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8. Representation theory


8.1 Ordinary characters

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())
CT2

 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

Here is another example: 

sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3)]])
sage: G.character_table()

[          1           1           1           1]
[          1           1 -zeta3 - 1      zeta3]
[          1           1      zeta3 -zeta3 - 1]
[          3          -1           0           0]
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)")
CT1

     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-ER(-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, 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: print 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: gap.eval("CG := ConjugacyClasses(G)")
'[ ()^G, (1,2)(3,4)^G, (1,2,3)^G, (1,2,4)^G ]'
sage: gap.eval("gamma := CG[3]")
'(1,2,3)^G'
sage: gap.eval("g := Representative(gamma)")
'(1,2,3)'
sage: gap.eval("chi := irr[2]")
'Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, 1, E(3)^2, E(3) ] )'
sage: gap.eval("g^chi")
'E(3)^2'
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)")
      CT1

     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-ER(-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


8.2 Brauer characters

The example below using the GAP interface illusrates the syntax. 

sage: gap.eval("G := Group((1,2)(3,4),(1,2,3))")
      'Group([ (1,2)(3,4), (1,2,3) ])'
sage: gap.eval("irr := IrreducibleRepresentations(G,GF(7))")
      '[ [ (1,2)(3,4), (1,2,3) ] -> [ [ [ Z(7)^0 ] ], [ [ Z(7)^4 ] ] ], \n
  [ (1,2)(3,4), (1,2,3) ] -> [ [ [ Z(7)^0 ] ], [ [ Z(7)^0 ] ] ], \n
  [ (1,2)(3,4), (1,2,3) ] -> [ [ [ Z(7)^0 ] ], [ [ Z(7)^2 ] ] ], \n
  [ (1,2)(3,4), (1,2,3) ] -> \n
    [ [ [ 0*Z(7), 0*Z(7), Z(7)^0 ], [ Z(7)^3, Z(7)^3, Z(7)^3 ], \n
          [ Z(7)^0, 0*Z(7), 0*Z(7) ] ], \n
      [ [ Z(7)^0, Z(7)^0, 0*Z(7) ], [ 0*Z(7), 0*Z(7), Z(7)^0 ], \n
          [ 0*Z(7), Z(7)^3, Z(7)^3 ] ] ] ]'
sage: gap.eval("brvals := List(irr,chi-> 
      List(ConjugacyClasses(G),c->BrauerCharacterValue(Image(chi, 
      Representative(c)))))")
      '[ [ 1, 1, E(3)^2, E(3) ], [ 1, 1, 1, 1 ], [ 1, 1, E(3), E(3)^2 ], 
         [ 3, -1, 0, 0 ] ]'
sage: gap.eval("Display(brvals)")
      '[ [       1,       1,  E(3)^2,    E(3) ],\n
  [       1,       1,       1,       1 ],\n
  [       1,       1,    E(3),  E(3)^2 ],\n
  [       3,      -1,       0,       0 ] ]'
sage: gap.eval("T := CharacterTable(G)")
      'CharacterTable( Alt( [ 1 .. 4 ] ) )'
sage: gap.eval("Display(T)")
      'CT2\n\n     2  2  2  .  .\n
     3  1  .  1  1\n
\n
       1a 2a 3a 3b\n
    2P 1a 1a 3b 3a\n
    3P 1a 2a 1a 1a\n
\n
X.1     1  1  1  1\n
X.2     1  1  A /A\n
X.3     1  1 /A  A\n
X.4     3 -1  .  .\n
\n
A = E(3)^2\n
  = (-1-ER(-3))/2 = -1-b3'
sage:

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