Bruhat Order Gives Schubert Closure Relations
With \(G,B,T,W\) as in the Bruhat decomposition, define the Schubert variety \[ X_w=\overline{BwB/B}\subseteq G/B. \] Then \[ X_w=\bigcup_{v\leq w} BvB/B, \] where \(\leq\) is the Bruhat order on the Coxeter group \(W\). In particular, \[ X_v\subseteq X_w \quad\Longleftrightarrow\quad v\leq w. \]
Proof
For a simple reflection \(s\), the rank-one subgroup associated to \(s\) controls the projection \(G/B\to G/P_s\). Its fibers are projective lines, and multiplying a Bruhat cell by \(P_s\) either preserves the cell or adds the adjacent cell indexed by \(ws\), according as \(\ell(ws)<\ell(w)\) or \(\ell(ws)>\ell(w)\).
Inducting on a reduced expression for \(w\) gives all cells indexed by subexpressions of that reduced word, exactly the elements \(v\leq w\) in Bruhat order. The dimension formula for Bruhat cells prevents additional cells from appearing in the closure.