Schubert variety

Remarks

$X_\lambda$ is a closed irreducible subvariety of codimension $\abs{\lambda} = \lambda_1 + \cdots + \lambda_d$.

Example: $\lambda = 0$ gives all of $\mathrm{Gr}(d, n)$. Indeed $\dim(\Lambda \cap F_{n - d + i}) \ge d + (n - d + i) - n = i$ holds for every $\Lambda$. For $d = 2$, $n = 4$ and $\lambda = (1, 0)$, the conditions reduce to $\dim(\Lambda \cap F_2) \ge 1$. In projective terms these are the lines in $\mathbb{P}^3$ that meet a fixed line, a hypersurface in $\mathrm{Gr}(2, 4)$.

Non-example: the set of $\Lambda \in \mathrm{Gr}(2, 4)$ with $\Lambda \cap F_2 = 0$ is open and dense, and it is not closed. So it is not a Schubert variety; it is the open Schubert cell.

General form: let $G$ be a connected reductive group with a Borel subgroup $B$ and a parabolic subgroup $P \supseteq B$. A Schubert variety in the flag variety $G / P$ is the Zariski closure of a $B$-orbit.