The set of Schubert varieties of in a flag variety is in one-to-one correspondence with elements of the Weyl groups group via left cells,the cells. There is also some relation between products of Schubert varieties product the and perverse sheaves on the flag variety ,
my question [this is my best attempt to make sense of the previous form of this sentence - ed.].
The relations in Weyl groups can reflect to are reflected in Schubert varieties and then perverse the intersection homology sheaves, but this relation is not $\le$,for example,$l(s*u)=l(u)+1$,where \leq$. For example, when $l(s*u)=l(u)+1$, where $s$ is a simple reflection,then reflection, then we have $C(s)C(u)=C(su)$ ,C(s)C(u)=C(su)$, where $C(?)$ is a left cell,what cell.
Question: What is the relation relationship between Schubert varieties labeled $s,u,su$ ,s,u,su$, and their intersection cohomologyof them
there must be people study this question ,but I donot know, If you know,please tell me ,Thank you very much.?