0, 0, 0 ] ), Character( m11, [ 220, 12, 4, 4, 0, 0, 0, 0, 0, 0 ] ), Character( m11, [ 220, 20, 4, 0, 0, 2, 0, 0, 0, 0 ] ) ]

If a record is given as the second argument `cond`, `PermChars`

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 `cond` contains a degree as value of the record component `degree`

then `PermChars`

will behave exactly as if this degree was entered as `cond`.

gap> deg220 = PermChars( m11, rec( degree:= 220 ) ); true

For the meaning of additional components of `cond` besides `degree`

, see `PermComb`

(72.14-4).

Instead of `degree`

, `cond` may have the component `torso`

bound to a list that contains some known values of the required characters at the right positions; at least the degree `cond``.torso[1]`

must be an integer. In this case, the algorithm described in [BP98, Section 3.3] is chosen. The component `chars`

, if present, holds a list of all those *rational* irreducible characters of `tbl` that might be constituents of the required characters.

(*Note*: If `cond``.chars`

is bound and does not contain *all* rational irreducible characters of `tbl`, **GAP** checks whether the scalar products of all class functions in the result list with the omitted rational irreducible characters of `tbl` 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.)

gap> PermChars( m11, rec( torso:= [ 220 ] ) ); [ Character( m11, [ 220, 4, 4, 0, 0, 4, 0, 0, 0, 0 ] ), Character( m11, [ 220, 20, 4, 0, 0, 2, 0, 0, 0, 0 ] ), Character( m11, [ 220, 12, 4, 4, 0, 0, 0, 0, 0, 0 ] ) ] gap> PermChars( m11, rec( torso:= [ 220,,,,, 2 ] ) ); [ Character( m11, [ 220, 20, 4, 0, 0, 2, 0, 0, 0, 0 ] ) ]

An additional restriction on the possible permutation characters computed can be forced if `con` contains, in addition to `torso`

, the components `normalsubgroup`

and `nonfaithful`

, with values a list of class positions of a normal subgroup \(N\) of the group \(G\) of `tbl` and a possible permutation character \(\pi\) of \(G\), respectively, such that \(N\) is contained in the kernel of \(\pi\). In this case, `PermChars`

returns the list of those possible permutation characters \(\psi\) of `tbl` coinciding with `torso`

wherever its values are bound and having the property that no irreducible constituent of \(\psi - \pi\) has \(N\) in its kernel. If the component `chars`

is bound in `cond` then the above statements apply. An interpretation of the computed characters is the following. Suppose there exists a subgroup \(V\) of \(G\) such that \(\pi = (1_V)^G\); Then \(N \leq V\), and if a computed character is of the form \((1_U)^G\), for a subgroup \(U\) of \(G\), then \(V = UN\).

gap> s4:= CharacterTable( "Symmetric", 4 );; gap> nsg:= ClassPositionsOfDerivedSubgroup( s4 );; gap> pi:= TrivialCharacter( s4 );; gap> PermChars( s4, rec( torso:= [ 12 ], normalsubgroup:= nsg, > nonfaithful:= pi ) ); [ Character( CharacterTable( "Sym(4)" ), [ 12, 2, 0, 0, 0 ] ) ] gap> pi:= Sum( Filtered( Irr( s4 ), > chi -> IsSubset( ClassPositionsOfKernel( chi ), nsg ) ) ); Character( CharacterTable( "Sym(4)" ), [ 2, 0, 2, 2, 0 ] ) gap> PermChars( s4, rec( torso:= [ 12 ], normalsubgroup:= nsg, > nonfaithful:= pi ) ); [ Character( CharacterTable( "Sym(4)" ), [ 12, 0, 4, 0, 0 ] ) ]

The class functions returned by `PermChars`

have the properties tested by `TestPerm1`

(72.14-2), `TestPerm2`

(72.14-2), and `TestPerm3`

(72.14-2). So they are possible permutation characters. See `TestPerm1`

(72.14-2) for criteria whether a possible permutation character can in fact be a permutation character.

`‣ TestPerm1` ( tbl, char ) | ( function ) |

`‣ TestPerm2` ( tbl, char ) | ( function ) |

`‣ TestPerm3` ( tbl, chars ) | ( function ) |

`‣ TestPerm4` ( tbl, chars ) | ( function ) |

`‣ TestPerm5` ( tbl, chars, modtbl ) | ( function ) |

The first three of these functions implement tests of the properties of possible permutation characters listed in Section 72.13, The other two implement test of additional properties. Let `tbl` be the ordinary character table of a group \(G\), `char` a rational character of `tbl`, and `chars` a list of rational characters of `tbl`. For applying `TestPerm5`

, the knowledge of a \(p\)-modular Brauer table `modtbl` of \(G\) is required. `TestPerm4`

and `TestPerm5`

expect the characters in `chars` to satisfy the conditions checked by `TestPerm1`

and `TestPerm2`

(see below).

