Let $K$ be an algebraically closed field and $G$ be the Grassmannian of $k$ planes in some $l$ dimensional vector space $V$ over $K$. Let $V_1\subsetneq ... \subsetneq V_l$ be a flag for $V$. A Schubert variety in $G$ for our flag is $S_{a_1,...,a_k}:=\{\Lambda\in G:dim(\Lambda\cap V_{l-k+i-a_i})\geq i,\ \forall i\}$. A Schubert variety has codimension $\sum a_i$ in $G$. Call a Schubert variety special if $a_i = 0$ for $i>1$.
Let $S_1,...,S_n$ be special Schubert varieties of $Gr$ and let $V_1,V_2$ be distinct irreducible components of $\cap_i S_i$. My question is, must $V_1\cap V_2 = \emptyset$? If so are there any conditions we can impose for this intersection must be empty?
Thanks