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I have read in a number of places that the lower Bruhat interval $[e, w]$ is rank-symmetric if and only if the KL-polynomial $P_{e, w}(q) = 1$. All of the proofs I've come across use "rationally smooth Schubert varieties", which I don't really understand.

The KL polynomials can be defined purely in terms of the Iwahori-Hecke algebra of the Coxeter group, and satisfy a number of identities involving sums over Bruhat intervals. I would like to know then if there is a more direct way to prove that $[e, w]$ is rank symmetric iff $P_{e, w}(q) = 1$, using only the Hecke algebra (and Bruhat order).

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Have you tried looking for a counterexample in a non-Weyl group? – Qiaochu Yuan Jul 13 '12 at 1:55
@Vlad: It would help to add a reference or two (maybe the 2009 Annals paper by Bjorner-Ekedahl?) and/or a bit more background. There are other open problems about non-crystallographic Coxeter groups and KL polynomials, e.g., the KL conjecture that coefficients of the polynomials should be non-negative. In the crystallographic case they can be interpreted as dimensions of cohomology groups. Alvis checked by computer the $H_3, H_4$ groups, while Soergel has proposed an ambitious program to show non-negativity without algebraic geometry. Your problem also looks very hard. – Jim Humphreys Jul 13 '12 at 16:02
@Qiaochu: I've checked it by computer for F4, H3, H4, A1-7, B1-6, D1-6, E6, E7. Not sure what else I'd check, aside from larger groups (which tend to take forever and cause memory overflows). @Jim: Most of what I know about Coxeter groups and Bruhat order is from Bjorner and Brenti's "Combinatorics of Coxeter Groups", and my main source for Hecke Algebra and KL-polynomials is James Humphreys' "Reflection Groups and Coxeter Groups". – Vlad Firoiu Jul 13 '12 at 17:13
@Vlad: To your question about Jim, yes they are the same person. To speculate, I doubt this is any easier than nonnegativity. Perhaps it might be possible to prove this assuming nonnegativity, but of course that is cheating since nonnegativity also requires geometry to prove. And, to give advice, I think you get further in this subject if you at least know enough of the algebraic geometry to understand the basic ideas coming from that side of the subject, even if you can't master the machinery well enough to use it to prove anything. The same can probably be said of the rep theory. – Alexander Woo Jul 13 '12 at 23:05
@Christian: I'm using a C++ program called "coxeter", written a while ago by Fokko du Cloux (…). The code is a bit outdated (doesn't use standard C++ data structures) but otherwise pretty clean and fast. – Vlad Firoiu Jul 14 '12 at 17:37

The statement does indeed hold in type H3; you can find the computation at The machine is still running (and I don't know how long it might take) doing type H4.

So, if no Coxeter theoretic proof is known, there is hope to find one...

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@Christian: This kind of experimentation for finite Coxeter groups is certainly essential, though $H_4$ is much bigger than $H_3$. (Be patient.) – Jim Humphreys Jul 13 '12 at 16:06
Running a few quick tests shows that the statement holds in A7, B6, D6, H4, E6, and E7. Larger groups overflow my memory :( – Vlad Firoiu Jul 13 '12 at 17:00
@Christian (and Vlad): Anyone concerned about Lie-theoretic computation should be aware of the software versions posted at the Atlas of Lie Groups site Fokko du Cloux was an essential part of the original team, but he died prematurely of ALS. – Jim Humphreys Jul 16 '12 at 15:01

The answer to the question is "yes", allowing for a generous interpretation of "direct way". This will follow from the recently posted work of Ben Elias and Geordie Williamson on non-negativity of coefficients of Kazhdan-Lusztig polynomials for an arbitrary Coxeter group here.

See the Update to my MO question here, which refers to the 1991 conference report by Jim Carrell (with Dale Peterson): in the first section, the equivalence you want is formulated for an arbitrary Coxeter group under the hypothesis that coefficients of relevant K-L polynomials are all non-negative. (This may be one of the sources you are referring to.)

At first I had overlooked this type of answer to my own question. (I'm still looking for other consequences of the non-negativity theorem, of course, but this one is interesting.) Note that for general Coxeter groups, one needs an approach which doesn't involve the geometry of Schubert varieties. What Elias and Williamson seem to do is avoid all that algebraic geometry by providing a sophisticated substitute.

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"Singular Loci of Schubert Varieties" by Billey and Lakshmibai is by far the best reference I've found so far for this question. Chapter 6 in particular deals with the combinatorial consequences of $P_{x, w}(q) = 1$.

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@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.) – Jim Humphreys Jul 20 '12 at 19:23
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. – Vlad Firoiu Jul 24 '12 at 19:54

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