The return values of the functions were chosen parallel to the tests listed in [NPP84].

`TestPerm1`

return `1`

or `2`

if `char` 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 `tbl`. If `char` satisfies the conditions, `0`

is returned.

`TestPerm2`

returns `1`

if `char` fails because of the criterion (c), it returns `3`

, `4`

, or `5`

if `char` fails because of (T3), (T4), or (T5), respectively; these tests correspond to (g), a weaker form of (h), and (j). If `char` satisfies the conditions, `0`

is returned.

`TestPerm3`

returns the list of all those class functions in the list `chars` that satisfy criterion (h); this is a stronger version of (T6).

`TestPerm4`

returns the list of all those class functions in the list `chars` that satisfy (T8) and (T9) for each prime divisor \(p\) of the order of \(G\); these tests use modular representation theory but do not require the knowledge of decomposition matrices (cf. `TestPerm5`

below).

(T8) implements the test of the fact that in the case that \(p\) divides \(|G|\) and the degree of a transitive permutation character \(\pi\) exactly once, the projective cover of the trivial character is a summand of \(\pi\). (This test is omitted if the projective cover cannot be identified.)

Given a permutation character \(\pi\) of a group \(G\) and a prime integer \(p\), the restriction \(\pi_B\) to a \(p\)-block \(B\) of \(G\) has the following property, which is checked by (T9). For each \(g \in G\) such that \(g^n\) is a \(p\)-element of \(G\), \(\pi_B(g^n)\) is a nonnegative integer that satisfies \(|\pi_B(g)| \leq \pi_B(g^n) \leq \pi(g^n)\). (This is [Sco73, Corollary A on p. 113].)

`TestPerm5`

requires the \(p\)-modular Brauer table `modtbl` of \(G\), for some prime \(p\) dividing the order of \(G\), and checks whether those characters in the list `chars` whose degree is divisible by the \(p\)-part of the order of \(G\) can be decomposed into projective indecomposable characters; `TestPerm5`

returns the sublist of all those characters in `chars` that either satisfy this condition or to which the test does not apply.

gap> tbl:= CharacterTable( "A5" );; gap> rat:= RationalizedMat( Irr( tbl ) ); [ Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ), Character( CharacterTable( "A5" ), [ 6, -2, 0, 1, 1 ] ), Character( CharacterTable( "A5" ), [ 4, 0, 1, -1, -1 ] ), Character( CharacterTable( "A5" ), [ 5, 1, -1, 0, 0 ] ) ] gap> tup:= Filtered( Tuples( [ 0, 1 ], 4 ), x -> not IsZero( x ) ); [ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 1 ], [ 0, 1, 0, 0 ], [ 0, 1, 0, 1 ], [ 0, 1, 1, 0 ], [ 0, 1, 1, 1 ], [ 1, 0, 0, 0 ], [ 1, 0, 0, 1 ], [ 1, 0, 1, 0 ], [ 1, 0, 1, 1 ], [ 1, 1, 0, 0 ], [ 1, 1, 0, 1 ], [ 1, 1, 1, 0 ], [ 1, 1, 1, 1 ] ] gap> lincomb:= List( tup, coeff -> coeff * rat );; gap> List( lincomb, psi -> TestPerm1( tbl, psi ) ); [ 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0 ] gap> List( lincomb, psi -> TestPerm2( tbl, psi ) ); [ 0, 5, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1 ] gap> Set( TestPerm3(tbl, lincomb), x -> Position(lincomb, x) ); [ 1, 4, 6, 7, 8, 9, 10, 11, 13 ] gap> tbl:= CharacterTable( "A7" ); CharacterTable( "A7" ) gap> perms:= PermChars( tbl, rec( degree:= 315 ) ); [ Character( CharacterTable( "A7" ), [ 315, 3, 0, 0, 3, 0, 0, 0, 0 ] ) , Character( CharacterTable( "A7" ), [ 315, 15, 0, 0, 1, 0, 0, 0, 0 ] ) ] gap> TestPerm4( tbl, perms ); [ Character( CharacterTable( "A7" ), [ 315, 15, 0, 0, 1, 0, 0, 0, 0 ] ) ] gap> perms:= PermChars( tbl, rec( degree:= 15 ) ); [ Character( CharacterTable( "A7" ), [ 15, 3, 0, 3, 1, 0, 0, 1, 1 ] ), Character( CharacterTable( "A7" ), [ 15, 3, 3, 0, 1, 0, 3, 1, 1 ] ) ] gap> TestPerm5( tbl, perms, tbl mod 5 ); [ Character( CharacterTable( "A7" ), [ 15, 3, 0, 3, 1, 0, 0, 1, 1 ] ) ]

