Let $V$ be a full flag, $\lambda$ a partition. Consider $$\sigma_\lambda(V) = \{ \Lambda \in G(k,n): \Lambda \cap V_{n-k+i-\lambda_i} \geq i \}.$$ If you have another full flag $V'$, are $\sigma_\lambda(V)$ and $\sigma_\lambda(V')$ isomorphic to each other? It seems that in intersection theory, they only care about the partition and not about the flag. Why is that?
Thanks.
Yes, $GL_n$ acts on the Grassmannian by linear change of coordinates, which changes the choice of auxiliary flag.
For intersection theory, it also matters that $GL_n$ is rationally connected. That is, the two subvarieties are not only isomorphic but rationally equivalent (essentially the algebraic version of being homotopic).