Let me give a very elemntary descriptiopn of the Morse-theoretical interpretation of the Bruhat decomposition in a simple case, of thecomplex Grasmannians. $\DeclareMathOperator{\Gr}{Gr}$ Denote by $\Gr_{k,n}$ the Grassmannian of complex $k$-dimensional subspaces of $\mathbb{C}^n$.
We can identify $\Gr_{k,n}$ with a submanifold of the vector space $\mathscr{S}_n$ of hermitian $n\times n$ matrices by associating to a subspace $L$ the orthogonal projection $P_L$. Let $A\in\mathscr{S}_n$ a hermitian matrix with distinct eigenvalues. Define $\DeclareMathOperator{\tr}{tr}$
$$f_A:\Gr_{k,n}\to\mathbb{R},\;\;L\mapsto f_A:{\Gr}_{k,n}\to\mathbb{R},\;\;L\mapsto \tr(AP_L).$$
Denote by $X_A$ the (negative) gradient of $f_AS$ f_A$with respect to the metric on$\Gr_{k,n}$induced by the Euclidean metric on$\mathscr{S}_n$. Then the unstable manifolds of the flow generated by$X_A$are the Schubert cells on$\Gr_{k,n}$with respect to a falg flag determined by the eigenvectors of$A$. 1 Let me give a very elemntary descriptiopn of the Morse-theoretical interpretation of the Bruhat decomposition in a simple case, of thecomplex Grasmannians.$\DeclareMathOperator{\Gr}{Gr}$Denote by$\Gr_{k,n}$the Grassmannian of complex$k$-dimensional subspaces of$\mathbb{C}^n$. We can identify$\Gr_{k,n}$with a submanifold of the vector space$\mathscr{S}_n$of hermitian$n\times n$matrices by associating to a subspace$L$the orthogonal projection$P_L$. Let$A\in\mathscr{S}_n$a hermitian matrix with distinct eigenvalues. Define$\DeclareMathOperator{\tr}{tr}$$$f_A:\Gr_{k,n}\to\mathbb{R},\;\;L\mapsto \tr(AP_L).$$ Denote by$X_A$the (negative) gradient of$f_AS$with respect to the metric on$\Gr_{k,n}$induced by the Euclidean metric on$\mathscr{S}_n$. Then the unstable manifolds of the flow generated by$X_A$are the Schubert cells on$\Gr_{k,n}$with respect to a falg determined by the eigenvectors of$A\$.