Weighted counting of Bruhat paths by shifted $R$-polynomials

TL;DR

This paper introduces shifted R-polynomials for finite Coxeter groups, applying them to weighted Bruhat path counting, establishing irregularity criteria, and providing bounds related to Jacobsthal numbers.

Contribution

It proposes shifted R-polynomials for all Bruhat intervals, offering new tools for weighted path counting and irregularity analysis in Coxeter groups.

Findings

Introduces shifted R-polynomials as Bruhat weights.

Provides a new irregularity criterion for lower intervals.

Establishes upper bounds of shifted R-polynomials using Jacobsthal numbers.

Abstract

We revisit $R$-polynomials with introducing the new idea ``shifted $R$-polynomials" (or Bruhat weight) for all Bruhat intervals in finite Coxeter groups. Then, we apply these polynomials to weighted counting of Bruhat paths. Further, we prove a new criterion of irregularity of lower intervals as analogy of Carrell-Peterson's and Dyer's results. Also, we present the upper bound of shifted $R$-polynomials for Bruhat intervals of fixed length by Jacobsthal numbers.

Tables

Table 1. Table 1. R 𝑅 R -polynomials in A 3 subscript 𝐴 3 A_{3} ; see Billey-Lakshmibai [ 3 , p.73]

$v$	$R_{ev}\left(q\right)$	$\left|v\right|$
1234	$1$	1
1243, 1324, 2134	$q-1$	1
1342, 1423, 2143, 2314, 3124	${\left(q-1\right)}^2$	1
1432, 3214	${\left(q-1\right)}^3+q\left(q-1\right)$	3
2341, 2413, 3142, 4123	${\left(q-1\right)}^3$	1
2431, 3241, 3412, 4132, 4213	${\left(q-1\right)}^4+q{\left(q-1\right)}^2$	3
4231	${\left(q-1\right)}^5+2q{\left(q-1\right)}^3+q^2\left(q-1\right)$	9
3421, 4312	${\left(q-1\right)}^5+2q{\left(q-1\right)}^3$	5
4321	${\left(q-1\right)}^6+3q{\left(q-1\right)}^4+q^2{\left(q-1\right)}^2$	11

Table 2. Table 2. dihedral polynomials and numbers

$n$	$d_n\left(q\right)$	$d_n$	$d_n^{\prime }$
$0$	1	1	0
1	$q$	1	1
2	$q^2$	1	2
3	$q^3+q^2+q$	3	6
4	$q^4+2q^3+2q^2$	5	14
5	$q^5+3q^4+4q^3+2q^2+q$	11	34
6	$q^6+4q^5+7q^4+6q^3+3q^2$	21	78
7	$q^7+5q^6+11q^5+13q^4+9q^3+3q^2+q$	43	178
8	$q^8+6q^7+16q^5+24q^4+22q^3+12q^2+4q$	85	398
$⋮$	$⋮$	$⋮$	$⋮$

Equations

$$\sum _{v\in [u,w]}{R}_{uv}(q)P_{vw}(q)=q^{\ell (u,w)}{P}_{vw}(q^{-1}).$$

$$[u,w]=\{v\in W\mid u\le v\le w\}.$$

$${\mathcal{P}}_w(q)=\sum _{v\le w}{q}^{\ell (v)}.$$

$${\mathcal{P}}_w(-1)=\sum _{v\le w}(-1{)}^{\ell (v)}=\{\begin{array}{ll}1& w=e,\\ 0& w\ne e.\end{array}$$

$$1+2(q+\cdots +q^{n-1})+q^n\le {\mathcal{P}}_w(q)\le (1+q{)}^n$$

$$|\{u\in [e,w]\mid \ell (u)=k\}|\ge 2$$

$$v=s_1\cdots \widehat{s_{i_1}}\cdots \widehat{s_{i_{n-k}}}\cdots s_n\text{(reduced)}.$$

$$\text{av}{\mathcal{P}}_w(q)=\frac{{\mathcal{P}}_w^{\prime }(1)}{{\mathcal{P}}_w(1)}=\frac{|E(w)|}{|V(w)|}.$$

$${\mathcal{P}}_w(q)=1+3q+5q^2+4q^3+q^4,$$

$$|V(w)|={\mathcal{P}}_w(1)=14,|E(w)|={\mathcal{P}}_w^{\prime }(1)=29,$$

$$\text{av}{\mathcal{P}}_w(q)=\frac{29}{14}>2=\frac{\ell (w)}{2}.$$

$$\mathbf{W}$$

$$\mathbf{M}$$

$$|f|=f(1),\Vert f\Vert =f^{\prime }(1),\text{av}(f)=\frac{\Vert f\Vert }{|f|},$$

$$\text{av}(fg)=\frac{(fg{)}^{\prime }(1)}{(fg)(1)}=\frac{f^{\prime }(1)g(1)+f(1)g^{\prime }(1)}{f(1)g(1)}=\frac{f^{\prime }(1)}{f(1)}+\frac{g^{\prime }(1)}{g(1)}=\text{av}(f)+\text{av}(g).$$

$$M(f)=|f|q^{\text{av}(f)}.$$

$$|M(f)|=|f|,\Vert M(f)\Vert =\Vert f\Vert ,\text{av}(M(f))=\text{av}(f).$$

$$M(M(f)+M(g))$$

$$=(|f|+|g|)q^{(f^{\prime }+g^{\prime })/(f+g)}$$

$$=(|f+g|)q^{\text{av}(f+g)}=M(f+g).$$

$$M(fg)=|f||g|q^{\text{av}(fg)}=|f||g|q^{\text{av}(f)+\text{av}(g)}=|f|q^{\text{av}(f)}|g|q^{\text{av}(g)}=M(f)M(g).$$

$${\mathcal{P}}_{4231}(q)=1+3q+5q^2+6q^3+4q^4+q^5=(1+q{)}^2(1+q+2q^2+q^3).$$

$$M({\mathcal{P}}_{4231}(q))=M(1+q{)}^{2}M(1+q+2q^2+q^3)=(2q^{1/2}{)}^2(5q^{8/5})=20q^{52/20}$$

$$\text{av}{\mathcal{P}}_{4231}(q)=\frac{52}{20}>\frac{50}{20}=\frac{5}{2}=\frac{\ell (4231)}{2}.$$

$$\Gamma :u=v_0\to v_1\to \cdots \to v_k=w$$

$$\chi (W^{\prime })=\{t^{\prime }\in T\mid T_L(t^{\prime })\cap W^{\prime }=\{t^{\prime }\}\}$$

