Schubert calculus on the arithmetic Grassmannian

Let G¯ be the arithmetic Grassmannian over Spec ℤ with the natural invariant Kähler metric on G¯(ℂ). We study the combinatorics of the arithmetic Schubert calculus in the Arakelov Chow ring CH(G¯). We obtain formulas for the arithmetic Littlewood- Richardson numbers and the Faltings height of G under the Plücker embedding, using “rim hook operations” on Young diagrams. An analysis of the duality involution leads to new combinatorial relations among Kostka numbers. © 1999 Applied Probability Trust.