info ginac

5.11 Symmetrization

The three methods

ex ex::symmetrize(const lst & l);
ex ex::antisymmetrize(const lst & l);
ex ex::symmetrize_cyclic(const lst & l);

symmetrize an expression by returning the sum over all symmetric, antisymmetric or cyclic permutations of the specified list of objects, weighted by the number of permutations.

The three additional methods

ex ex::symmetrize();
ex ex::antisymmetrize();
ex ex::symmetrize_cyclic();

symmetrize or antisymmetrize an expression over its free indices.

Symmetrization is most useful with indexed expressions but can be used with almost any kind of object (anything that is subs()able):

{
    idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
    symbol A("A"), B("B"), a("a"), b("b"), c("c");
                                           
    cout << indexed(A, i, j).symmetrize() << endl;
     // -> 1/2*A.j.i+1/2*A.i.j
    cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
     // -> -1/2*A.j.i.k+1/2*A.i.j.k
    cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
     // -> 1/3*{a,b,c}+1/3*{b,c,a}+1/3*{c,a,b}
}