Linear representation theory of special linear group:SL(2,Z4)

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This article gives specific information, namely, linear representation theory, about a particular group, namely: special linear group:SL(2,Z4).
View linear representation theory of particular groups | View other specific information about special linear group:SL(2,Z4)

Summary

Item Value
degrees of irreducible representations over a splitting field (such as or ) 1,1,1,1,2,2,3,3,3,3
grouped: 1 (4 times), 2 (2 times), 3 (4 times)
number: 10, maximum: 3, lcm: 6, sum of squares: 48

GAP implementation

The degrees of irreducible representations can be determined using GAP's CharacterDegrees function: 

gap> CharacterDegrees(SL(2,ZmodnZ(4)));
[ [ 1, 4 ], [ 2, 2 ], [ 3, 4 ] ]

The character table can be computed using the Irr and CharacterTable functions: 

gap> Irr(CharacterTable(SL(2,ZmodnZ(4))));
[ Character( CharacterTable( SL(2,Z/4Z) ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ),
  Character( CharacterTable( SL(2,Z/4Z) ), [ 1, 1, -1, 1, -1, -1, -1, 1, 1, 1
     ] ), Character( CharacterTable( SL(2,Z/4Z) ),
    [ 1, -1, -E(4), 1, -E(4), E(4), E(4), -1, 1, -1 ] ),
  Character( CharacterTable( SL(2,Z/4Z) ),
    [ 1, -1, E(4), 1, E(4), -E(4), -E(4), -1, 1, -1 ] ),
  Character( CharacterTable( SL(2,Z/4Z) ), [ 2, 1, 0, -1, 0, 0, 0, -2, 2, -2
     ] ), Character( CharacterTable( SL(2,Z/4Z) ),
    [ 2, -1, 0, -1, 0, 0, 0, 2, 2, 2 ] ),
  Character( CharacterTable( SL(2,Z/4Z) ), [ 3, 0, -1, 0, 1, -1, 1, -1, -1, 3
     ] ), Character( CharacterTable( SL(2,Z/4Z) ),
    [ 3, 0, 1, 0, -1, 1, -1, -1, -1, 3 ] ),
  Character( CharacterTable( SL(2,Z/4Z) ),
    [ 3, 0, -E(4), 0, E(4), E(4), -E(4), 1, -1, -3 ] ),
  Character( CharacterTable( SL(2,Z/4Z) ),
    [ 3, 0, E(4), 0, -E(4), -E(4), E(4), 1, -1, -3 ] ) ]

The irreducible representations can be computed using the IrreducibleRepresentations function: 

gap> IrreducibleRepresentations(SL(2,ZmodnZ(4)));
[ CompositionMapping( [ (1,8,2,7)(3,6,9,10)(4,11,12,5), (1,2)(3,9)(4,12)(5,11)(6,10)(7,8), (1,3,12)(2,9,4)(5,6,8)(7,11,10),
      (1,2)(3,11)(4,10)(5,9)(6,12)(7,8), (1,8)(2,7)(3,9)(4,6)(5,11)(10,12) ] -> [ [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ]
     ], <action isomorphism> ), CompositionMapping( [ (1,8,2,7)(3,6,9,10)(4,11,12,5), (1,2)(3,9)(4,12)(5,11)(6,10)(7,8), (1,3,12)(2,9,4)(5,6,8)(7,11,10),
      (1,2)(3,11)(4,10)(5,9)(6,12)(7,8), (1,8)(2,7)(3,9)(4,6)(5,11)(10,12) ] -> [ [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ]
     ], <action isomorphism> ), CompositionMapping( [ (1,8,2,7)(3,6,9,10)(4,11,12,5), (1,2)(3,9)(4,12)(5,11)(6,10)(7,8), (1,3,12)(2,9,4)(5,6,8)(7,11,10),
      (1,2)(3,11)(4,10)(5,9)(6,12)(7,8), (1,8)(2,7)(3,9)(4,6)(5,11)(10,12) ] -> [ [ [ E(4) ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ]
     ], <action isomorphism> ), CompositionMapping( [ (1,8,2,7)(3,6,9,10)(4,11,12,5), (1,2)(3,9)(4,12)(5,11)(6,10)(7,8), (1,3,12)(2,9,4)(5,6,8)(7,11,10),
      (1,2)(3,11)(4,10)(5,9)(6,12)(7,8), (1,8)(2,7)(3,9)(4,6)(5,11)(10,12) ] ->
     ], <action isomorphism> ), CompositionMapping( [ (1,8,2,7)(3,6,9,10)(4,11,1
      (1,2)(3,11)(4,10)(5,9)(6,12)(7,8), (1,8)(2,7)(3,9)(4,6)(5,11)(10,12) ] ->
      [ [ E(3), 0 ], [ 0, E(3)^2 ] ], [ [ 1, 0 ], [ 0, 1 ] ], [ [ 1, 0 ], [ 0, 1
  CompositionMapping( [ (1,8,2,7)(3,6,9,10)(4,11,12,5), (1,2)(3,9)(4,12)(5,11)(6
      (1,2)(3,11)(4,10)(5,9)(6,12)(7,8), (1,8)(2,7)(3,9)(4,6)(5,11)(10,12) ] ->
      [ [ E(3), 0 ], [ 0, E(3)^2 ] ], [ [ 1, 0 ], [ 0, 1 ] ], [ [ 1, 0 ], [ 0, 1
  CompositionMapping( [ (1,8,2,7)(3,6,9,10)(4,11,12,5), (1,2)(3,9)(4,12)(5,11)(6
      (1,2)(3,11)(4,10)(5,9)(6,12)(7,8), (1,8)(2,7)(3,9)(4,6)(5,11)(10,12) ] ->
      [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ], [ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0
      [ [ 1, 0, 0 ], [ 0, -1, 0 ], [ 0, 0, -1 ] ] ], <action isomorphism> ),
  CompositionMapping( [ (1,8,2,7)(3,6,9,10)(4,11,12,5), (1,2)(3,9)(4,12)(5,11)(6
      (1,2)(3,11)(4,10)(5,9)(6,12)(7,8), (1,8)(2,7)(3,9)(4,6)(5,11)(10,12) ] ->
      [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ], [ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0
      [ [ 1, 0, 0 ], [ 0, -1, 0 ], [ 0, 0, -1 ] ] ], <action isomorphism> ),
  CompositionMapping( [ (1,8,2,7)(3,6,9,10)(4,11,12,5), (1,2)(3,9)(4,12)(5,11)(6
      (1,2)(3,11)(4,10)(5,9)(6,12)(7,8), (1,8)(2,7)(3,9)(4,6)(5,11)(10,12) ] ->
      [ [ -1, 0, 0 ], [ 0, -1, 0 ], [ 0, 0, -1 ] ], [ [ 0, 0, 1 ], [ 1, 0, 0 ],
      [ [ 1, 0, 0 ], [ 0, -1, 0 ], [ 0, 0, -1 ] ] ], <action isomorphism> ),
  CompositionMapping( [ (1,8,2,7)(3,6,9,10)(4,11,12,5), (1,2)(3,9)(4,12)(5,11)(6
      (1,2)(3,11)(4,10)(5,9)(6,12)(7,8), (1,8)(2,7)(3,9)(4,6)(5,11)(10,12) ] ->
      [ [ -1, 0, 0 ], [ 0, -1, 0 ], [ 0, 0, -1 ] ], [ [ 0, 0, 1 ], [ 1, 0, 0 ],
      [ [ 1, 0, 0 ], [ 0, -1, 0 ], [ 0, 0, -1 ] ] ], <action isomorphism> ) ]
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