Identities involving (doubly) symmetric polynomials and integrals over Grassmannians

TL;DR

This paper derives identities involving symmetric polynomials to facilitate calculations of characteristic class integrals over Grassmannians, providing new proofs and formulas relevant to geometric and topological applications.

Contribution

It introduces novel identities for symmetric and doubly symmetric polynomials that simplify integral computations over Grassmannians and offers a rigorous proof of the Martin formula.

Findings

Derived identities for symmetric polynomials related to Grassmannian integrals

Formulas for characteristic class integrals over Grassmannians

Provided a proof of the Martin formula for classical Grassmannians

Abstract

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.