50.43 ContainedSpecialVectors

ContainedSpecialVectors( tbl, chars, parachar, func )

returns the list of all elements vec of the parametrized character parachar (see More about Maps and Parametrized Maps), that have integral norm and integral scalar product with the principal character of the character table tbl and that satisfy func( tbl, chars, vec ), i.e., for that true is returned.

    gap> s:= CharTable( "HSM12" );; hs:= CharTable( "HS" );;
    gap> fus:= InitFusion( s, hs );;
    gap> rest:= CompositionMaps( hs.irreducibles[8], fus );;
    # no further condition\:
    gap> ContainedSpecialVectors( s, s.irreducibles, rest,
    >                      function(tbl,chars,vec) return true; end );;
    gap> Length( last );
    24
    # better filter\:\ those with integral scalar products
    gap> ContainedSpecialVectors( s, s.irreducibles, rest,
    >                             IntScalarProducts );
    [ [ 231, 7, -9, -9, 6, 15, 15, -1, -1, 1, 6, 6, 1, 1, -2, 1, 2, 2, 0,
          0, 1, 0, 0, 0, 0 ],
      [ 231, 7, -9, 7, 6, 15, 15, -1, -1, 1, 6, 6, 1, 1, -2, 1, 2, 2, 0,
          0, 1, 0, 0, 0, 0 ],
      [ 231, 7, -9, -9, 6, 15, 15, 15, 15, 1, 6, 6, 1, 1, -2, 1, 2, 2, 0,
          0, 1, 0, 0, 0, 0 ],
      [ 231, 7, -9, 7, 6, 15, 15, 15, 15, 1, 6, 6, 1, 1, -2, 1, 2, 2, 0,
          0, 1, 0, 0, 0, 0 ] ]
    # better filter\:\ the scalar products must be nonnegative
    gap> ContainedSpecialVectors( s, s.irreducibles, rest,
    >                             NonnegIntScalarProducts );
    [ [ 231, 7, -9, -9, 6, 15, 15, -1, -1, 1, 6, 6, 1, 1, -2, 1, 2, 2, 0,
          0, 1, 0, 0, 0, 0 ],
      [ 231, 7, -9, 7, 6, 15, 15, -1, -1, 1, 6, 6, 1, 1, -2, 1, 2, 2, 0,
          0, 1, 0, 0, 0, 0 ] ]

Special cases of ContainedSpecialVectors are ContainedPossibleCharacters ContainedPossibleCharacters and ContainedPossibleVirtualCharacters ContainedPossibleVirtualCharacters.

ContainedSpecialVectors successively examines all vectors contained in parachar, thus it might not be useful if the indeterminateness exceeds 10^6. For another strategy that works for rational characters only, see ContainedDecomposables.

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