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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.

Identities involving (doubly) symmetric polynomials and integrals over Grassmannians