3 Reducible Representations

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

`> 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.

`> IsReducibleRepresentation` ( rep ) | ( function ) |

If `rep` is a representation of a group G then `IsReducibleRepresentation`

returns `true`

if `rep` is a reducible representation of G.

`> 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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