53.13 OrbitsCharacters

OrbitsCharacters( irr )

returns a list of orbits of the characters irr under the action of Galois automorphisms and multiplication with linear characters in irr. This is used for functions that need to consider only representatives under the operation of this group, like TestSubnormallyMonomial.

OrbitsCharacters works also for irr a list of character value lists. In this case the result contains orbits of these lists.

Note that OrbitsCharacters does not require that irr is closed under the described action, so the function may also be used to complete the orbits.

    gap> irr:= Irr( SolvableGroup( "S4" ) );;
    gap> OrbitsCharacters( irr );
    [ [ Character( S4, [ 1, 1, 1, -1, -1 ] ),
          Character( S4, [ 1, 1, 1, 1, 1 ] ) ],
      [ Character( S4, [ 2, 2, -1, 0, 0 ] ) ],
      [ Character( S4, [ 3, -1, 0, -1, 1 ] ),
          Character( S4, [ 3, -1, 0, 1, -1 ] ) ] ]
    gap> OrbitsCharacters( List( irr{ [1,2,4] }, x -> x.values ) );
    [ [ [ 1, 1, 1, -1, -1 ], [ 1, 1, 1, 1, 1 ] ],
      [ [ 3, -1, 0, 1, -1 ], [ 3, -1, 0, -1, 1 ] ] ] 

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