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:
"sort"
:LLL
sorts characters and the remainders
component of the
result according to the degrees.
"linearcomb"
: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 := [ ] )
GAP 3.4.4