  
  [1X3 Reducible Representations[0X
  
  In  this  chapter  we  introduce  some  functions  which deal with a complex
  reducible representation R of a finite group G.
  
  
  [1X3.1 Constituents of Representations[0X
  
  [1X3.1-1 ConstituentsOfRepresentation[0m
  
  [2X> ConstituentsOfRepresentation( [0X[3Xrep[0X[2X ) ______________________________[0Xfunction
  
  called  with a representation [3Xrep[0m of a group G. This function returns a list
  of irreducible representations of G which are constituents of [3Xrep[0m, and their
  corresponding  multiplicities.  For example, if [3Xrep[0m is a representation of G
  affording a character X such that X = mY + nZ, where Y and Z are irreducible
  characters  of  G,  and  m  and n are the corresponding multiplicities, then
  [10XConstituentsOfRepresentation[0m  returns  [[m,  S],  [n,  T]] where S and T are
  irreducible  representations  of  G  affording  Y  and Z, respectively. This
  function call can be quite expensive when G is a large group.
  
  [1X3.1-2 IsReducibleRepresentation[0m
  
  [2X> IsReducibleRepresentation( [0X[3Xrep[0X[2X ) _________________________________[0Xfunction
  
  If  [3Xrep[0m  is  a  representation  of  a group G then [10XIsReducibleRepresentation[0m
  returns [10Xtrue[0m if [3Xrep[0m is a reducible representation of G.
  
  
  [1X3.2 Block Representations[0X
  
  [1X3.2-1 EquivalentBlockRepresentation[0m
  
  [2X> EquivalentBlockRepresentation( [0X[3Xrep[0X[2X ) _____________________________[0Xfunction
  [2X> EquivalentBlockRepresentation( [0X[3Xlist[0X[2X ) ____________________________[0Xfunction
  
