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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