The set of Schubert varieties in a flag variety is in one-to-one correspondence with elements of the Weyl group via left cells. There is also some relation between products of Schubert varieties and perverse sheaves on the flag variety [this is my best attempt to make sense of the previous form of this sentence - ed.].
The relations in Weyl groups are reflected in Schubert varieties and the intersection homology sheaves, but this relation is not $\leq$. For example, when $l(s*u)=l(u)+1$, where $s$ is a simple reflection, then we have $C(s)C(u)=C(su)$, where $C(?)$ is a left cell.
Question: What is the relationship between Schubert varieties labeled $s,u,su$, and their intersection cohomology?