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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?

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    $\begingroup$ Could you repair your English at least roughly ? The first sentence is completely incomprehensible, not a sentence at all; besides no capital letters, no period...I can not parse this text at all. $\endgroup$
    – Zoran Skoda
    Commented Nov 7, 2011 at 15:03
  • $\begingroup$ Yes many things about this are known, though it is still an active area of research. Are you looking for a reference to a book that introduces this material? If so, from the combinatorial or representation theoretic or geometric point of view? How much do you know already? Or are you asking about something specific in the theory? $\endgroup$
    – Alexander Woo
    Commented Nov 7, 2011 at 18:09

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