Chern classes of tautological bundle over the Grassmannian G(2,4)
The cohomology ring of $G(k,n)$ is isomorphic to $$\frac{\mathbb{Z}[c_1(T),...,c_k(T),c_1(Q),..., c_{n−k} (Q)]}{(c(T)c(Q) = 1)},$$ where $T$ and $Q$ are respectively the tautological and the quotient bundle. The chern classes of the tautological and the quotient bundle are given in terms of Schubert cycles by the following formulas:
- $c_i(T) = (−1)^i\sigma_{1,...,1}$,
- $c_i(Q) = \sigma_i$.