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GroupTheory

  

Character

  

construct a finite group character from a character table

 

Calling Sequence

Parameters

Description

Methods

Examples

Compatibility

Calling Sequence

Character( ct, k )

Parameters

ct

-

character table

k

-

positive integer

Description

• 

The Character( ct, k ) command creates the kth character of the character table ct, which may be constructed by using the CharacterTable command from a finite group.

Methods

• 

Characters are implemented as Maple objects and support several object methods, outlined below.

Indicator( chi, k )

returns the kth higher indicator of the character chi

Indicator( chi )

returns the Frobenius-Schur indicator of the character chi

Kernel( chi )

returns the kernel of the character chi

chi1 . chi2

returns the inner product of characters chi1 and chi2

Examples

withGroupTheory:

GAlt4

GroupTheory:-AlternatingGroup4

(1)

ctCharacterTableG

GroupTheory:-CharacterTablemodule...end module

(2)

Retrieve the fourth character (i.e., the last row) from the character table.

Characterct,4

GroupTheory:-Charactermodule...end module,4

(3)

The next statement assigns all the characters from the table.

c1,c2,c3,c4opmap2Character,ct,seq1..4

c1,c2,c3,c4character: 1a1,2a1,3a1,3b1 for A4,character: 1a1,2a1,3a12I32,3b12+I32 for A4,character: 1a1,2a1,3a12+I32,3b12I32 for A4,character: 1a3,2a−1,3a0,3b0 for A4

(4)

c4·c4

1

(5)

c3·c4

0

(6)

Since the irreducible characters form an orthonormal basis, the following produces an identity matrix.

Matrix4,4,i,jci·cj

1000010000100001

(7)

KKernelc3

GroupTheory:-PermutationGroupmodule...end module,module...end module,module...end module,degree=4,supergroup=GroupTheory:-PermutationGroupmodule...end module,module...end module,degree=4

(8)

IsNormalK,G

true

(9)

GetValuesc3

1,1,12+I32,12I32

(10)

Notice that, although the non-Abelian groups of order 8 have identical character tables, they are distinguished by the Frobenius-Schur indicator.

ctQCharacterTableQuaternionGroup:

c1Q,c2Q,c3Q,c4Q,c5Qopmap2Character,ctQ,seq1..5:

ctD4CharacterTableDihedralGroup4:

c1D4,c2D4,c3D4,c4D4,c5D4opmap2Character,ctD4,seq1..5:

mapIndicator,c1Q,c2Q,c3Q,c4Q,c5Q

1,1,1,1,−1

(11)

mapIndicator,c1D4,c2D4,c3D4,c4D4,c5D4

1,1,1,1,1

(12)

Compatibility

• 

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

• 

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

See Also

GroupTheory

GroupTheory[AlternatingGroup]

GroupTheory[CharacterTable]

GroupTheory[DihedralGroup]

GroupTheory[QuaternionGroup]