$$r<rsr<\cdots <srs<s\text{ or }s<srs<\cdots <rsr<r.$$

$${\mathbf{x}}_i=\{\begin{array}{ll}\text{a}& \text{if ti<ti+1}\\ \text{b}& \text{if ti>ti+1},\end{array}\psi (\Gamma )={\mathbf{x}}_1\cdots {\mathbf{x}}_k\text{ and }{\Psi }_{uw}(\text{a},\text{b})=\sum _{\Gamma }\psi (\Gamma )$$

$${\Phi }_{uw}(\text{a}+\text{b},\text{a}\text{b}+\text{b}\text{a})={\Psi }_{uw}(\text{a},\text{b}).$$

$${\mathbf{x}}_i=\{\begin{array}{ll}\text{a}& \text{if ti<ti+1}\\ \text{b}& \text{if ti>ti+1},\end{array}\tilde{\psi }(\Gamma )={\mathbf{x}}_1\cdots {\mathbf{x}}_k\text{ and }{\tilde{\Psi }}_{uw}(\text{a},\text{b})=\sum _{\Gamma }\tilde{\psi }(\Gamma )$$

$${\tilde{\Phi }}_{uw}(\text{a}+\text{b},\text{a}\text{b}+\text{b}\text{a})={\tilde{\Psi }}_{uw}(\text{a},\text{b}).$$

$$R_{uw}(q)=\{\begin{array}{ll}{R}_{us,ws}(q)& \text{mbox}if\ell (us)<\ell (u),\\ (q-1)R_{u,ws}(q)+q{R}_{us,ws}(q)& \text{mbox}if\ell (u)<\ell (us).\end{array}$$

Full text

*Department of Engineering, Kanagawa University, Japan

1. 1 Introduction

2. 1.1 Kazhdan-Lusztig polynomials and $R$-polynomials

3. 1.2 Main results

4. 1.3 organization of this article

5. 2 Irregularity of Bruhat graphs

6. 2.1 preliminaries on Coxeter groups

7. 2.2 Boolean, dihedral posets and Poincaré polynomial

8. 2.3 Bruhat graphs

9. 2.4 example: 3412

10. 2.5 monomialization technique

11. 3 Weighted counting of Bruhat paths

12. 3.1 ab-, cd-index

13. 3.2 complete index

14. 3.3 $R$-polynomials

15. 3.4 $\widetilde{R}$-polynomials

16. 3.5 Bruhat size and total

17. 3.6 Bruhat weight for edges

18. 3.7 shifted $R$-polynomials

19. 3.8 sum of $R$-polynomials

20. 3.9 examples

21. 3.10 new criterion of irregularity of lower intervals

22. 3.11 higher Deodhar inequality

23. 4 Lower and upper bounds of shifted ${R}$-polynomials

24. 4.1 lower and upper bounds of $\widetilde{R}$-polynomials

25. 4.2 lower and upper bounds of shifted ${R}$-polynomials

26. 5 Concluding remarks

27. 5.1 double $R$-polynomials

28. 5.2 Bruaht size on Bruhat graph

29. 5.3 extension of higher Deodhar inequality

1. Introduction

1.1. Kazhdan-Lusztig polynomials and $R$-polynomials

The motivation of this article is to better understand Kazhdan-Lusztig (KL) polynomials which they introduced in 1979 [17]. This is a family of polynomials over nonnegative integer coefficients. Although these polynomials originated from representation theory of Coxeter groups, Hecke algebras and geometry of Schubert varieties, they have been an important topic in algebraic combinatorics as well since then. In particular, Bruhat intervals forms a nice subclass of Eulerian posets so that the framework of Eulerian posets ($f$-vector, ab-index, $\dots$) works well. Here, let us mention the five family of polynomials which play some role to investigate KL polynomials:

• •

$R$-polynomial

• •

$\widetilde{R}$-polynomial

• •

ab-, cd-index

• •

complete ab-, cd-index

• •

Poincaré polynomial

Among these polynomials, only $R$-polynomials have negative coefficients. However, $R$-polynomials satisfy some relations together with KL polynomials:

[TABLE]

Thus, it is crucial to better understand coefficients of $R$-polynomials as well. For this reason, we decided to revisit classical $R$-polynomials hoping to find some interpretation by nonnegative integers. Our idea is simple:we introduce shifted $R$-polynomials (or “Bruhat weight”); this is just shifting of its variable $q\mapsto q+1$. We will then show the connection between this shifted $R$-polynomials and $\widetilde{R}$-polynomials which have nonnegative coefficients so that we can discuss weighted counting of Bruhat paths.

1.2. Main results

Main results of this article are the following:

• •

Theorem 3.21: property of Bruhat weight for lower intervals

• •

Theorem 3.30: another criterion of irregularity of lower intervals

• •

Theorem 3.32: higher Deodhar inequality

• •

Theorem 4.3: the upper bound of shifted $R$-polynomials

• •

Corollary 4.7: the upper bound of Bruhat size of shifted $R$-polynomials by Jacobsthal (dihedral) numbers

Theorems 3.21, 4.3, Corollary 4.7 are new while we present Theorems 3.30, 3.32 as new interpretations of several known results (Carrell-Peterson, Dyer, the author).

1.3. organization of this article

Section 2 begins the topic with irregularity of Bruhat graph and Poincaré polynomials. Section 3 is all devoted to the main discussions on $R$-polynomials, $\widetilde{R}$-polynomials, shifted $R$-polynomials, and Bruhat weight for edges, Bruhat paths and intervals. Along the way, we provide many examples. Section 4 proves the upper bound of shifted $R$-polynomials as an analogy of the upper bound of $\widetilde{R}$-polynomials by Fibonacci polynomials. We end in Section 5 with recording several ideas for further development of our ideas.

2. Irregularity of Bruhat graphs

2.1. preliminaries on Coxeter groups

Throughout this article, we denote by $W=(W,S,T,\ell,\leq)$ a Coxeter system with $W$ the underlying Coxeter group, $S$ its Coxeter generators, $T$ the set of its reflections, $\ell$ the length function, $\leq$ Bruhat order. Moreover, assume that $W$ is finite. Unless otherwise noticed, $u,v,w,x,y$ are elements of $W$, $r,s\in S$, $t\in T$ and $e$ is the unit of $W$. The symbol $\ell(u,v)$ means $\ell(v)-\ell(u)$ for $u\leq v$. A Bruhat interval is a subposet of $W$ of the form

[TABLE]

By $f\leq g$ for polynomials $f,g\in\mathbf{N}[q]$, we mean $[q^{i}](f)\leq[q^{i}](g)$ for each $i$ where $[q^{i}](P(q))$ denotes the coefficient of $q^{i}$ in a polynomial $P(q)$.

