# Representation theory#

## Ordinary characters#

How can you compute character tables of a finite group in Sage? The Sage-GAP interface can be used to compute character tables.

You can construct the table of character values of a permutation group $$G$$ as a Sage matrix, using the method character_table of the PermutationGroup class, or via the pexpect interface to the GAP command CharacterTable.

sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4)]])
sage: G.order()
8
sage: G.character_table()
[ 1  1  1  1  1]
[ 1 -1 -1  1  1]
[ 1 -1  1 -1  1]
[ 1  1 -1 -1  1]
[ 2  0  0  0 -2]
sage: CT = gap(G).CharacterTable()
sage: print(gap.eval("Display(%s)"%CT.name()))
CT1

2  3  2  2  2  3

1a 2a 2b 4a 2c
2P 1a 1a 1a 2c 1a
3P 1a 2a 2b 4a 2c

X.1     1  1  1  1  1
X.2     1 -1 -1  1  1
X.3     1 -1  1 -1  1
X.4     1  1 -1 -1  1
X.5     2  .  .  . -2

>>> from sage.all import *
>>> G = PermutationGroup([[(Integer(1),Integer(2)),(Integer(3),Integer(4))], [(Integer(1),Integer(2),Integer(3),Integer(4))]])
>>> G.order()
8
>>> G.character_table()
[ 1  1  1  1  1]
[ 1 -1 -1  1  1]
[ 1 -1  1 -1  1]
[ 1  1 -1 -1  1]
[ 2  0  0  0 -2]
>>> CT = gap(G).CharacterTable()
>>> print(gap.eval("Display(%s)"%CT.name()))
CT1
<BLANKLINE>
2  3  2  2  2  3
<BLANKLINE>
1a 2a 2b 4a 2c
2P 1a 1a 1a 2c 1a
3P 1a 2a 2b 4a 2c
<BLANKLINE>
X.1     1  1  1  1  1
X.2     1 -1 -1  1  1
X.3     1 -1  1 -1  1
X.4     1  1 -1 -1  1
X.5     2  .  .  . -2


Here is another example:

sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3)]])
sage: G.character_table()
[         1          1          1          1]
[         1 -zeta3 - 1      zeta3          1]
[         1      zeta3 -zeta3 - 1          1]
[         3          0          0         -1]
sage: gap.eval("G := Group((1,2)(3,4),(1,2,3))")
'Group([ (1,2)(3,4), (1,2,3) ])'
sage: gap.eval("T := CharacterTable(G)")
'CharacterTable( Alt( [ 1 .. 4 ] ) )'
sage: print(gap.eval("Display(T)"))
CT2

2  2  .  .  2
3  1  1  1  .

1a 3a 3b 2a
2P 1a 3b 3a 1a
3P 1a 1a 1a 2a

X.1     1  1  1  1
X.2     1  A /A  1
X.3     1 /A  A  1
X.4     3  .  . -1

A = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3

>>> from sage.all import *
>>> G = PermutationGroup([[(Integer(1),Integer(2)),(Integer(3),Integer(4))], [(Integer(1),Integer(2),Integer(3))]])
>>> G.character_table()
[         1          1          1          1]
[         1 -zeta3 - 1      zeta3          1]
[         1      zeta3 -zeta3 - 1          1]
[         3          0          0         -1]
>>> gap.eval("G := Group((1,2)(3,4),(1,2,3))")
'Group([ (1,2)(3,4), (1,2,3) ])'
>>> gap.eval("T := CharacterTable(G)")
'CharacterTable( Alt( [ 1 .. 4 ] ) )'
>>> print(gap.eval("Display(T)"))
CT2
<BLANKLINE>
2  2  .  .  2
3  1  1  1  .
<BLANKLINE>
1a 3a 3b 2a
2P 1a 3b 3a 1a
3P 1a 1a 1a 2a
<BLANKLINE>
X.1     1  1  1  1
X.2     1  A /A  1
X.3     1 /A  A  1
X.4     3  .  . -1
<BLANKLINE>
A = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3


where $$E(3)$$ denotes a cube root of unity, $$ER(-3)$$ denotes a square root of $$-3$$, say $$i\sqrt{3}$$, and $$b3 = \frac{1}{2}(-1+i \sqrt{3})$$. Note the added print Python command. This makes the output look much nicer.

