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).

Annalspaper 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. $\endgroup$4more comments