Let $(W,S)$ be a Coxeter system with length function $\ell$ and $T=\bigcup_{w\in W}wSw^{-1}$.

Set

$N(u,v):=\{t\in T: u< tu \le v\}$, $\overline{\ell}(u,v):=|N(u,v)|$,

$\ell(u,v):=\ell(v)-\ell(u)$ and $df(u,v):=\overline{\ell}(u,v)-\ell(u,v)$.

Following [1], we call $df(u,v)$ the defect of the interval $[u,v]$.

By [2, Theorem C] and the fact that coefficients of Kazhdan-Lusztig polynomials are nonnegative, we have the following:

Let $x,w\in W$ and $x\le w$. Then we have the following equivalent statements:

$P_{x,w}(q)=1$.

$P_{y,w}(q)=1$ for all $y\in [x,w]$.

$df(y,w)=0$ for all $y\in [x,w]$.

Conjecture 1: $df(x,w)=0$ $\iff$ $df(y,w)=0$ for all $y\in [x,w]$.

There are two evidences to support my conjecture 1.

First, as noted in [1], there are similarities between $P_{x,w}(1)$ and $df(x,w)$ for $x\le w$.

Suppose $ws<w$ and $rw<w$. We have $P_{xs,w}(1)=P_{x,w}(1)=P_{rx,w}(1)$ and $df(xs,w)=df(x,w)=df(rx,w)$.

We have $P_{x,w}(1)\ge 1$ and $df(x,w)\ge 0$.

Now the equivalence (1)$\iff$(2) suggests that the conjecture may be true.

Apart from that, it is well-known that $[q^i](P_{x,w}(q))\ge [q^i](P_{y,w}(q))$ for all $y\in [x,w]$.

This implies that $P_{x,w}(1)\ge P_{y,w}(1)$ for all $y\in [x,w]$.

This suggests my second conjecture:

Conjecture 2: $df(x,w)\ge df(y,w)$ for all $y\in [x,w]$.

Clearly, conjecture 2 is true would imply conjecture 1 is true.

Second, the conjecture 1 is true for the case $W$ is a Weyl group of type $A$.

Let $e_x=x\cdot B$.

Theorem 1: $X_w$ is smooth at $e_x\iff \dim X_w=\dim T_{e_x} X_w$.

*Proof of Theorem 1:* See [3, Remark 13.1.4].

Theorem 2: $\dim X_w=\ell(w)$.

*Proof of Theorem 2:* See [3, Remark after Definition 10.8.1].

Theorem 3: $X_w$ is rationally smooth at $e_x\iff df(y,w)=0$ for all $y\in [x,w]$.

*Proof of Theorem 3:* See [4, Section 13.2].

Theorem 4: $X_w$ is smooth at $e_x \implies X_w$ is rationally smooth at $e_x$.

*Proof of Theorem 4:* See [4, Chapter 6].

Theorem 5: Suppose $W$ is a Weyl group of type A. Then $\dim T_{e_x} X_w=|\{r\in T: r x\le w\}|$.

*Proof of Theorem 5:*

By [3, Theorem 13.2.6], we get $\dim T_{e_x} X_w=|\{s_\alpha: \alpha\in x\Phi^+, s_\alpha x\le w\}|$.

It suffices to show $\{s_\alpha: \alpha\in x\Phi^+, s_\alpha x\le w\} =\{s_\alpha\in T: s_\alpha x\le w\}$. This follows from $\varphi:x\Phi^+\cong T$ defined by $\varphi(\alpha)=s_\alpha$.

Theorem 6: Suppose $W$ is a Weyl group of type $A$. Then $df(x,w)=0\implies df(y,w)=0$ for all $y\in [x,w]$.

*Proof of Theorem 6:*

Suppose $df(x,w)=0$. Then $|\{r\in T: x<rx\le w\}|=\ell(w)-\ell(x)$. Note that $\ell(x)=|\{r\in T: rx<x\}|$. This implies that $\ell(w)=|\{r\in T: rx\le w\}|$. Then $\dim X_{w}=\dim T_{e_x}X_w$ by Theorems 2,5. By Theorem 1, we get $X_w$ is smooth at $e_x$. By Theorem 4, we get $X_w$ is rationally smooth at $e_x$. By Theorem 3, we get $df(y,w)=0$ for all $y\in [x,w]$. The claim follows.

How to prove/disprove Conjectures 1 and 2?

[1]: Masato Kobayashi, **Combinatorics on Bruhat Graphs and
Kazhdan-Lusztig Polynomials**, Pure Mathematical Sciences, Vol. 3, 2014, no. 1, 23 - 33

[2]: J. Carrell, **The Bruhat graph of a Coxeter group, a conjecture of Deodhar
and rational smoothness of Schubert varieties**, Proc. Symp. Pure Math.,
56 (1994), 53-61.

[3]: Lakshmibai, Venkatramani, and Justin Brown. **Flag varieties: an interplay of geometry, combinatorics, and representation theory.** Vol. 53. Springer, 2018.

[4]: Sarason, I. G., Billey, S., Sarason, S., & Lakshmibai, V. (2000). **Singular loci of Schubert varieties** (Vol. 182). Springer Science & Business Media.