What are big cells in Grasmannians? I found that in some papers the term ''big cells'' is used, for example, A categorification of Grassmannian cluster algebras. What are ''big cells'' and ``non-big cells'' in a Grassmannian? Thank you very much.
 A: The Grassmannian $\mathrm{Gr}_{k,n}(F)$ of $k$ planes in $F^n$ is not an affine space, but you can parametrize a dense open subset by an affine space. Here is how: if you fix a particular $n-k$ plane $V$ in $F^n$, then the set of all $k$ planes projecting isomorphically onto the quotient $F^n / V$ may be put in bijection with an affine space. This set is a big open cell (and, as noted in the comments, is the unique open $B$-orbit on the Grassmannian, where $B$ is a choice of Borel subgroup compatible with our choice of $V$; however in the special case of the Grassmannian it is perhaps clearer not to use the more general language of partial flag varieties).
To obtain such a parametrization, you fix a complementary subspace $U$ to $V$, so that $F^n=U \oplus V$. Each $k$-plane projecting isomorphically onto $V$ is then the graph of a unique linear function $\phi:V \rightarrow U$. This identifies the big open cell with $\mathrm{Hom}_F(V,U)$, an affine space of dimension $k(n-k)$, and shows that the Grassmannian is of dimension $k(n-k)$, with tangent space at $V$ naturally isomorphic to $\mathrm{Hom}_F(V,F^n/V)$. 
The rest of the Grassmannian breaks up into smaller affine spaces, corresponding to the position of the $k$-plane with respect to a choice of complete flag in $F^n$. You might have a look at Griffiths-Harris for this; each integer partition whose Young diagram fits into a $k$ by $n-k$ box gives rise to one such affine space, and each of these is called a Schubert cell. 
