Bruhat order and graph structures of lower intervals in Coxeter groups

TL;DR

This paper investigates the structure of lower Bruhat intervals in Coxeter groups, revealing they are unions of two-sided cosets and introducing quotient structures to analyze their properties.

Contribution

It introduces the concepts of quotient lower intervals and quotient Bruhat graphs, providing new tools to understand Coxeter group interval structures.

Findings

Lower Bruhat intervals are unions of two-sided cosets

Quotient lower intervals and quotient Bruhat graphs are introduced

Structural descriptions in terms of graphs and posets are provided

Abstract

We show that any lower Bruhat interval in a Coxeter group is a disjoint union of certain two-sided cosets as a consequence of Lifting Property and Subword Property. Furthermore, we describe these details in terms of Bruhat graphs, graded posets, and two-sided quotients altogether. For this purpose, we introduce some new ideas, quotient lower intervals and quotient Bruhat graphs.

Tables

Table 1. Table 1. C ​ ( w ) 𝐶 𝑤 C(w) for w = 3412 𝑤 3412 w=3412

$C$	$P^{\uparrow }\left(C\right)$	$\ell \left(C\right)$	$P_{\downarrow }\left(C\right)$	${\text{mid}}_w\left(C\right)$	${\text{side}}_w\left(C\right)$
$\{3412,3142,2413,2143\}$	3412	4	2143	2	2
$\{3214,2314,3124,2134\}$	3214	3	2134	1	2
$\{1432,1342,1423,1243\}$	1432	3	1243	1	2
$\{1324,1234\}$	1324	1	1234	0	1

Table 2. Table 2. C ​ ( w ) 𝐶 𝑤 C(w) for w = 45312 𝑤 45312 w=45312

$P^{\uparrow }\left(C\right)$	$P_{\downarrow }\left(C\right)$	$\ell \left(C\right)$	${\text{mid}}_w\left(C\right)$	${\text{side}}_w\left(C\right)$
$45312=s_2{s}_3{s}_2\left|s_1{s}_4\right|s_2{s}_3{s}_2$	$s_1{s}_4$	8	2	6
$43215=s_2{s}_3{s}_2\left|s_1\right|s_2{s}_3$	$s_1$	6	1	5
$15432=s_2{s}_3{s}_2\left|s_4\right|s_3{s}_2$	$s_4$	6	1	5
$14325=s_2{s}_3{s}_2\left|e\right|e$	$e$	3	0	3

Table 3. Table 3. C ​ ( w ) 𝐶 𝑤 C(w) for w = 52341 𝑤 52341 w=52341

$P^{\uparrow }\left(C\right)$	$P_{\downarrow }\left(C\right)$	$\ell \left(C\right)$	${\text{mid}}_w\left(C\right)$	${\text{side}}_w\left(C\right)$
$52341=s_1{s}_4\left|s_2{s}_3{s}_2\right|s_1{s}_4$	$s_2{s}_3{s}_2$	7	3	4
$52143=s_1{s}_4\left|s_3{s}_2\right|s_1{s}_4$	$s_3{s}_2$	6	2	4
$32541=s_1{s}_4\left|s_2{s}_3\right|s_1{s}_4$	$s_2{s}_3$	6	2	4
$21543=s_1{s}_4\left|s_3\right|s_4$	$s_3$	4	1	3
$32154=s_1\left|s_2\right|s_1{s}_4$	$s_2$	4	1	3
$21354=s_1\left|e\right|s_4$	$e$	2	0	2

