8.2 Brauer characters

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

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
<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-ER(-3))/2 = -1-b3