50.35 LLL

LLL( tbl, characters [, y] [, "sort"] [, "linearcomb"] )

calls the LLL algorithm (see LLLReducedBasis) in the case of lattices spanned by (virtual) characters characters of the character table tbl (see ScalarProduct). By finding shorter vectors in the lattice spanned by characters, i.e. virtual characters of smaller norm, in some cases LLL is able to find irreducible characters.

LLL returns a record with at least components irreducibles (the list of found irreducible characters), remainders (a list of reducible virtual characters), and norms (the list of norms of remainders). irreducibles together with remainders span the same lattice as characters.

There are some optional parameters:

y:

controls the sensitivity of the algorithm; the value of y must be between 1/4 and 1, the default value is 3/4.

"sort":

LLL sorts characters and the remainders component of the result according to the degrees.

"linearcomb":

The returned record contains components irreddecomp and reddecomp which are decomposition matrices of irreducibles and remainders, with respect to characters.

    gap> s4:= CharTable( "Symmetric", 4 );;
    gap> chars:= [ [ 8, 0, 0, -1, 0 ], [ 6, 0, 2, 0, 2 ],
    >     [ 12, 0, -4, 0, 0 ], [ 6, 0, -2, 0, 0 ], [ 24, 0, 0, 0, 0 ],
    >     [ 12, 0, 4, 0, 0 ], [ 6, 0, 2, 0, -2 ], [ 12, -2, 0, 0, 0 ],
    >     [ 8, 0, 0, 2, 0 ], [ 12, 2, 0, 0, 0 ], [ 1, 1, 1, 1, 1 ] ];;
    gap> LLL( s4, chars );
    rec(
      irreducibles :=
       [ [ 2, 0, 2, -1, 0 ], [ 1, 1, 1, 1, 1 ], [ 3, 1, -1, 0, -1 ],
          [ 3, -1, -1, 0, 1 ], [ 1, -1, 1, 1, -1 ] ],
      remainders := [  ],
      norms := [  ] )

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