  We obtain identities involving symmetric and doubly symmetric polynomials.
These identities provide a way of handling expressions appearing in the
Atiyah-Bott-Berline-Vergne formula for Grassmannians. As corollaries, we obtain
formulas for integrals over Grassmannians of characteristic classes of the
tautological bundles. Moreover, we provide a valid proof of the Martin formula
for the classical Grassmannian.
