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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