Let (W,S) be a Coxeter system, and let T=bigcupwinW,sinSwsw−1. Associated to every element tinT is a unique positive root alphatinPhi+ considered as a vector in the standard geometric representation V of W. A total order on T is a reflection order if, whenever alphat1<alphat2, it follows that alphat1<xalphat1+yalphat2<alphat2 whenever the middle term is a positive root with x>0,y>0. (See, for example, Bjorner and Brenti's book.)
Fix a reflection order and let [u,v] be a Bruhat interval. A maximal chain u=w0tow1to...towm=v in the Bruhat order is what I'll call monotonic if wiw−1i−1>wi+1w−1i in the reflection order.
There is a nonrecursive formula for the Kazhdan-Lusztig polynomials Pu,v(q) which implies that Pu,v(0) is equal to the number of monotonic maximal chains in [u,v]. This number is known by other means to be equal to 1, so I know that there is a unique monotonic maximal chain; however, I can't prove this directly. So far all I've been able to do is use the greedy algorithm to prove that at least one monotonic maximal chain exists.
Does anyone have a direct proof of this fact?
Edit: No progress, but now I have a more general conjecture which I no longer know by other means is true. Fix a sequence a1,...am of odd positive integers such that sumiai=ell(v)−ell(u). Then there exists at most one monotonic chain (not necessarily maximal) such that wiw−1i−1inT and such that ell(wi−1)−ell(wi)=ai.
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