`‣ PermBounds` ( tbl, d ) | ( function ) |

Let `tbl` be the ordinary character table of the group \(G\). All \(G\)-characters \(\pi\) satisfying \(\pi(g) > 0\) and \(\pi(1) = \textit{d}\), for a given degree `d`, lie in a simplex described by these conditions. `PermBounds`

computes the boundary points of this simplex for \(d = 0\), from which the boundary points for any other `d` are easily derived. (Some conditions from the power maps of `tbl` are also involved.) For this purpose, a matrix similar to the rational character table of \(G\) has to be inverted. These boundary points are used by `PermChars`

(72.14-1) to construct all possible permutation characters (see 72.13) of a given degree. `PermChars`

(72.14-1) either calls `PermBounds`

or takes this information from the `bounds`

component of its argument record.

`‣ PermComb` ( tbl, arec ) | ( function ) |

`PermComb`

computes possible permutation characters of the character table `tbl` by the improved combinatorial approach described at the end of [BP98, Section 3.2].

For computing the possible linear combinations *without* prescribing better bounds (i.e., when the computation of bounds shall be suppressed), enter

,`arec`:= rec( degree := `degree`, bounds := false )

where `degree` is the character degree; this is useful if the multiplicities are expected to be small, and if this is forced by high irreducible degrees.

A list of upper bounds on the multiplicities of the rational irreducibles characters can be explicitly prescribed as a `maxmult`

component in `arec`.

`‣ Inequalities` ( tbl, chars[, option] ) | ( operation ) |

Let `tbl` be the ordinary character table of a group \(G\). The condition \(\pi(g) \geq 0\) for every possible permutation character \(\pi\) of \(G\) places restrictions on the multiplicities \(a_i\) of the irreducible constituents \(\chi_i\) of \(\pi = \sum_{{i = 1}}^r a_i \chi_i\). For every element \(g \in G\), we have \(\sum_{{i = 1}}^r a_i \chi_i(g) \geq 0\). The power maps provide even stronger conditions.

This system of inequalities is kind of diagonalized, resulting in a system of inequalities restricting \(a_i\) in terms of \(a_j\), \(j < i\). These inequalities are used to construct characters with nonnegative values (see `PermChars`

(72.14-1)). `PermChars`

(72.14-1) either calls `Inequalities`

or takes this information from the `ineq`

component of its argument record.

The number of inequalities arising in the process of diagonalization may grow very strongly.

There are two ways to organize the projection. The first, which is chosen if no `option` 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 `"small"`

is entered as third argument `option`, tries to keep the number of intermediate inequalities small by eventually changing the order of characters.

gap> tbl:= CharacterTable( "M11" );; gap> PermComb( tbl, rec( degree:= 110 ) ); [ Character( CharacterTable( "M11" ), [ 110, 6, 2, 2, 0, 0, 2, 2, 0, 0 ] ), Character( CharacterTable( "M11" ), [ 110, 6, 2, 6, 0, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( "M11" ), [ 110, 14, 2, 2, 0, 2, 0, 0, 0, 0 ] ) ] gap> # Now compute only multiplicity free permutation characters. gap> bounds:= List( RationalizedMat( Irr( tbl ) ), x -> 1 );; gap> PermComb( tbl, rec( degree:= 110, maxmult:= bounds ) ); [ Character( CharacterTable( "M11" ), [ 110, 6, 2, 2, 0, 0, 2, 2, 0, 0 ] ) ]

`‣ FrobeniusCharacterValue` ( value, p ) | ( function ) |

Let `value` be a cyclotomic whose coefficients over the rationals are in the ring \(ℤ_{\textit{p}}\) of `p`-local numbers, where `p` is a prime integer. Assume that `value` lies in \(ℤ_{\textit{p}}[\zeta]\) for \(\zeta = \exp(\textit{p}^n-1)\), for some positive integer \(n\).