2.2. Boolean, dihedral posets and Poincaré polynomial

The set of all Bruhat intervals forms a subclass of Eulerian posets. In particular, each lower interval $[e,w]$ is Eulerian graded by the length function $v\mapsto\ell(v)$.

Definition 2.1**.**

The Poincaré polynomial for $w$ is

[TABLE]

This is the rank generating function of $[e,w]$. Observe that

[TABLE]

There are two important classes of Eulerian posets: Boolean and dihedral. Let $B_{n}$ and $D_{n}$ denote the Boolean and dihedral poset of rank $n$, respectively; we understand that the Boolean or dihedral poset of rank 0 is the trivial poset. Note that $B_{n}=D_{n}$ for $n=0,1,2$ while $B_{n}\neq D_{n}$ for $n\geq 3$. These posets can be realized as Bruhat intervals (in fact, as lower intervals). Indeed, Boolean and dihedral intervals are “extremal” lower intervals in the following sense:

Proposition 2.2**.**

For any $w$ such that $\ell(w)=n\geq 1$, we have

[TABLE]

coefficientwise. In particular, $|D_{n}|=2n\leq|[e,w]|\leq 2^{n}=|B_{n}|$.

Proof.

Let $v\in[e,w]$ such that $0<\ell(v)=k<n$. Then there exist some $v_{0},v_{1}\in[e,w]$ such that $v_{0}<v<v_{1}$ and $\ell(v_{0},v)=\ell(v,v_{1})=1$ since $[e,w]$ is graded. Now $[v_{0},v_{1}]$ is an interval of length 2 and every such an interval in any Eulerian poset consists of exactly four elements. So there exists a unique $v^{\prime}$ such that $v_{0}<v^{\prime}<v_{1}$ and $v^{\prime}\neq v$. Thus we have

[TABLE]

which proves the first inequality. To show the second one, choose a reduced word $s_{1}\cdots s_{n}$ for $w$. For each $v\in[e,w]$ with $\ell(v)=k$, there is a reduced subword of this word for $v$ with $n-k$ simple reflections deleted:

[TABLE]

The number of such words is at most $\binom{n}{n-k}=\binom{n}{k}$. ∎

2.3. Bruhat graphs

Definition 2.3**.**

The Bruhat graph of $W$ is a directed graph for vertices $w\in W$ and for edges $u\to v$. For each subset $V\subseteq W$, we can also consider the induced subgraph with the vertex set $V$ (Bruhat subgraph). An edge $u\to v$ is short if $\ell(u,v)=1$. By $a(u,w)$ we mean the directed-graph-theoretic distance from $u$ to $w$.

We can make use of Poincaré polynomials even for edge counting on Bruhat graphs. Let $V(w)=[e,w]$ and $E(w)=\{u\to v\mid u,v\in[e,w]\}$ be the vertex and edge set of $[e,w]$, respectively. Observe that $|V(w)|=\mathcal{P}_{w}(1)$. What is more, each vertex $v\in[e,w]$ is incident to exactly $\ell(v)$ incoming edges so that $|E(w)|=\mathcal{P}_{w}^{\prime}(1)$ (where $\mathcal{P}^{\prime}_{w}(q)$ is the (formal) derivative of $\mathcal{P}_{w}(q)$). It follows from Proposition 2.2 that $2\ell(w)\leq|V(w)|\leq 2^{\ell(w)}$ and $\ell(w)^{2}\leq|E(w)|\leq\ell(w)2^{\ell(w)-1}$. In this way, $\mathcal{P}_{w}(q)$ contains subtle information on edges of Bruhat graphs on $[e,w]$.

Definition 2.4**.**

The average of $\mathcal{P}_{w}(q)$ is $\mathcal{P}_{w}^{\prime}(1)/\mathcal{P}_{w}(1)$.

Often, we write $\textnormal{av}\mathcal{P}_{w}(q)=\mathcal{P}_{w}^{\prime}(1)/\mathcal{P}_{w}(1)$. As seen above,

[TABLE]

Fact 2.5** (Carrell-Peterson [10]).**

The following are equivalent:

1. (1)

$\textnormal{av}\mathcal{P}_{w}(q)=\ell(w)/2$. 2. (2)

$[e,w]$ is regular.

Remark 2.6**.**

Carrell-Peterson (1994) assumed that the Kazhdan-Lusztig polynomial $P_{uw}(q)$ has nonnegative coefficients for all $u\leq w$. This is now (2019 at the time of writing) true due to Elias-Williamson [15] in 2014.

In fact, $B_{n}$ and $D_{n}$ are both regular. Equivalently, they have same average which is $n/2$.

2.4. example: 3412

Example 2.7**.**

Let $e=1234$ and $w=3412$ in the type $A_{3}$ Coxeter group. The lower interval $[e,w]$ consists of 14 vertices and 29 edges (Figure 1):

[TABLE]

[TABLE]

[TABLE]

Due to Carrell-Peterson, $[e,w]$ is an irregular graph; Precisely two of 14 vertices, $1234$ and $1324$, have degree 5 while all others have degree $4=\ell(w)$.

2.5. monomialization technique

If we are interested in only the average of a Poincaré polynomial (or more generally a polynomial over nonnegative integer coefficients), there is a useful technique to express it by a monomial as shown below. Let $\mathbf{N}$ denote the set of all nonnegative integers and $\mathbf{Q}_{\geq 0}$ the set of nonnegative rational numbers.

Definition 2.8**.**

[TABLE]

We call each element of $\mathbf{W}$ ($\mathbf{M}$) a weight (monomial weight).

For example, $q^{\ell(v)}$ is a monomial weight (we call it the Poincaré weight of $v$ for convenience).

Let $f\in\mathbf{W}$ with $f(1)\neq 0$ (i.e. $f\neq 0$). Define the size, total, average of $f$ by

[TABLE]

respectively. Set $|0|=\|0\|=0$ and let us not define $\textnormal{av}(0)$.

Proposition 2.9**.**

For all $f,g\in\mathbf{W}$, we have the following:

1. (1)

$|f+g|=|f|+|g|$. 2. (2)

$\|f+g\|=\|f\|+\|g\|$. 3. (3)

$\textnormal{av}(fg)=\textnormal{av}(f)+\textnormal{av}(g)$ ($f,g\neq 0$).

