I am trying to learn how to understand the cell-decomposition for the grassmannian and am following these notes: http://www.math.drexel.edu/~jblasiak/grassmannian.pdf. On page 2 the author considers the cell $e(\lambda)$ in $G(3,7)$ for $\lambda = (3,3,1)$. From the definition of $e(\lambda)$ you can find that \begin{align} \text{dim}(W \cap F_1) = 0 && \text{dim}(W \cap F_2) = 1 && \text{dim}(W \cap F_3) = 2 \\ \text{dim}(W \cap F_5) = 2 && \text{dim}(W \cap F_6) = 3 \end{align} where $F_\bullet$ is a maximal flag in $\mathbb{C}^{n+m}$. The author claims that $e(\lambda)$ is the set of matrices $$ \begin{bmatrix} * & 1 & 0 & 0 & 0 & 0 & 0 \\ * & 0 & 1 & 0 & 0 & 0 & 0 \\ * & 0 & 0 & * & * & 1 & 0 \end{bmatrix} $$ but I am not sure how this argument works. How can I show the dimension conditions imply that every $3$-plane looks like this matrix?
What I've been able to figure out is that if we consider a basis $v_1,\ldots, v_7$ defining the standard flag, then we have \begin{align} v_2 \in & W\cap F_2,\ldots F_7 \\ v_3 \in & W \cap F_3, \ldots F_7 \\ v_6 \in & W \cap F_6, F_7 \end{align}