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Let $(W, S)$ be a Coxeter system, and let $T = \bigcup_{w \in W, s \in S} wsw^{-1}$. Associated to every element $t \in T$ is a unique positive root $\alpha_t \in \Phi^{+}$ considered as a vector in the standard geometric representation $V$ of $W$. A total order on $T$ is a reflection order if, whenever $\alpha_{t_1} < \alpha_{t_2}$, it follows that $\alpha_{t_1} < x \alpha_{t_1} + y \alpha_{t_2} < \alpha_{t_2}$ 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 $u = w_0 \to w_1 \to ... \to w_m = v$ in the Bruhat order is what I'll call monotonic if $w_i w_{i-1}^{-1} > w_{i+1} w_i^{-1}$ in the reflection order.

There is a nonrecursive formula for the Kazhdan-Lusztig polynomials $P_{u,v}(q)$ which implies that $P_{u,v}(0)$ 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 $a_1, ... a_m$ of odd positive integers such that $\sum_i a_i = \ell(v) - \ell(u)$. Then there exists at most one monotonic chain (not necessarily maximal) such that $w_i w_{i-1}^{-1} \in T$ and such that $\ell(w_{i-1}) - \ell(w_i) = a_i$.

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Is "monotonic" a standard term? It would seem more logical to call it "convex". –  Victor Protsak Jul 14 '10 at 1:14
    
Victor, maybe that would be true if the reflection order was inherited from an order on the vertices, but it's not. I am thinking here of plotting the sequence w_i w_{i-1}^{_1}, not of plotting the sequence w_i. –  Qiaochu Yuan Jul 14 '10 at 1:46
    
Hmm. The original question more or less follows from Theorems 5.3.4 and 5.3.7 in Bjorner and Brenti. But I would still be interested in a proof or a counterexample to the conjecture. –  Qiaochu Yuan Jul 19 '10 at 3:02

1 Answer 1

up vote 4 down vote accepted

Hi, you may want to try to peruse these two papers:

Dyer, M. J. Hecke algebras and shellings of Bruhat intervals. Compositio Math. 89 (1993), no. 1, 91--115.

Dyer, M. J. Hecke algebras and shellings of Bruhat intervals. II. Twisted Bruhat orders. Kazhdan-Lusztig theory and related topics (Chicago, IL, 1989), 141--165, Contemp. Math., 139, Amer. Math. Soc., Providence, RI, 1992.

Dyer proves that Bruhat intervals are EL-shellable, and this gives the answer to your first question. As to your more general question, there may be answers to those as well (indirectly of course).

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