Equations

$$u=v_0\to v_1\to v_2\to \cdots \to v_k=w.$$

$$\ell (u,w)\stackrel{\text{def}}{=}\ell (w)-\ell (u)$$

$$[u,w]\stackrel{\text{def}}{=}\{v\in W\mid u\le v\le w\}$$

$$B(w)\stackrel{\text{def}}{=}[e,w]=\{v\in W\mid v\le w\}.$$

$$V=W_{I}u{W}_J\stackrel{\text{def}}{=}\{xuy\mid x\in W_I,y\in W_J\}$$

$$W_{I}u{W}_{\varnothing }=W_{I}u\text{ and }{W}_{\varnothing }u{W}_J=u{W}_J.$$

$${}^{\varnothing }{W}^J=W^J\text{ and }{}^I{W}^{\varnothing }={}^{I}W.$$

$$v$$

$$\ell (v)$$

$$v$$

$$\ell (v)$$

$$\ell (v_0)\le \ell (v)\le \ell (v_1)$$

$$u<w,\ell (u,w)\ge 2$$

$$B(w)=\bigcup _{u\in B(w)}{C}_w(u).$$

$$D_L(w)\subseteq D_L(u)\text{ and }{D}_R(w)\subseteq D_R(u).$$

$$D_L(w)=D_L(u)=\{s_2\},D_R(w)=D_R(u)=\{s_2\}.$$

$$D_L(w)=D_L(u)=\{s_2,s_3\},D_R(w)=D_R(u)=\{s_2,s_3\}.$$

$$B^{\uparrow }(w)$$

$$B_{\downarrow }(w)$$

$$P^{\uparrow }(u)=\max C_w(u),P_{\downarrow }(u)=\min C_w(u).$$

$$u\le v⟹P^{\uparrow }(u)\le P^{\uparrow }(v)⟺P_{\downarrow }(u)\le P_{\downarrow }(v).$$

$$P^{\uparrow }(C)=\max C,P_{\downarrow }(C)=\min C.$$

$$(C(w),\le )\cong (B^{\uparrow }(w),\le )\cong (B_{\downarrow }(w),\le )$$

$${\text{mid}}_w(u)=\ell (P_{\downarrow }(u)).$$

$${\text{side}}_w(u)=\ell (u)-\ell (P_{\downarrow }(u)).$$

$$u\le v⟹{\text{mid}}_w(u)\le {\text{mid}}_w(v).$$

$$u\le v,u{\sim }_{w}v⟹{\text{side}}_w(u)\le {\text{side}}_w(v))$$

$${\text{mid}}_w(C)$$

$${\text{side}}_w(C)$$

$$\ell (C)$$

$$C\le D⟹{\text{mid}}_w(C)\le {\text{mid}}_w(D).$$

$$C\le D⟹{\text{side}}_w(C)\le {\text{side}}_w(D)$$

$$B^{\uparrow }(w)$$

Full text

1. 1 Preliminaries

2. 1.1 Bruhat graph

3. 1.2 main results

4. 2 Quotient lower interval

5. 2.1 two-sided cosets and quotients

6. 2.2 Bruhat cosets

7. 2.3 quotient lower interval

8. 2.4 side, middle length

9. 2.5 subposets in a graded poset

10. 2.6 main theorem 1

11. 3 Quotient Bruhat graph

12. 3.1 separated element

13. 3.2 quotient Bruhat graph

14. 3.3 main theorem 2

15. 3.4 example 1: 45312

16. 3.5 example 2: 52341

17. 4 Future research

18. A Basics on Coxeter groups

19. A.1 reduced words, weak orders, subword property

20. A.2 one-sided cosets and quotients

1.

Preliminaries

Throughout this article, we denote by $W=(W,S,T,\ell)$ a Coxeter system with $W$ the underlying Coxeter group and $S$ its Coxeter generators. By $T$ we mean the set of its reflections; $\ell$ is the length function. Unless otherwise noticed below, symbols $u,v,w,x,y$ are elements of $W$, $s\in S$, $t\in T$, $e$ is the group-theoretic unit of $W$ and $I,J$ are subsets of $S$. See Appendix for more details on those definitions and notation.

1.1. Bruhat graph

By $u\to v$ we mean $\ell(u)<\ell(v)$ and $v=tu$ for some $t\in T$. Define Bruhat order $u\leq w$ on $W$ if there is a directed path

[TABLE]

For convenience, we use notation

[TABLE]

whenever $u\leq w$.

Definition 1.1**.**

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).

1.2. main results

Any subset $V$ of $W$ in the form

[TABLE]

is called a Bruhat interval. In this article, we mainly study lower intervals:

[TABLE]

Although some results on such intervals are known in the books [1, 2], there are still many unsolved problems.

Our main results (Theorems 2.17 and 3.9) give a new description of poset structures $B(w)$ from aspects of Bruhat graphs and two-sided cosets/quotients.

2. Quotient lower interval

2.1. two-sided cosets and quotients

First, we discuss two-sided cosets and quotients, a little unfamiliar topic in the context of Coxeter groups; for one-sided cosets and quotients, see Appendix.

Let $V$ be a subset of $W$.

Definition 2.1**.**

Say $V$ is a two-sided coset if

[TABLE]

for some $u,I,J$.

Every one-sided coset is a special case of two-sided cosets:

[TABLE]

Definition 2.2** (two-sided quotient).**

${\vphantom{W}}^{I}{W}^{J}\overset{\text{def}}{=}{\vphantom{W}}^{I}{W}\cap W^{J}$.

Similarly, every one-sided quotient is a special case of two-sided quotients:

[TABLE]

Remark 2.3**.**

Here, we would like to make sure some difference between one-sided cosets/quotients and two-sided ones.

