Notes on the K-theory of Complex Grassmannians

generators. Our goal is to make this description more concrete. Note that G(k, n) = U(n)/(U(k)×U(n− k)) = St(k, n)/U(k) is the quotient by U(k) of the complex Stiefel manifold St(k, n) of partial orthonormal k-frames in C. Thus any finite-dimensional representation ρ : U(k) −→ GL(W ) of U(k) gives rise to the natural vector bundle St(k, n)×ρ W −→ G(k, n), where St(k, n) −→ G(k, n) is viewed as a principal U(k)bundle. This vector bundle, which we will denote by Φ(ρ), is most concretely visualized as a quotient by U(k) of the trivial U(k)-equivariant vector bundle St(k, n) ×W(ρ) −→ St(k, n). (Here (ρ) denotes the action of U(k) on W by ρ.) This trivial vector bundle is simply the pullback under the natural map St(k, n) −→ G(k, n) of Φ(ρ). It is an easy exercise to verify that given two representations ρ1, ρ2 of U(k), Φ(ρ1 ⊗ ρ2) = Φ(ρ1) ⊗ Φ(ρ2). Hence the map ρ 7→ Φ(ρ) gives a semiring homomorphism from representations of U(k) to vector bundles over G(k, n), which extends by universality to a ring homomorphism Φ : R(U(k)) ⊗ Q −→ K(G(k, n)) ⊗ Q