TITLE: Indexing the line bundles over a Grassmannian.
QUESTION [7 upvotes]: As is well known, the line bundles over *CP*$^1$ are indexed by the integers. My question is how are the line bundles over *CP*$^n$, $n > 1$, and *Gr*$(n,k)$ indexed? Moreover, do there exist any other interesting classifications of line bundles over spaces (I remember something about Atiyah and elliptic curves)?

REPLY [4 votes]: As a compact homogeneous Kahler manifold the Grassmannian is a coset space of the general linear group by a maximal parabolic subgroup. Homogeneous line bundles over the Grassmannian are in a one to one correspondence with the character representations of the maximal parabolic, which are indexed by one integer. According to the Bott-Borel-Weil theorem, the space of holomorphic sections of the line bundle carries an irreducible representation of the special unitary group SU(n). In the case of the Grassmannian Gr(n,k), these representations correpond to Young tableaux with k rows. The integer characterizing the line bundle is just the number of columns in this tableaux.