Proof.

We only confirm (3).

[TABLE]

∎

Definition 2.10**.**

Define the monomialization $M:\mathbf{W}\to\mathbf{M}$ as follows: set $M(0)=0$. For $f\neq 0$, define

[TABLE]

As we can easily see, the monomialization preserves size, total and average:

[TABLE]

Proposition 2.11**.**

For each $f,g\in\mathbf{W}$, all of the following are true:

1. (1)

$M(f+g)=M(M(f)+M(g))$. 2. (2)

$M(fg)=M(f)M(g)$. 3. (3)

If $f$ is a monomial, then $M(f)=f$. In particular, $M(M(f))=M(f)$.

Proof.

[TABLE]

[TABLE]

(3) is clear. ∎

Example 2.12**.**

Let $w=4231$ in $A_{3}$. Figure 2 shows that

[TABLE]

Then

[TABLE]

so that

[TABLE]

Again, due to Carrell-Peterson, [1234, 4231] is irregular.

Remark 2.13**.**

These examples above come from the characterization of irregular lower intervals in terms of pattern avoidance. Say a permutation $w$ of $\{1,2,\dots,n\}$ contains 3412 (4231) if there exist $i,j,k,l$ such that $i<j<k<l$ and $w(k)<w(l)<w(i)<w(j)$ ($w(l)<w(j)<w(k)<w(i)$); say $w$ is singular if it contains 3412 or 4231. Then, the following are equivalent:

1. (1)

$w$ is singular. 2. (2)

$[e,w]$ is irregular.

See Billey-Lakshmibai [3] for more details on this topic.

3. Weighted counting of Bruhat paths

Definition 3.1**.**

A Bruhat path is a directed path $\Gamma$ such as

[TABLE]

(Below, a “path” always means a directed path). Say $\Gamma$ is short (maximal) if all its edges are short; it is long otherwise. For each Bruhat path $\Gamma$ as above, we can consider two kinds of length: $k$ is the absolute length of $\Gamma$; $\ell(u,w)$ is the Coxeter length. We write $a(\Gamma)=k$ and $\ell(\Gamma)=\ell(u,w)$. By $x\overset{t}{\to}y$ we mean $x\to y$ and $y=xt$, $t\in T$.

Definition 3.2**.**

By a reflection subgroup of $W$, we mean an algebraic subgroup of $W$ generated by a subset of $T$.

Every reflection subgroup $W^{\prime}$ is itself a Coxeter system with the canonical generator

[TABLE]

where $T_{L}(t^{\prime})=\{t\in T\mid\ell(tt^{\prime})<\ell(t^{\prime})\}$. A reflection subgroup $W^{\prime}$ is dihedral if $|\chi(W^{\prime})|=2$.

Definition 3.3**.**

Let $<$ be a total order on $T$. Say $<$ is a reflection order if for all dihedral reflection subgroup $W^{\prime}$ of $W$ with $\chi(W^{\prime})=\{r,s\}$ $(r\neq s)$, we have

[TABLE]

3.1. ab-, cd-index

Let $\texttt{a},\texttt{b}$ be noncommutative variables and $\Gamma:u\to{v_{1}}{}\to\cdots\to{v_{k}}=w$ a short path. Define

[TABLE]

where the sum is taken over all short paths $\Gamma$ from $u$ to $w$. The ab-polynomial $\Psi_{uw}(\texttt{a},\texttt{b})$ is called the ab*-index* of $[u,w]$.

Fact 3.4**.**

$\Psi_{uw}(\texttt{a},\texttt{b})$ is a polynomial of $\texttt{a}+\texttt{b}$ and $\texttt{a}\texttt{b}+\texttt{b}\texttt{a}$. That is, there exists a unique noncommutative two-variable polynomial $\Phi_{uw}(\texttt{c},\texttt{d})$ such that

[TABLE]

The homogeneous cd-polynomial $(\deg\texttt{c}=1,\deg\texttt{d}=2)$ $\Phi_{uw}(\texttt{c},\texttt{d})$ is called the cd*-index* of $[u,w]$.

3.2. complete index

Let $\Gamma:u\to{v_{1}}{}\to\cdots\to{v_{k}}=w$ be a (not necessarily short) path. Similarly, define

[TABLE]

where the sum is taken over all paths $\Gamma$ from $u$ to $w$. Again, there exists a unique two-variable polynomial $\widetilde{\Phi}_{uw}(\texttt{c},\texttt{d})$ such that

[TABLE]

$\widetilde{\Phi}_{uw}(\texttt{c},\texttt{d})$ is called the complete cd*-index* of $[u,w]$.

Remark 3.5**.**

These indices do not depend on the choice of a reflection order.

Remark 3.6**.**

In 1990’s, the theory on ab-, cd-index for polytopes and Eulerian posets has been developed by many researchers such as Bayer, Fine, Klapper and Stanley, for example. Later Reading [20] proved (with Karu’s work) that all coefficients of cd-index for a lower interval $[e,w]$ is nonnegative: ${\Phi}_{ew}(\texttt{c},\texttt{d})\geq 0$. A complete index for a Bruhat interval is a more recent idea in 2010’s: See Billera [1], Billera-Brenti [2], Blanco [5] and Karu [16].

Conjecture 3.7**.**

1. (1)

Reading [20]:${\Phi}_{ew}(\texttt{c},\texttt{d})\leq{\Phi}_{B_{\ell(w)}}(\texttt{c},\texttt{d})$. 2. (2)

Billera-Brenti [2] strong conjecture: $\widetilde{\Phi}_{ew}(\texttt{c},\texttt{d})\leq{\Phi}_{B_{\ell(w)}}(\texttt{c},\texttt{d})$.

There is one demerit of such indices: From an ab- or a cd-monomial $\mathbf{x}=\mathbf{x}_{1}\cdots\mathbf{x}_{k}$ alone, we cannot recover the Coxeter length of a path. Unlike this, we will later on introduce a weight (Bruhat weight) which contains some information on both of absolute and Coxeter length of paths.

3.3. $R$-polynomials

Following Björner-Brenti [4], we introduce $R$-polynomials.

