For $G$ a simple linear algebraic group and $B$ a fixed Borel subgroup, we have the Bruhat decomposition $G = \coprod_{w \in W} B\dot{w}B$, where $W$ is the Weyl group and $\dot{w}$ is any representative of $w$ in $N_G(T)$.

If $s_\alpha \in W$ is a simple reflection, then the following rule is known:

- $(B\dot{s_\alpha} B)(B\dot{w}B) = B\dot{s_\alpha}\dot{ w} B,\hspace{5mm}$ if $\hspace{3mm}l(s_\alpha w) = l(w) + 1$,
- $(B\dot{s_\alpha} B)(B\dot{w}B) = B\dot{w}B \cup B\dot{s_\alpha}\dot{ w} B,\hspace{5mm}$ if $\hspace{3mm}l(s_\alpha w) = l(w) - 1$,

where $l(w)$ is the length of $w \in W$.

This rule for simple reflections implies that for any $w_1, w_2 \in W$

$(B\dot{w}_1 B)(B\dot{w}_2B) = \coprod_{w \in U} B\dot{w}B$,

where $U \subset W$. However, knowing what this subset $U$ is seems quite complicated (I think one can put some partial conditions on $U$ involving Bruhat order).

My question is if anyone has seen a description of which cells appear in $(B\dot{w}_1 B)(B\dot{w}_2B)$ (i.e., what does $U$ look like)? Or, does anyone know of some nice connections between any aspects of this problem and some other theory? It seems likely to me that the decomposition of $(B\dot{w}_1 B)(B\dot{w}_2B)$ is much to complicated to have any single rule for arbitrary $w_1, w_2$. But, I thought I'd pose this question to M.O. just in case. Thanks!