GAP Manual: 50.10. Reduced

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_chiinconstituents fracScalarProduct( tbl, chi, reducibles[i] ) ScalarProduct( tbl, chi, constituents[j] ) cdotchi]

from reducibles[i] in the first case or subtraction of

[ sum_j leqi fracScalarProduct( tbl, reducibles[j], reducibles[i] ) ScalarProduct( tbl, reducibles[j], reducibles[j] ) cdotreducibles[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 ] ] )