Fact 3.8**.**

There exists a unique family of polynomials $\{R_{uw}(q)\mid u,w\in W\}\subseteq\mathbf{Z}[q]$ ($R$-polynomials) such that

1. (1)

$R_{uw}(q)=0$ if $u\not\leq w$, 2. (2)

$R_{uw}(q)=1$ if $u=w$, 3. (3)

if $s\in S$ and $\ell(ws)<\ell(w)$, then

[TABLE]

Example 3.9**.**

$R$-polynomials involve many negative coefficients. For example, suppose $u\leq w$. We can show that

[TABLE]

It is tempting to say that coefficients of $R$-polynomials alternate in sign. However, Boe [7] found the following counterexample:

[TABLE]

We wish to understand $R$-polynomials as ones over nonnegative integer coefficients with some combinatorial interpretation. For this purpose, we have to mention Deodhar’s work [12] first: he showed that $R_{uw}(q)$ ($u\leq w$) is the sum of $q^{m}(q-1)^{n}$ with $m,n$ nonnegative integers:

[TABLE]

where $\mathcal{D}$ is the set of distinguished subexpressions $\underline{\sigma}$ of some fixed reduced expression $s_{1}\cdots s_{\ell(w)}$ for $w$ and $\pi$ is a certain map (which we do not need to discuss here). Shifting the variable $q$ to $q+1$, it is immediate to obtain a polynomial of nonnegative integer coefficients. On the other hand, there is an interesting property of $R$-polynomials as characteristic functions of a vertex and an edge at $q=1$ [4, Chapter 5, Exercise 35]:

[TABLE]

An easy guess is that $R$-polynomials are “counting something” implicitly in Bruhat graphs since vertices and edges are special cases of Bruhat paths of absolute length 0 and 1. Thus, it is natural to ask if $R$-polynomials somehow count paths of absolute length $\geq 2$. Further, if this is the case, then its weighting should be something like $q^{m}(q-1)^{n}$. We will see that this guess is right and make this point more explicit after discussing $\widetilde{R}$-polynomials and shifted $R$-polynomials.

Remark 3.10**.**

Caselli [11] also proved certain nonnegativity of $R$-polynomials. We have not found any concrete connection yet, though.

3.4. $\widetilde{R}$-polynomials

Next, following [4], we introduce another family of polynomials associated to $R$-polynomials. They have nonnegative integer coefficients:

Fact 3.11**.**

There exists a unique family of polynomials $\{\widetilde{R}_{uw}(q)\mid u,w\in W\}\subseteq\mathbf{N}[q]$ ($\widetilde{R}$-polynomials) such that

1. (1)

$\widetilde{R}_{uw}(q)=0$ if $u\not\leq w$, 2. (2)

$\widetilde{R}_{uw}(q)=1$ if $u=w$, 3. (3)

if $s\in S$ and $\ell(ws)<\ell(w)$, then

[TABLE] 4. (4)

$\widetilde{R}_{uw}(q)$ ($u\leq w$) is a monic polynomial of degree $\ell(u,w)$, 5. (5)

$R_{uw}(q)=q^{\ell(u,w)/2}\widetilde{R}_{uw}(q^{1/2}-q^{-1/2})$.

We remark that although $q^{1/2}$ and $q^{-1}$ appear in the definition above, $\widetilde{R}_{uw}(q)$ is indeed a polynomial in $q.$ To give a precise description of this family of polynomials, we need the following idea:

Definition 3.12**.**

Let $<$ be a reflection order and

[TABLE]

a path. Say $\Gamma$ is $<$-increasing if $t_{1}<t_{2}<\cdots<t_{k}$. We understand that any Bruhat path of absolute length 0 or 1 is $<$-increasing for all $<$.

Fact 3.13** (Dyer [14]).**

[TABLE]

where the sum is taken all over $<$-increasing paths $\Gamma$ from $u$ to $w$. Moreover, this sum does not depend on the choice of a reflection order.

3.5. Bruhat size and total

Lemma 3.14**.**

Let $u<w$, $a=a(u,w)$ and $\ell=\ell(u,w)$. Then there exist positive integers $\gamma_{\ell}\,(=1)$, $\gamma_{\ell-2},\dots,\gamma_{a}$ such that

[TABLE]

Consequently, we have

[TABLE]

Proof.

The first statement is a well-known property of $\widetilde{R}$-polynomials. As a result,

[TABLE]

∎

Hence shifting the variable by one,

[TABLE]

is a polynomial of nonnegative integer coefficients. It turns out that

[TABLE]

is an appropriate choice for a weight of $\Gamma$.

Definition 3.15**.**

Let

[TABLE]

be a Bruhat path. Define the Bruhat weight of $\Gamma$:

[TABLE]

In particular, $\rho(\Gamma)$ equals a monomial $q^{k}$ if $\Gamma$ is a short path of length $k$.

3.6. Bruhat weight for edges

Let us introduce a weight also for edges. The height of an edge $u\to v$ is $(\ell(u,v)+1)/2$. In particular, $u\to v$ is short if and only if its height is 1; otherwise it is long. Write $h(u\to v)=\displaystyle\frac{\ell(u,v)+1}{2}$.

Definition 3.16**.**

The Bruhat weight of an edge of height $h$ is $(q+1)^{h-1}q.$ For convenience, we use this symbol:

[TABLE]

This weighting is “multiplicative” in the following sense: If $\Gamma$ is $v_{0}\to v_{1}\to v_{2}\to\cdots\to v_{k-1}\to v_{k}$, then

[TABLE]

As we see, $\to_{h}$ is a polynomial of degree $h$. For example,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Remark 3.17**.**

Paths with the same absolute and Coxeter length have an identical weight: For example, $\to_{1}\to_{3}=(q+1)^{2}q^{2}=\to_{2}\to_{2}.$

3.7. shifted $R$-polynomials

Definition 3.18**.**

The shifted $R$-polynomial ($\overrightarrow{R}$-polynomial) for $(u,w)$ is

[TABLE]

In particular, for $u\not\leq w$, $\overrightarrow{R}_{uw}(q)=0$ and for $u\leq w$, it is monic of degree $\ell(u,w)$.

Definition 3.19**.**

The Bruhat size of $[u,w]$ is $|\overrightarrow{R}_{uw}(q)|(=R_{uw}(2))$. The Bruhat total of $[u,w]$ is $\|\overrightarrow{R}_{uw}(q)\|(=R^{\prime}_{uw}(2)).$ For convenience, we sometimes write

[TABLE]

Example 3.20** (Table 1).**

[TABLE]

Let us simply say $\overrightarrow{R}_{ev}(q)$ is the $\overrightarrow{R}$-polynomial of $v$. The Bruhat size of $v$ is $|v|=|\overrightarrow{R}_{ev}(q)|$ and the Bruhat total of $v$ is $\|v\|=\|\overrightarrow{R}_{ev}^{\prime}(q)\|$. For example, $|1234|=1,|3412|=3,|4231|=9,|4321|=11$ (Table 1).