sage: print(gap.eval("irr := Irr(G)"))
[ Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, 1, 1, 1 ] ),
Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, E(3)^2, E(3), 1 ] ),
Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, E(3), E(3)^2, 1 ] ),
Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 3, 0, 0, -1 ] ) ]
sage: print(gap.eval("Display(irr)"))
[ [       1,       1,       1,       1 ],
[       1,  E(3)^2,    E(3),       1 ],
[       1,    E(3),  E(3)^2,       1 ],
[       3,       0,       0,      -1 ] ]
sage: gap.eval("CG := ConjugacyClasses(G)")
'[ ()^G, (2,3,4)^G, (2,4,3)^G, (1,2)(3,4)^G ]'
sage: gap.eval("gamma := CG[3]")
'(2,4,3)^G'
sage: gap.eval("g := Representative(gamma)")
'(2,4,3)'
sage: gap.eval("chi := irr[2]")
'Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, E(3)^2, E(3), 1 ] )'
sage: gap.eval("g^chi")
'E(3)'

>>> from sage.all import *
>>> print(gap.eval("irr := Irr(G)"))
[ Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, 1, 1, 1 ] ),
Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, E(3)^2, E(3), 1 ] ),
Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, E(3), E(3)^2, 1 ] ),
Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 3, 0, 0, -1 ] ) ]
>>> print(gap.eval("Display(irr)"))
[ [       1,       1,       1,       1 ],
[       1,  E(3)^2,    E(3),       1 ],
[       1,    E(3),  E(3)^2,       1 ],
[       3,       0,       0,      -1 ] ]
>>> gap.eval("CG := ConjugacyClasses(G)")
'[ ()^G, (2,3,4)^G, (2,4,3)^G, (1,2)(3,4)^G ]'
>>> gap.eval("gamma := CG[3]")
'(2,4,3)^G'
>>> gap.eval("g := Representative(gamma)")
'(2,4,3)'
>>> gap.eval("chi := irr[2]")
'Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, E(3)^2, E(3), 1 ] )'
>>> gap.eval("g^chi")
'E(3)'


This last quantity is the value of the character chi at the group element g.

Alternatively, if you turn IPython “pretty printing” off, then the table prints nicely.

sage: %Pprint
Pretty printing has been turned OFF
sage: gap.eval("G := Group((1,2)(3,4),(1,2,3))")
'Group([ (1,2)(3,4), (1,2,3) ])'
sage: gap.eval("T := CharacterTable(G)")
'CharacterTable( Alt( [ 1 .. 4 ] ) )'
sage: gap.eval("Display(T)")
CT3

2  2  2  .  .
3  1  .  1  1

1a 2a 3a 3b
2P 1a 1a 3b 3a
3P 1a 2a 1a 1a

X.1     1  1  1  1
X.2     1  1  A /A
X.3     1  1 /A  A
X.4     3 -1  .  .

A = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3
sage: gap.eval("irr := Irr(G)")
[ 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 ] ),
Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 3, -1, 0, 0 ] ) ]
sage: gap.eval("Display(irr)")
[ [       1,       1,       1,       1 ],
[       1,       1,  E(3)^2,    E(3) ],
[       1,       1,    E(3),  E(3)^2 ],
[       3,      -1,       0,       0 ] ]
sage: %Pprint
Pretty printing has been turned ON

>>> from sage.all import *
>>> %Pprint
Pretty printing has been turned OFF
>>> gap.eval("G := Group((1,2)(3,4),(1,2,3))")
'Group([ (1,2)(3,4), (1,2,3) ])'
>>> gap.eval("T := CharacterTable(G)")
'CharacterTable( Alt( [ 1 .. 4 ] ) )'
>>> gap.eval("Display(T)")
CT3
<BLANKLINE>
2  2  2  .  .
3  1  .  1  1
<BLANKLINE>
1a 2a 3a 3b
2P 1a 1a 3b 3a
3P 1a 2a 1a 1a
<BLANKLINE>
X.1     1  1  1  1
X.2     1  1  A /A
X.3     1  1 /A  A
X.4     3 -1  .  .
<BLANKLINE>
A = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3
>>> gap.eval("irr := Irr(G)")
[ 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 ] ),
Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 3, -1, 0, 0 ] ) ]
>>> gap.eval("Display(irr)")
[ [       1,       1,       1,       1 ],
[       1,       1,  E(3)^2,    E(3) ],
[       1,       1,    E(3),  E(3)^2 ],
[       3,      -1,       0,       0 ] ]
>>> %Pprint
Pretty printing has been turned ON


## Brauer characters#

The Brauer character tables in GAP do not yet have a “native” interface. To access them you can directly interface with GAP using pexpect and the gap.eval command.