`FrobeniusCharacterValue`

returns the image of `value` under the ring homomorphism from \(ℤ_{\textit{p}}[\zeta]\) to the field with \(\textit{p}^n\) elements that is defined with the help of Conway polynomials (see `ConwayPolynomial`

(59.5-1)), more information can be found in [JLPW95, Sections 2-5].

If `value` is a Brauer character value in characteristic `p` 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.

If the result of `FrobeniusCharacterValue`

cannot be expressed as an element of a finite field in **GAP** (see Chapter 59) then `FrobeniusCharacterValue`

returns `fail`

.

If the Conway polynomial of degree \(n\) is required for the computation then it is computed only if `IsCheapConwayPolynomial`

(59.5-2) returns `true`

when it is called with `p` and \(n\), otherwise `fail`

is returned.

`‣ BrauerCharacterValue` ( mat ) | ( attribute ) |

For an invertible matrix `mat` over a finite field \(F\), `BrauerCharacterValue`

returns the Brauer character value of `mat` if the order of `mat` is coprime to the characteristic of \(F\), and `fail`

otherwise.

The *Brauer character value* of a matrix is the sum of complex lifts of its eigenvalues.

gap> g:= SL(2,4);; # 2-dim. irreducible representation of A5 gap> ccl:= ConjugacyClasses( g );; gap> rep:= List( ccl, Representative );; gap> List( rep, Order ); [ 1, 2, 5, 5, 3 ] gap> phi:= List( rep, BrauerCharacterValue ); [ 2, fail, E(5)^2+E(5)^3, E(5)+E(5)^4, -1 ] gap> List( phi{ [ 1, 3, 4, 5 ] }, x -> FrobeniusCharacterValue( x, 2 ) ); [ 0*Z(2), Z(2^2), Z(2^2)^2, Z(2)^0 ] gap> List( rep{ [ 1, 3, 4, 5 ] }, TraceMat ); [ 0*Z(2), Z(2^2), Z(2^2)^2, Z(2)^0 ]

`‣ SizeOfFieldOfDefinition` ( val, p ) | ( function ) |

For a cyclotomic or a list of cyclotomics `val`, and a prime integer `p`, `SizeOfFieldOfDefinition`

returns the size of the smallest finite field in characteristic `p` that contains the `p`-modular reduction of `val` if this can be determined, and `fail`

otherwise.

The latter happens if `val` is not closed under Galois conjugacy and if the `p`-modular reduction of some value cannot be determined via the function `FrobeniusCharacterValue`

(72.15-1). Note that the reduction map is defined as in [JLPW95], that is, the complex \((\textit{p}^d-1)\)-th root of unity \(\exp(\textit{p}^d-1)\) is mapped to the residue class of the indeterminate, modulo the ideal spanned by the Conway polynomial (see `ConwayPolynomial`

(59.5-1)) of degree \(d\) over the field with \(p\) elements.

If `val` is closed under Galois conjugacy then the result can be determined without explicitly computing the `p`-modular reduction of `val`. This happens for example if `val` is a Brauer character.

If `val` is an irreducible Brauer character then the value returned is the size of the smallest finite field in characteristic `p` over which the corresponding representation lives.

`‣ RealizableBrauerCharacters` ( matrix, q ) | ( function ) |

For a list `matrix` of absolutely irreducible Brauer characters in characteristic \(p\), and a power `q` of \(p\), `RealizableBrauerCharacters`

returns a duplicate-free list of sums of Frobenius conjugates of the rows of `matrix`, each irreducible over the field with `q` elements.

gap> irr:= Irr( CharacterTable( "A5" ) mod 2 ); [ Character( BrauerTable( "A5", 2 ), [ 1, 1, 1, 1 ] ), Character( BrauerTable( "A5", 2 ), [ 2, -1, E(5)+E(5)^4, E(5)^2+E(5)^3 ] ), Character( BrauerTable( "A5", 2 ), [ 2, -1, E(5)^2+E(5)^3, E(5)+E(5)^4 ] ), Character( BrauerTable( "A5", 2 ), [ 4, 1, -1, -1 ] ) ] gap> List( irr, phi -> SizeOfFieldOfDefinition( phi, 2 ) ); [ 2, 4, 4, 2 ] gap> RealizableBrauerCharacters( irr, 2 ); [ Character( BrauerTable( "A5", 2 ), [ 1, 1, 1, 1 ] ), ClassFunction( BrauerTable( "A5", 2 ), [ 4, -2, -1, -1 ] ), Character( BrauerTable( "A5", 2 ), [ 4, 1, -1, -1 ] ) ]

**GAP** supports groups, vector spaces, and algebras generated by class functions.

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