Theorem 3.21**.**

1. (1)

$u\leq v\Longrightarrow|u|\leq|v|$. 2. (2)

$|v|$ is odd.

Lemma 3.22**.**

Let $f,g,h\in\mathbf{N}[q]$.

1. (1)

$|u|=2^{\ell(u)/2}\widetilde{R}_{eu}(2^{-1/2})$. 2. (2)

$g\leq h\Longrightarrow fg\leq fh\Longrightarrow(fg)(2^{-1/2})\leq(fh)(2^{-1/2})$. 3. (3)

$\widetilde{R}_{eu}(q)q^{\ell(u,v)}\leq\widetilde{R}_{ev}(q)$ if $u\leq v$. In particular, $\widetilde{R}_{eu}(2^{-1/2})\leq 2^{\ell(u,v)/2}\widetilde{R}_{ev}(2^{-1/2})$.

Proof.

1. (1)

Recall that

[TABLE]

Now let $q=1$.

[TABLE] 2. (2)

Suppose $g\leq h$. Then $h-g=\sum_{i=0}^{d}a_{i}q^{i}$ for some nonnegative integers $(a_{i})$. Obviously, $(fh-fg)(q)=f(q)\left({\displaystyle\sum_{i=0}^{d}a_{i}q^{i}}\right)$ and $q={2^{-1/2}}>0$ yields a nonnegative real number. 3. (3)

Blanco [6, Theorem 4] proved that $u\leq x\leq v$ $\Longrightarrow$ $\widetilde{R}_{ux}(q)\widetilde{R}_{xv}(q)\leq\widetilde{R}_{uv}(q)$. Let $u\mapsto e,x\mapsto u,v\mapsto v$ so that $\widetilde{R}_{eu}(q)\widetilde{R}_{uv}(q)\leq\widetilde{R}_{ev}(q)$. Together with (2) and $q^{\ell(u,v)}\leq\widetilde{R}_{uv}(q)$, we have

[TABLE]

Finally, set $q=2^{-1/2}$.

∎

Proof of Theorem 3.21.

1. (1)

Suppose $u\leq v$. With the Lemma above, we have

[TABLE] 2. (2)

Let $a=a(e,v)$, $\ell=\ell(v)$ and $\gamma_{j}=[q^{j}](\widetilde{R}_{ev}(q))$. Because $\gamma_{\ell}=1$, we see that

[TABLE]

is odd.

∎

3.8. sum of $R$-polynomials

Fact 3.23**.**

Bruhat order with a reflection order is Edge-Labeling shellable (EL-shellable) (Dyer [14]): let $<$ be an arbitrary reflection order.

1. (1)

For each $[u,w]$, there is a unique $<$-increasing short path from $u$ to $w$, say

[TABLE] 2. (2)

Moreover, $(t_{1},t_{2},\dots,t_{\ell(u,w)})\in T^{\ell(u,w)}$ is lexicographically first among all short paths from $u$ to $w$.

Consequently, for each $v$, there exists a unique $<$-increasing short path $\Gamma:e\to\dots\to v$ such that $\rho(\Gamma)=q^{\ell(v)}$ (corresponding to the Poincaré weight for $v$).

Definition 3.24**.**

Define the Bruhat-Poincaré polynomial for $w$:

[TABLE]

Thanks to EL-shellability, it splits into two parts:

[TABLE]

In particular, $\mathcal{P}_{w}(q)\leq\overrightarrow{\mathcal{P}_{w}}(q)$ and $\overrightarrow{\mathcal{P}_{w}}(-1)=\mathcal{P}_{w}(-1)+0=\mathcal{P}_{w}(-1)$.

3.9. examples

Example 3.25** (Figure 3).**

1. (1)

$B_{3}:$ Let $s_{1},s_{2},s_{3}$ be distinct simple reflections such that they all commute. Let $w=s_{1}s_{2}s_{3}$ so that $[e,w]\cong B_{3}$ as Bruhat graphs.

[TABLE] 2. (2)

$D_{3}=[e,w],w=s_{1}s_{2}s_{1},(s_{1}s_{2})^{3}=e$:

[TABLE]

Definition 3.26**.**

Say $[u,w]$ is Bruhat-Boolean if

[TABLE]

As seen above, $B_{3}$ and $D_{3}$ are both Bruhat-Boolean.

Fact 3.27** ([3, p.209]).**

The following are equivalent:

1. (1)

$[u,w]$ is regular. 2. (2)

Each upper subinterval $[v,w]$ of $[u,w]$ $(v\in[u,w])$ is Bruhat-Boolean.

In particular, if $[e,w]$ is regular, then $[e,w]$ itself must be Bruhat-Boolean:

[TABLE]

Observation 3.28**.**

The finite Coxeter group $W=[e,w_{0}]$ ($w_{0}$ longest element) is Bruhat-Boolean. In particular, $\sum_{v\in W}|v|=2^{\ell(w_{0})}$. This is because $W=[e,w_{0}]$ is $|\ell(w_{0})|$-regular and

[TABLE]

For example, $\sum_{v\in A_{4}}|v|=2^{6}=64$.

Example 3.29**.**

Let us see two intervals $[u,w]$ such that $u\to w$ and $\ell(u,w)=5$. One is regular and the other is irregular.

1. (1)

$D_{5}$ (Figure 3): Let $D_{5}=[e,w]$ with $w=s_{1}s_{2}s_{1}s_{2}s_{1},s_{i}\in S,(s_{1}s_{2})^{5}=e$. Also, let $u=e$ and $v_{i},v_{i}^{\prime}$ be two elements of level $i$ $(1\leq i\leq 4)$. Thus $\overrightarrow{R}_{uv_{i}}(q)=\overrightarrow{R}_{uv^{\prime}_{i}}(q)$ for such $i$ because of combinatorial invariance of $R$-polynomials for dihedral intervals (if $[u,w]\cong[x,y]$ as posets and they are both dihedral, then $R_{uw}(q)=R_{xy}(q)$ as a consequence of the combinatorial invariance of complete cd-index for dihedral intervals by Blanco [5, Lemma 3.2]). We can compute the following by induction (see also Table 2):

[TABLE]

Altogether, the Bruhat-Poincaré polynomial of $D_{5}$ is

[TABLE] 2. (2)

$[1234,4231]$ (Figure 2 and Table 1):

[TABLE]

Although $M\left({\overrightarrow{\mathcal{P}}_{4231}(q)}\right)=(2q^{1/2})^{3}(5q^{5/5})=40q^{100/40}$ (the average here is $100/40=5/2=\ell(w)/2$), $[1234,4231]$ is not Bruhat-Boolean and hence irregular; $\overrightarrow{\mathcal{P}}_{4231}(q)$ is “slightly larger” than $(1+q)^{5}$.