  If  [3Xrep[0m  is a reducible representation of a group G, this function returns a
  block diagonal representation of G equivalent to [3Xrep[0m. If [3X list [0m = [[m1, R1],
  [m2,  R2],  ... , [mt, Rt]] is a list of irreducible representations R1, R2,
  ...   ,   Rt   of   G   with   multiplicities   m1,   m2,  ...  ,  mt,  then
  [10XEquivalentBlockRepresentation[0m  returns  a block diagonal representation of G
  containing the blocks R1, R2, ... , Rt.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> G := AlternatingGroup( 5 );;
[0X
    [4Xgap> H := SylowSubgroup( G, 2 );;
[0X
    [4Xgap> chi := TrivialCharacter( H );;
[0X
    [4Xgap> Hrep := IrreducibleAffordingRepresentation( chi );;
[0X
    [4Xgap> rep := InducedSubgroupRepresentation( G, Hrep );;
[0X
    [4Xgap> IsReducibleRepresentation( rep );
[0X
    [4Xtrue
[0X
    [4Xgap> con := ConstituentsOfRepresentation( rep );
[0X
    [4X[ [ 1, [ (1,2,3,4,5), (3,4,5) ] -> [ [ [ 1 ] ], [ [ 1 ] ] ] ],
[0X
    [4X  [ 1, [ (1,2,3,4,5), (3,4,5) ] ->
[0X
    [4X        [ [ [ E(3), -1/3*E(3)-2/3*E(3)^2, 0, 1/3*E(3)-1/3*E(3)^2 ],
[0X
    [4X            [ 1, -4/3*E(3)+1/3*E(3)^2, E(3), -2/3*E(3)-1/3*E(3)^2 ],
[0X
    [4X            [ 1, -E(3), E(3), 0 ],
[0X
    [4X            [ 1, -1/3*E(3)+1/3*E(3)^2, 1, 1/3*E(3)+2/3*E(3)^2 ] ],
[0X
    [4X          [ [ 1, -2/3*E(3)-1/3*E(3)^2, 0, 2/3*E(3)+1/3*E(3)^2 ],
[0X
    [4X            [ 0, -E(3), E(3), 1 ],
[0X
    [4X            [ 0, -4/3*E(3)-2/3*E(3)^2, E(3), -2/3*E(3)-1/3*E(3)^2 ],
[0X
    [4X            [ 0, 0, 1, 0 ] ] ] ],
[0X
    [4X  [ 2, [ (1,2,3,4,5), (3,4,5) ] -> 
[0X
    [4X        [ [ [ -1, 1, 1, 1, -1 ], 
[0X
    [4X            [ 0, 0, 0, 0, 1 ],
[0X
    [4X            [ -1, 0, 0, 1, -1 ],
[0X
    [4X            [ 0, 0, 1, 0, 0 ], 
[0X
    [4X            [ 0, -1, 0, -1, 1 ] ],
[0X
    [4X          [ [ 0, 0, 0, 0, 1 ],
[0X
    [4X            [ 0, -1, -1, -1, 0 ],
[0X
    [4X            [ 0, 1, 0, 0, 0 ],
[0X
    [4X            [ 0, 0, 0, 1, 0 ],
[0X
    [4X            [ -1, 0, 0, 1, -1 ] ] ] ] ]
[0X
    [4Xgap> EquivalentBlockRepresentation( con );
[0X
    [4X[ (1,2,3,4,5), (3,4,5) ] ->
[0X
    [4X[ [ [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[0X
    [4X    [ 0, E(3), -1/3*E(3)-2/3*E(3)^2, 0, 1/3*E(3)-1/3*E(3)^2, 0, 
[0X
    [4X      0, 0, 0, 0,  0, 0, 0, 0, 0 ],
[0X
    [4X    [ 0, 1, -4/3*E(3)+1/3*E(3)^2, E(3), -2/3*E(3)-1/3*E(3)^2, 0, 
[0X
    [4X      0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[0X
    [4X    [ 0, 1, -E(3), E(3), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[0X
    [4X    [ 0, 1, -1/3*E(3)+1/3*E(3)^2, 1, 1/3*E(3)+2/3*E(3)^2, 0, 0, 
[0X
    [4X      0, 0, 0, 0, 0, 0, 0, 0 ], 
[0X
    [4X    [ 0, 0, 0, 0, 0, -1, 1, 1, 1, -1, 0, 0, 0, 0, 0 ],
[0X
    [4X    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ],
[0X
    [4X    [ 0, 0, 0, 0, 0, -1, 0, 0, 1, -1, 0, 0, 0, 0, 0 ],
[0X
    [4X    [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ],
[0X
    [4X    [ 0, 0, 0, 0, 0, 0, -1, 0, -1, 1, 0, 0, 0, 0, 0 ],
[0X
    [4X    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 1, 1, -1 ],
[0X
    [4X    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ],
[0X
    [4X    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, -1 ],
[0X
    [4X    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 ],
[0X
    [4X    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 1 ] ],
[0X
    [4X  [ [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[0X
    [4X    [ 0, 1, -2/3*E(3)-1/3*E(3)^2, 0, 2/3*E(3)+1/3*E(3)^2, 0, 0, 
[0X
    [4X      0, 0, 0, 0, 0, 0, 0, 0 ],
[0X
    [4X    [ 0, 0, -E(3), E(3), 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[0X
    [4X    [ 0, 0, -4/3*E(3)-2/3*E(3)^2, E(3), -2/3*E(3)-1/3*E(3)^2, 0, 
[0X
    [4X      0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[0X
    [4X    [ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[0X
    [4X    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ],
[0X
    [4X    [ 0, 0, 0, 0, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0, 0 ],
[0X
    [4X    [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ],
[0X
    [4X    [ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ],
[0X
    [4X    [ 0, 0, 0, 0, 0, -1, 0, 0, 1, -1, 0, 0, 0, 0, 0 ],
[0X
    [4X    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ],
[0X
    [4X    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, 0 ],
[0X
    [4X    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ],
[0X
    [4X    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ],
[0X
    [4X    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, -1 ] ] ]
[0X
    [4X [0X
  [4X------------------------------------------------------------------[0X
  
