51.23 PowermapsAllowedBySymmetrisations

PowermapsAllowedBySymmetrisations( tbl, subchars, chars, pow,
prime, parameters )

More about Maps and Parametrized Maps) maps map which are contained in the parametrized map pow and which have the property that for all chi in the list chars of characters of the character table tbl, the symmetrizations [ chi^p- = MinusCharacter( chi, map, prime ) ] (see MinusCharacter) have nonnegative integral scalar products with all characters in the list subchars.

parameters must be a record with fields

maxlen:

an integer that controls the position where branches take place

contained:

a function, usually ContainedCharacters or ContainedPossibleCharacters; for a symmetrization minus, it returns the list contained( tbl, subchars, minus )

minamb, maxamb:

two arbitrary objects; contained is called only for symmetrizations minus with [ minamb < Indeterminateness( minus ) < maxamb ]

quick:

a boolean; if it is true, the scalar products of uniquely determined symmetrizations are not checked.

pow will be improved, i.e. is changed by the algorithm.

If there is no character left which allows an immediate improvement but there are characters in chars with indeterminateness of the symmetrizations bigger than parameters.minamb, a branch is necessary. Two kinds of branches may occur: If parameters.contained( tbl, subchars, minus ) has length at most parameters.maxlen, the union of maps allowed by the characters in minus is computed; otherwise a suitable class c is taken which is significant for some character, and the union of all admissible maps with image x on c is computed, where x runs over pow[c].

    # see example in "ConsiderKernels"
    gap> t := CharTable( "U4(3).4" );;
    gap> PowermapsAllowedBySymmetrisations(t,t.irreducibles,t.irreducibles,
    >      pow, 2, rec( maxlen:=10, contained:=ContainedPossibleCharacters,
    >      minamb:= 2, maxamb:= "infinity", quick:= false ) );
    [ [ 1, 1, 3, 4, 5, 2, 2, 8, 3, 4, 11, 12, 6, 14, 9, 1, 1, 2, 2, 3, 4,
          5, 6, 8, 9, 9, 10, 11, 12, 16, 16, 16, 16, 17, 17, 18, 18, 18,
          18, 20, 20, 20, 20, 22, 22, 24, 24, 25, 26, 28, 28, 29, 29 ] ]
    gap> t.powermap[2] = last[1];
    true

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