GroupTheory - Maple Programming Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Group Theory : GroupTheory/CharacterTable

GroupTheory

  

CharacterTable

  

construct the character table of a finite group

 

Calling Sequence

Parameters

Description

Methods

Examples

Compatibility

Calling Sequence

CharacterTable( G )

Parameters

G

-

finite group

Description

• 

The CharacterTable( G ) command computes the (ordinary) character table of the finite group G, which must be either a permutation group or a Cayley table group.

• 

The CharacterTable command uses an implementation of the Burnside-Dixon-Schneider algorithm to compute the character table.

• 

Character tables are printed using a Matrix so their display is sensitive to the value of the rtablesize interface option.  (See the Examples section.)

Methods

• 

Character tables support the following methods, in which ct denotes a character table returned by the CharacterTable command.

ClassNumber( ct )

returns the class number of the group for which this is a character table

ClassNames( ct )

returns the (ordered) list of conjugacy class names

ClassRepresentatives( ct )

returns the (ordered) list of conjugacy class representatives

CharacterDegrees( ct )

returns a sorted list of pairs [d,m] of the degrees d of the irreducible characters and their respective multiplicities m

Exponent( ct )

returns the exponent of the group

GetMatrix( ct )

returns the matrix of character values

PowerMap( ct, p )

returns the pth power map

Display( ct, opts )

prints the character table in a more readable format

• 

The ClassNumber( ct ) command returns the class number of the underlying group of the character table ct, which is the number of conjugacy classes of the group. In the case when the group is Abelian, this is simply the order of the group but, for non-Abelian groups, the number of conjugacy classes is less than the order of the group.

• 

The ClassNames( ct ) command returns an ordered list of the names given to the conjugacy classes.  These have the form of an integer, indicating the order of the class representative (and hence, of all of its members) followed by one or more letters used to distinguish classes whose elements have the same order.  For example, the class names "4a" and "4b" distinguish two distinct conjugacy classes of elements of order 4.

• 

The ClassRepresentatives( ct ) command returns an Array containing the (ordered) class representatives for the conjugacy classes of the underlying group of the character table ct.

• 

The CharacterDegrees( ct ) command returns a sorted list of pairs [d, m] of the degrees d of the irreducible characters and their respective multiplicities m.

• 

The Exponent( ct ) command returns the exponent of the underlying group, using data stored in the character table.

• 

The GetMatrix( ct ) command returns the character values of the character table ct in a Matrix.

• 

For a prime number p, the PowerMap( ct, p ) command returns the pth power map of the character table ct.  This is an Array a with k elements, where k is the class number of the underlying group of ct, such that the class of the pth power of an element of the ith class is in the class a[ i ], for i = 1 .. k.

• 

The Display( ct, opts ) command prints the character table in a nicer format, and optionally allows you to include additional information, such as the Frobenius-Schur indicator values of the characters (using the showindicator option) and power maps (using the showpowermaps option) for the prime divisors of the group order.

Examples

withGroupTheory:

GSymm3

GroupTheory:-SymmetricGroup3

(1)

ctCharacterTableG

GroupTheory:-CharacterTablemodule...end module

(2)

Displayct

C

1a

2a

3a

|C|

1

3

2

 

 

 

 

χ__1

1

1

1

χ__2

1

−1

1

χ__3

2

0

−1

ClassNumberct

3

(3)

ClassNamesct

1a,2a,3a

(4)

ctCharacterTableCyclicGroup4

GroupTheory:-CharacterTablemodule...end module

(5)

Displayct

C

1a

2a

4a

4b

|C|

1

1

1

1

 

 

 

 

 

χ__1

1

1

1

1

χ__2

1

−1

I

−I

χ__3

1

−1

−I

I

χ__4

1

1

−1

−1

DisplayCharacterTableAlt4,showindicator

 

C

1a

2a

3a

3b

 

|C|

1

3

4

4

ν2

 

 

 

 

 

1

χ__1

1

1

1

1

0

χ__2

1

1

12I32

12+I32

0

χ__3

1

1

12+I32

12I32

1

χ__4

3

−1

0

0

DisplayCharacterTableAlt4,showpowermaps

C

1a

2a

3a

3b

|C|

1

3

4

4

C2

1a

1a

3b

3a

C3

1a

2a

1a

1a

 

 

 

 

 

χ__1

1

1

1

1

χ__2

1

1

12I32

12+I32

χ__3

1

1

12+I32

12I32

χ__4

3

−1

0

0

DisplayCharacterTablePSL2,7

C

1a

2a

3a

4a

7a

7b

|C|

1

21

56

42

24

24

 

 

 

 

 

 

 

χ__1

1

1

1

1

1

1

χ__2

3

−1

0

1

−147−127+−1171

−127+−147−117

χ__3

3

−1

0

1

−127+−147−117

−147−127+−1171

χ__4

6

2

0

0

−1

−1

χ__5

7

−1

1

−1

0

0

χ__6

8

0

−1

0

1

1

ctCharacterTableDihedralGroup12:

The following command indicates that the dihedral group of degree 12 has four linear characters (of degree 1) and 5 irreducible characters of degree 2.

CharacterDegreesct

1,4,2,5

(6)

pmapPowerMapct,2

pmap1,1,1,1,5,2,5,7,7

(7)

ccConjugacyClassesct:

gRepresentativecc7

Perm1,3,5,7,9,11,2,4,6,8,10,12

(8)

evalbg2incc5

true

(9)

It may be necessary to set the value of the rtablesize interface variable to see the character values for larger character tables.

ctCharacterTableElementaryGroup2,5

GroupTheory:-CharacterTablemodule...end module

(10)

interfacertablesize=35:

ct

GroupTheory:-CharacterTablemodule...end module

(11)

Compatibility

• 

The GroupTheory[CharacterTable] command was introduced in Maple 2017.

• 

For more information on Maple 2017 changes, see Updates in Maple 2017.

• 

The GroupTheory[CharacterTable] command was updated in Maple 2020.

See Also

GroupTheory

GroupTheory[AlternatingGroup]

GroupTheory[ConjugacyClasses]

GroupTheory[CyclicGroup]

GroupTheory[Exponent]

GroupTheory[PSL]

GroupTheory[SymmetricGroup]

interface