3.10. new criterion of irregularity of lower intervals

Recall that Carrell-Peterson proved if $\textnormal{av}(\mathcal{P}_{w}(q))=\displaystyle\frac{|E(w)|}{|V(w)|}\neq\displaystyle\frac{\ell(w)}{2}$, then $[e,w]$ is irregular. We can now generalize this result to $\textnormal{av}(\overrightarrow{\mathcal{P}}_{w}(q))=\displaystyle\frac{\sum_{v\leq w}\|v\|}{\sum_{v\leq w}|v|}$.

Theorem 3.30**.**

If $\textnormal{av}(\overrightarrow{\mathcal{P}}_{w}(q))\neq\displaystyle\frac{\ell(w)}{2}$, then $[e,w]$ is irregular.

Proof.

Suppose $[e,w]$ is regular. Then, all upper subintervals of $[e,w]$ are Bruhat-Boolean. In particular,

[TABLE]

Therefore, $\textnormal{av}(\overrightarrow{\mathcal{P}_{w}}(q))=\textnormal{av}((1+q)^{\ell(w)})=\displaystyle\frac{\ell(w)}{2}$. ∎

Example 3.31**.**

[TABLE]

[TABLE]

Clearly,

[TABLE]

Therefore, apart from the characterization of singular permutations, [1234, 3412] is irregular (note: $\textnormal{av}\overrightarrow{\mathcal{P}}_{3412}(q)<2$ while $\textnormal{av}{\mathcal{P}}_{3412}(q)>2$).

3.11. higher Deodhar inequality

For each $[u,w]$, define the following integer sequence $(\widetilde{f}_{i})$: For each $i$ with $0\leq i\leq\ell(u,w)$, let

[TABLE]

where $[q^{i}](P(q))$ denotes the coefficient of $q^{i}$ in the polynomial $P(q)$. Clearly, $\widetilde{f}_{0}=1$ since the only $v=u$ term contributes to the constant term and $\overrightarrow{R}_{uu}(q)=1$. What about

[TABLE]

Recall that the weight of a Bruhat path $\Gamma$ is $(q+1)^{(\ell(\Gamma)-a(\Gamma))/2}q^{a(\Gamma)}$; only the weight of Bruhat paths involving $q$-term is one for length 1 (i.e. an edge) with

[TABLE]

Denoting by $\textnormal{out}_{w}(u)$ the out-degree of $u$ in $[u,w]$, that is,

[TABLE]

we have

[TABLE]

Thanks to Deodhar inequality (Dyer [13]), there is the simple lower bound of $\widetilde{f}_{1}$ as

[TABLE]

In fact, this $\geq$ is strict if and only if $[u,w]$ is irregular. Next, it is natural to ask about $q^{2}$-term: Paths $\Gamma$ whose weight involving $q^{2}$-term are only ones of absolute length 1 or 2. Those weights are of the form

[TABLE]

where $\ell=\ell(\Gamma)$. Note that

[TABLE]

This leads us to some weighted counting of edges and paths of length 2. Put

[TABLE]

so that $\widetilde{f}_{2}=p_{1}+p_{2}$.

Theorem 3.32** (higher Deodhar inequality).**

For all $[u,w]$, we have

[TABLE]

Moreover, if $\widetilde{f}_{2}\gneq\binom{\ell(u,w)}{2}$, then $[u,w]$ is irregular.

Proof.

This is a consequence of Kobayashi [18, Theorem 6.2] ($x=u$ case). ∎

Example 3.33**.**

Let $u=1234,w=3412$.

[TABLE]

Therefore,

[TABLE]

Again, apart from pattern avoidance, we can now say that $[1234,3412]$ is irregular.

4. Lower and upper bounds of shifted ${R}$-polynomials

In this section, we will prove Theorem 4.3 on the sharp lower and upper bounds of $\overrightarrow{R}$-polynomials.

4.1. lower and upper bounds of $\widetilde{R}$-polynomials

First, let us review several results on $\widetilde{R}$-polynomials proved by Brenti [8, Theorem 5.4, Corollary 5.5, Theorem 5.6].

Fact 4.1**.**

Let $u\leq v$.

1. (1)

$u\leq x\leq v$ $\Longrightarrow$ $q^{\ell(x,v)}\widetilde{R}_{ux}(q)\leq\widetilde{R}_{uv}(q)$. 2. (2)

Suppose $W$ is finite. Then $u\leq x\leq y\leq v$ $\Longrightarrow$ $q^{\ell(u,x)+\ell(y,v)}\widetilde{R}_{xy}(q)\leq\widetilde{R}_{uv}(q)$. 3. (3)

Let $x\leq y$ in a weak order and $y\leq z$. Then, $q^{\ell(x,y)}\widetilde{R}_{yz}(q)\leq\widetilde{R}_{xz}(q)$.

These inequalities all follow from the simple fact that $\widetilde{R}_{uw}(q)=q^{\ell(u,w)}$ whenever $[u,w]$ is Boolean; in addition, since each $\widetilde{R}_{uw}(q)$ is either 0 or monic of degree $\ell(u,w)$, $q^{n}$ is the least polynomial among

[TABLE]

in coefficientwise order.

On the other hand, Fibonacci polynomials ($F_{0}(q)=1,F_{1}(q)=q,F_{2}(q)=q^{2},F_{n}(q)=qF_{n-1}(q)+F_{n-2}(q)$ for $n\geq 3$) give an upper bound of such polynomials. In fact, this upper bound is also best possible: $F_{n}(q)$ is the $\widetilde{R}$-polynomial for any dihedral interval of rank $n$ (Brenti [9, Proposition 5.3]). Together, there always holds

[TABLE]

for $[u,w]$ such that $\ell(u,w)=n$. Now it is reasonable to ask what corresponds to these inequalities for shifted $R$-polynomials.

4.2. lower and upper bounds of shifted ${R}$-polynomials

Define a sequence of polynomials $(d_{n}(q))_{n=0}^{\infty}$ by $d_{0}(q)=1,d_{1}(q)=q,d_{2}(q)=q^{2}$ and

[TABLE]

Call $(d_{n}(q))_{n=0}^{\infty}$ dihedral polynomials.

