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Jul 24, 2012 at 19:54 comment added xpilot The book was not one of my original sources. While I don't have an outright answer, it does says many interesting things regarding my original question. For example, another equivalent condition to $P_{e, w}(q) = 1$ is that $|T \bigcap [e, w]| = l(w)$, which is the equality case of Deodhar's inequality. This kind of statement is the kind of thing I was looking for, and should be very useful in my research.
Jul 20, 2012 at 19:23 comment added Jim Humphreys @Vlad: I'm not sure whether this is given as another comment or as an answer to your question. Does the book do anything to answer that or is it one of the unnamed sources you started with? II realize the comments were already getting numerous.)
Jul 20, 2012 at 17:57 history answered xpilot CC BY-SA 3.0