The Connection between Representation Theory and Schubert Calculus

The structure constants cλμ determine the classical Schubert calculus on G(m,n). It has been known for some time that the integers cλμ in formulas (1) and (2) coincide. Following the work of Giambelli [G1] [G2], this is proved formally by relating both products to the multiplication of Schur S-polynomials; a precise argument along these lines was given by Lesieur [Les]. It is natural to ask for a more direct, conceptual explanation of this fact. This question has appeared every so often in print; some recent examples are [F2, §6.2] and [Len, §1]. Our aim here is to describe a direct and natural connection between the representation theory of GLn and the Schubert calculus, which goes via the Chern-Weil theory of characteristic classes. Indeed, since the Grassmannian is a universal carrier for the Chern classes of principal GLn-bundles, it is not so surprising that the cohomology ring of G(m,n) is related to the representation ring of GLn. From this point of view, we can also understand why a result of this sort fails to hold for other types of Lie groups: what makes GLn special is the fact that it sits naturally as a dense open subset of its own Lie algebra (see Sec. 2). The relation between Schubert calculus and the multiplication of Schur polynomials has been investigated before by Horrocks [Ho] and Carrell [C]. Although the approach in [Ho] is closest to the one here, the main ideas go back to the fundamental works of Chern [Ch1], Weil [W], and H. Cartan [Car]. We provide an exposition where the various ingredients from representation theory, differential geometry, topology of fiber bundles, and Schubert calculus are each presented in