It is easy to see that $d_{n}(q)$ is a monic polynomial of degree $n$ and morerover it is a weight: $d_{n}(q)\in\mathbf{W}$. Let $d_{n}=|d_{n}(q)|,d^{\prime}_{n}=\|d_{n}(q)\|$ denote its size and total (Table 2).

Lemma 4.2**.**

If $[u,w]$ is dihedral, then $\overrightarrow{R}_{uw}(q)=d_{\ell(u,w)}(q)$.

Proof.

Induction on $n=\ell(u,w)$. The cases for $n=\ell(u,w)\leq 2$ coincide with Boolean ones: $\overrightarrow{R}_{uw}(q)=q^{\ell(u,w)}=d_{\ell(u,w)}(q)$. Now suppose $n=\ell(u,w)\geq 3$. Thanks to the combinatorial invariance of $R$-polynomials for dihedral intervals, we may assume that $[u,w]$ is dihedral, $\ell(us)>\ell(u)$ and $\ell(ws)<\ell(w)$ for some $s\in S$:

[TABLE]

that is,

[TABLE]

The inequality $us<ws$ now holds since

[TABLE]

(in a dihedral interval, $x<y\iff\ell(x)<\ell(y)$). It follows from the property of dihedral intervals that subintervals $[us,ws]$ and $[u,ws]$ are also dihedral posets of length $n-1,n-2$, respectively. By inductive hypothesis, $R$-polynomials of those are $d_{n-1}(q)$ and $d_{n-2}(q)$ so that

[TABLE]

∎

Theorem 4.3**.**

Let $[u,w]$ be a Bruhat interval such that $\ell(u,w)=n\geq 1$. Then,

[TABLE]

Moreover,

[TABLE]

We give a proof after three lemmas.

Lemma 4.4**.**

Let $x<y$. Then, there exist some $x^{\prime},y^{\prime}$ such that $x^{\prime}\leq x,$ $y^{\prime}\leq y,$ $R_{x^{\prime}y^{\prime}}(q)=R_{xy}(q)$ and $\ell(x^{\prime}s)>\ell(x^{\prime}),\ell(y^{\prime}s)<\ell(y^{\prime})$ for some $s\in S$.

Proof.

Suppose $x<y$. We know that $\ell(y)\geq 1$ implies $\ell(ys)<\ell(y)$ for some $s\in S$. If further $\ell(xs)>\ell(x)$, then we are done. Otherwise, $\ell(xs)<\ell(x)$. Let $x_{1}=xs$, $y_{1}=ys$ ($R_{x_{1},y_{1}}(q)=R_{xy}(q)$). Now ask if there exists $s_{1}\in S$ such that $\ell(y_{1}s_{1})<\ell(y_{1})$ and $\ell(x_{1}s_{1})>\ell(x_{1})$. If this is the case, then we are done. Otherwise, let $x_{2}=x_{1}s_{1}$, $y_{2}=y_{1}s_{1}\dots$. This algorithm will end at most $\ell(x)$ steps since $\ell(x)>\ell(x_{1})=\ell(x)-1>\cdots>\ell(e)=0$ and $\ell(es)>\ell(e)$ for all $s$. ∎

Lemma 4.5**.**

If $f\leq g\leq h$ in $\mathbf{N}[q]$, then $f^{\prime}\leq g^{\prime}\leq h^{\prime}$.

Proof.

If $f\leq g$, then $[q^{i}](f^{\prime})=(i+1)[q^{i+1}](f)\leq(i+1)[q^{i+1}](g)=[q^{i}](g)$ for each $i$ which means $f^{\prime}\leq g^{\prime}$. The same is true for $g$ and $h$. ∎

To find out a closed formula for $d_{n}(q)$, we take the formal power series method.

Lemma 4.6**.**

[TABLE]

Proof.

We wish to find

[TABLE]

First, let us compute $D^{\geq 3}_{q}(z):=\displaystyle\sum_{n\geq 3}d_{n}(q)z^{n}.$

[TABLE]

Thus,

[TABLE]

and

[TABLE]

∎

Proof of Theorem 4.3.

It is easy to check for $n=1,2$. Suppose that $n=\ell(u,w)\geq 3$. By Lemma 4.4, we may assume that $\ell(us)>\ell(u),\ell(ws)<\ell(w)$ for some $s$ and

[TABLE]

By inductive hypothesis, the upper bounds for $\overrightarrow{R}$-polynomials of length $\ell(u,ws)=n-1,\ell(us,ws)=n-2$ intervals are $d_{n-1}(q)$ and $d_{n-2}(q)$ so that

[TABLE]

For the second inequalities, just differentiate this as in Lemma 4.5. Finally, Lemma 4.6 implies the last part as follows:

[TABLE]

and hence we conclude that

[TABLE]

for $n\geq 1$. ∎

Corollary 4.7**.**

Let $[u,w]$ be a Bruhat interval such that $n=\ell(u,w)\geq 1$. Then

[TABLE]

Moreover,

[TABLE]

The sequence $(J_{n})_{n=0}^{\infty}$ with $J_{0}=0,J_{1}=1$ and

[TABLE]

is known as Jacobsthal sequence (The On-line Encyropedia of Integer Sequences A001045 [19]) in combinatorics and number theory. The only difference between $J_{n}$ and our $d_{n}$ is the initial value: $d_{0}=1\neq 0=J_{0}$. For $n\geq 1$,

[TABLE]

5. Concluding remarks

We end with recording several ideas for our future research.

5.1. double $R$-polynomials

Let $p,q$ be commutative variables.

Definition 5.1**.**

Define the double $R$-polynomial for $(u,w)$ by

[TABLE]

where $(\gamma_{j})$ are positive integers such that

[TABLE]

as in Lemma 3.14.

Many polynomials in this articles are disguises of this double $R$-polynomials.

Observation 5.2**.**

[TABLE]

Question 5.3**.**

What is the recurrence of double $R$-polynomials?

5.2. Bruaht size on Bruhat graph

We proved that $u\leq v\Longrightarrow|u|\leq|v|$. Since Bruhat order is the transitive closure of edge relations, it is reasonable to ask this:

Question 5.4**.**

Suppose $u\to v$. When $|u|\lneq|v|$ and when not?

It is probably the easiest to try the type A case first.

5.3. extension of higher Deodhar inequality

We showed that for each interval $[u,w]$, we have

[TABLE]

for $i=0,1,2$. We do not know if the similar inequalities hold for all $i\geq 3$. If this is the case, then we always have

[TABLE]

which looks very nice. Prove or disprove it.