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Let G be the Grassmannian which parametrizes m-dimensional subspaces of affine (m+n)-space for any base field, thought of as a scheme over the ring of rational integers. We study the multiplicative structure of the Arakelov Chow ring of G, the so-called "arithmetic Schubert calculus". This talk will focus on the action of the Lefschetz operator in the context of combinatorics, topology, differential geometry, and Arakelov theory. Applications include: an explicit formula for the Faltings height of G under its Pluecker embedding in projective space, and a proof of the arithmetic Hodge index inequalities when m = 1 or 2. The latter are related to good estimates for a class of special functions called Racah polynomials.