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3 Reducible Representations
 3.1 Constituents of Representations
  3.1-1 ConstituentsOfRepresentation

  3.1-2 IsReducibleRepresentation
 3.2 Block Representations
  3.2-1 EquivalentBlockRepresentation

3 Reducible Representations

In this chapter we introduce some functions which deal with a complex reducible representation R of a finite group G.

3.1 Constituents of Representations

3.1-1 ConstituentsOfRepresentation
> ConstituentsOfRepresentation( rep )( function )

called with a representation rep of a group G. This function returns a list of irreducible representations of G which are constituents of rep, and their corresponding multiplicities. For example, if rep 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 ConstituentsOfRepresentation 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.

3.1-2 IsReducibleRepresentation
> IsReducibleRepresentation( rep )( function )

If rep is a representation of a group G then IsReducibleRepresentation returns true if rep is a reducible representation of G.

3.2 Block Representations

3.2-1 EquivalentBlockRepresentation
> EquivalentBlockRepresentation( rep )( function )
> EquivalentBlockRepresentation( list )( function )

If rep is a reducible representation of a group G, this function returns a block diagonal representation of G equivalent to rep. If list = [[m1, R1], [m2, R2], ... , [mt, Rt]] is a list of irreducible representations R1, R2, ... , Rt of G with multiplicities m1, m2, ... , mt, then EquivalentBlockRepresentation returns a block diagonal representation of G containing the blocks R1, R2, ... , Rt.

gap> G := AlternatingGroup( 5 );;
gap> H := SylowSubgroup( G, 2 );;
gap> chi := TrivialCharacter( H );;
gap> Hrep := IrreducibleAffordingRepresentation( chi );;
gap> rep := InducedSubgroupRepresentation( G, Hrep );;
gap> IsReducibleRepresentation( rep );
true
gap> con := ConstituentsOfRepresentation( rep );
[ [ 1, [ (1,2,3,4,5), (3,4,5) ] -> [ [ [ 1 ] ], [ [ 1 ] ] ] ],
  [ 1, [ (1,2,3,4,5), (3,4,5) ] ->
        [ [ [ E(3), -1/3*E(3)-2/3*E(3)^2, 0, 1/3*E(3)-1/3*E(3)^2 ],
            [ 1, -4/3*E(3)+1/3*E(3)^2, E(3), -2/3*E(3)-1/3*E(3)^2 ],
            [ 1, -E(3), E(3), 0 ],
            [ 1, -1/3*E(3)+1/3*E(3)^2, 1, 1/3*E(3)+2/3*E(3)^2 ] ],
          [ [ 1, -2/3*E(3)-1/3*E(3)^2, 0, 2/3*E(3)+1/3*E(3)^2 ],
            [ 0, -E(3), E(3), 1 ],
            [ 0, -4/3*E(3)-2/3*E(3)^2, E(3), -2/3*E(3)-1/3*E(3)^2 ],
            [ 0, 0, 1, 0 ] ] ] ],
  [ 2, [ (1,2,3,4,5), (3,4,5) ] -> 
        [ [ [ -1, 1, 1, 1, -1 ], 
            [ 0, 0, 0, 0, 1 ],
            [ -1, 0, 0, 1, -1 ],
            [ 0, 0, 1, 0, 0 ], 
            [ 0, -1, 0, -1, 1 ] ],
          [ [ 0, 0, 0, 0, 1 ],
            [ 0, -1, -1, -1, 0 ],
            [ 0, 1, 0, 0, 0 ],
            [ 0, 0, 0, 1, 0 ],
            [ -1, 0, 0, 1, -1 ] ] ] ] ]
gap> EquivalentBlockRepresentation( con );
[ (1,2,3,4,5), (3,4,5) ] ->
[ [ [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
    [ 0, E(3), -1/3*E(3)-2/3*E(3)^2, 0, 1/3*E(3)-1/3*E(3)^2, 0, 
      0, 0, 0, 0,  0, 0, 0, 0, 0 ],
    [ 0, 1, -4/3*E(3)+1/3*E(3)^2, E(3), -2/3*E(3)-1/3*E(3)^2, 0, 
      0, 0, 0, 0, 0, 0, 0, 0, 0 ],
    [ 0, 1, -E(3), E(3), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
    [ 0, 1, -1/3*E(3)+1/3*E(3)^2, 1, 1/3*E(3)+2/3*E(3)^2, 0, 0, 
      0, 0, 0, 0, 0, 0, 0, 0 ], 
    [ 0, 0, 0, 0, 0, -1, 1, 1, 1, -1, 0, 0, 0, 0, 0 ],
    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ],
    [ 0, 0, 0, 0, 0, -1, 0, 0, 1, -1, 0, 0, 0, 0, 0 ],
    [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ],
    [ 0, 0, 0, 0, 0, 0, -1, 0, -1, 1, 0, 0, 0, 0, 0 ],
    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 1, 1, -1 ],
    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ],
    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, -1 ],
    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 ],
    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 1 ] ],
  [ [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
    [ 0, 1, -2/3*E(3)-1/3*E(3)^2, 0, 2/3*E(3)+1/3*E(3)^2, 0, 0, 
      0, 0, 0, 0, 0, 0, 0, 0 ],
    [ 0, 0, -E(3), E(3), 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
    [ 0, 0, -4/3*E(3)-2/3*E(3)^2, E(3), -2/3*E(3)-1/3*E(3)^2, 0, 
      0, 0, 0, 0, 0, 0, 0, 0, 0 ],
    [ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ],
    [ 0, 0, 0, 0, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0, 0 ],
    [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ],
    [ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ],
    [ 0, 0, 0, 0, 0, -1, 0, 0, 1, -1, 0, 0, 0, 0, 0 ],
    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ],
    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, 0 ],
    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ],
    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ],
    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, -1 ] ] ]
 
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