Reduced( tbl, constituents, reducibles )
Reduced( tbl, reducibles )
returns a record with fields remainders
and irreducibles
, both
lists: Let rems
be the set of nonzero characters obtained from
reducibles by subtraction of
[ sum_chiin constituents
fracScalarProduct( tbl, chi, reducibles[i] )
ScalarProduct( tbl, chi, constituents[j] )
cdot chi ]
from reducibles[i]
in the first case or subtraction of
[ sum_j leq i
fracScalarProduct( tbl, reducibles[j], reducibles[i] )
ScalarProduct( tbl, reducibles[j], reducibles[j] )
cdot reducibles[j] ]
in the second case.
Let irrs
be the list of irreducible characters in rems
. rems
is
reduced with irrs
and all found irreducibles until no new irreducibles
are found. Then irreducibles
is the set of all found irreducible
characters, remainders
is the set of all nonzero remainders.
If one knows that reducibles are ordinary characters of tbl and
constituents are irreducible ones, ReducedOrdinary ReducedOrdinary
may be faster.
Note that elements of remainders
may be only virtual characters even if
reducibles are ordinary characters.
gap> t:= CharTable( "A5" );; gap> chars:= Sublist( t.irreducibles, [ 2 .. 4 ] );; gap> chars:= Set( Tensored( chars, chars ) );; gap> Reduced( t, chars ); rec( remainders := [ ], irreducibles := [ [ 1, 1, 1, 1, 1 ], [ 3, -1, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ], [ 3, -1, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ], [ 4, 0, 1, -1, -1 ], [ 5, 1, -1, 0, 0 ] ] )
GAP 3.4.4