Let us consider $G=GL_4(k)$, where $k=\overline{\mathbb{F}_p}$. Consider the set $S$ of the following kind of elements in $G$:
$$ \left( \begin{matrix} 1 & 0 & \ast &\ast \\ 0 & 1 & \ast &\ast \\ 0 & 0 & 1 &\ast \\ 0 & 0 & 0 &1 \end{matrix}\right) \left( \begin{matrix} A & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \\ \end{matrix}\right) \left( \begin{matrix} 0 & 1 & 0 &0 \\ 0 & 0 & 0 &1 \\ 0 & 0 & 1 &0 \\ 1 & 0 & 0 &0 \end{matrix}\right) \left( \begin{matrix} c^{-p} & 0 & 0 \\ 0 & A^{-p} & 0 \\ 0 & 0 & b^{-p} \\ \end{matrix}\right) \left( \begin{matrix} 1 & \ast & \ast &\ast \\ 0 & 1 & 0 &\ast \\ 0 & 0 & 1 &\ast \\ 0 & 0 & 0 &1 \end{matrix}\right).$$
Here, $A\in GL_2(k)$. I would like to know is $S$ or $\overline{S}$ equal to $G$?
Let me explain why I have ask a weird question. Actually I am reading some papers about groups with additional structures. It is predicted by the authors that the closure of the set $S$ above should be a codimension 1 subvariety of $G$, but I am not able to find any equation cutting out $\overline{S}$.
As you can see, the middle term corresponds to some Weyl group element. If it is replaced by \begin{equation} \left( \begin{matrix} 1 & 0 & 0 &0 \\ 0 & 0 & 0 &1 \\ 0 & 0 & 1 &0 \\ 0 & 1 & 0 &0 \end{matrix}\right) \end{equation} for example, then I can check that the closure of this set is $\{a_{31}=a_{41}=0\}$, which is predicted by the paper. I am able to verify their theory in some other cases but is stuck by this example. Can anyone help me on this. It would be better if you could tell me how you find the relations. Thanks a lot!