The example below using the GAP interface illustrates the syntax.

sage: print(gap.eval("G := Group((1,2)(3,4),(1,2,3))"))
Group([ (1,2)(3,4), (1,2,3) ])
sage: print(gap.eval("irr := IrreducibleRepresentations(G,GF(7))"))   # random arch. dependent output
[ [ (1,2)(3,4), (1,2,3) ] -> [ [ [ Z(7)^0 ] ], [ [ Z(7)^4 ] ] ],
[ (1,2)(3,4), (1,2,3) ] -> [ [ [ Z(7)^0 ] ], [ [ Z(7)^2 ] ] ],
[ (1,2)(3,4), (1,2,3) ] -> [ [ [ Z(7)^0 ] ], [ [ Z(7)^0 ] ] ],
[ (1,2)(3,4), (1,2,3) ] ->
[ [ [ Z(7)^2, Z(7)^5, Z(7) ], [ Z(7)^3, Z(7)^2, Z(7)^3 ],
[ Z(7), Z(7)^5, Z(7)^2 ] ],
[ [ 0*Z(7), Z(7)^0, 0*Z(7) ], [ 0*Z(7), 0*Z(7), Z(7)^0 ],
[ Z(7)^0, 0*Z(7), 0*Z(7) ] ] ] ]
sage: gap.eval("brvals := List(irr,chi->List(ConjugacyClasses(G),c->BrauerCharacterValue(Image(chi,Representative(c)))))")
''
sage: print(gap.eval("Display(brvals)"))              # random architecture dependent output
[ [       1,       1,  E(3)^2,    E(3) ],
[       1,       1,    E(3),  E(3)^2 ],
[       1,       1,       1,       1 ],
[       3,      -1,       0,       0 ] ]
sage: print(gap.eval("T := CharacterTable(G)"))
CharacterTable( Alt( [ 1 .. 4 ] ) )
sage: print(gap.eval("Display(T)"))
CT3

2  2  .  .  2
3  1  1  1  .

1a 3a 3b 2a
2P 1a 3b 3a 1a
3P 1a 1a 1a 2a

X.1     1  1  1  1
X.2     1  A /A  1
X.3     1 /A  A  1
X.4     3  .  . -1

A = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3

>>> from sage.all import *
>>> print(gap.eval("G := Group((1,2)(3,4),(1,2,3))"))
Group([ (1,2)(3,4), (1,2,3) ])
>>> print(gap.eval("irr := IrreducibleRepresentations(G,GF(7))"))   # random arch. dependent output
[ [ (1,2)(3,4), (1,2,3) ] -> [ [ [ Z(7)^0 ] ], [ [ Z(7)^4 ] ] ],
[ (1,2)(3,4), (1,2,3) ] -> [ [ [ Z(7)^0 ] ], [ [ Z(7)^2 ] ] ],
[ (1,2)(3,4), (1,2,3) ] -> [ [ [ Z(7)^0 ] ], [ [ Z(7)^0 ] ] ],
[ (1,2)(3,4), (1,2,3) ] ->
[ [ [ Z(7)^2, Z(7)^5, Z(7) ], [ Z(7)^3, Z(7)^2, Z(7)^3 ],
[ Z(7), Z(7)^5, Z(7)^2 ] ],
[ [ 0*Z(7), Z(7)^0, 0*Z(7) ], [ 0*Z(7), 0*Z(7), Z(7)^0 ],
[ Z(7)^0, 0*Z(7), 0*Z(7) ] ] ] ]
>>> gap.eval("brvals := List(irr,chi->List(ConjugacyClasses(G),c->BrauerCharacterValue(Image(chi,Representative(c)))))")
''
>>> print(gap.eval("Display(brvals)"))              # random architecture dependent output
[ [       1,       1,  E(3)^2,    E(3) ],
[       1,       1,    E(3),  E(3)^2 ],
[       1,       1,       1,       1 ],
[       3,      -1,       0,       0 ] ]
>>> print(gap.eval("T := CharacterTable(G)"))
CharacterTable( Alt( [ 1 .. 4 ] ) )
>>> print(gap.eval("Display(T)"))
CT3
<BLANKLINE>
2  2  .  .  2
3  1  1  1  .
<BLANKLINE>
1a 3a 3b 2a
2P 1a 3b 3a 1a
3P 1a 1a 1a 2a
<BLANKLINE>
X.1     1  1  1  1
X.2     1  A /A  1
X.3     1 /A  A  1
X.4     3  .  . -1
<BLANKLINE>
A = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3