• •

Unlike one-sided cosets, there is a choice of indices for an expression of a two-sided coset: it is possible that $W_{I}uW_{J}=W_{I^{\prime}}uW_{J^{\prime}}$ for some $(I,J)\neq(I^{\prime},J^{\prime})$.

• •

As analogy of Fact A.13, it is tempting to say that for each $v\in W_{I}uW_{J}$, there exists a unique pair $(x,y)\in W_{I}\times W_{J}$ such that

[TABLE]

But this uniqueness is not always true; it is possible that there exists a pair $(x^{\prime},y^{\prime})\in W_{I}\times W_{J}$ such that

[TABLE]

and moreover $(x^{\prime},y^{\prime})\neq(x,y)$. However, it makes sense to speak of the minimal and maximal length coset representatives of $W_{I}uW_{J}$, say $v_{0}$ and $v_{1}$:

[TABLE]

for all $v\in W_{I}uW_{J}$. Indeed, $W_{I}uW_{J}$ is a Bruhat interval $[v_{0},v_{1}]$ and hence a graded poset of rank $\ell(v_{0},v_{1})$; moreover, its Bruhat graph is $\ell(v_{0},v_{1})$-regular.

• •

Unlike one-sided quotients, a two-sided quotient $({\vphantom{W}}^{I}{W}^{J},\leq)$ is not necessarily graded with the rank function $\ell$ (it may be graded with some other rank function, though); that is, there may exist $u$, $w\in{}^{I}W^{J}$ such that

[TABLE]

and there does not exist $v\in{}^{I}W^{J}$ such that $u<v<w$.

2.2. Bruhat cosets

Let $B(w)\overset{\text{def}}{=}[e,w]$, the lower interval determined by $w$. Below, we study details on the graph structure of this interval in terms of our new idea, Bruhat cosets:

Definition 2.4**.**

A Bruhat coset of $w$ is a two-sided coset of the form $W_{D_{L}(w)}uW_{D_{R}(w)}$ for some $u\in B(w)$. Denote this coset by $C_{w}(u)$ for simplicity.

In particular, call $C_{w}(e)$ the bottom coset and $C_{w}(w)$ the top coset.

Several combinatorial properties of Bruhat cosets are in order.

Proposition 2.5**.**

We have $C_{w}(u)\subseteq B(w)$ for each $u\in B(w)$.

Proof.

If $u\in B(w)$ and $s\in D_{L}(w)$, then $su$ stays in $B(w)$ because of Lifting Property (it does not matter whether $s\in D_{L}(u)$ or not). As a consequence, if $x\in W_{D_{L}(w)}$, then $xu\in B(w)$. The same is true on right. ∎

Write $u\sim_{w}v$ if $C_{w}(u)=C_{w}(v)$; thus, $\sim_{w}$ defines an equivalent class on $B(w)$. In other words, Bruhat cosets of $w$ $\{C_{w}(u)\mid u\in B(w)\}$ give a partition of $B(w)$:

[TABLE]

But, this is just a set-theoretical result. We will go into more details of this partition with the ideas of graded posets and Bruhat graphs.

2.3. quotient lower interval

Definition 2.6**.**

Let $u\in B(w)$. We say that $(u,w)$ is a critical pair if

[TABLE]

(It seems that this terminology “critical pair” sometimes appears only in the context of Kazhdan-Lusztig polynomials)

Example 2.7**.**

We work on the type $A_{3}$ Coxeter group ($s_{i}$ is the transposition of $i$ and $i+1$). For example, $(u,w)=(1324,3412)=(s_{2},s_{2}s_{1}s_{3}s_{2})$ is a critical pair with

[TABLE]

Another example is $(u,w)=(14325,45312)=(s_{2}s_{3}s_{2},s_{2}s_{1}s_{3}s_{2}s_{1}s_{4}s_{3}s_{2})$ with

[TABLE]

Definition 2.8**.**

Let

[TABLE]

Note that each $u\in B^{\uparrow}(w)$ ($B_{\downarrow}(w)$) is the maximal (minimal) length coset representative of $C_{w}(u)$. Note also that $w\in B^{\uparrow}(w)$ and $e\in B_{\downarrow}(w)$ so that these sets are nonempty.

Definition 2.9**.**

Define projections $P^{\uparrow}:B(w)\to B^{\uparrow}(w)$ and $P_{\downarrow}:B(w)\to B_{\downarrow}(w)$ by

[TABLE]

(these $P^{\uparrow},P_{\downarrow}$ indeed depend on $w$; however, we suppress the letter $w$ for simplicity)

Proposition 2.10**.**

Both of $P^{\uparrow},P_{\downarrow}$ are surjections and weakly order-preserving in Bruhat order. In other words, in $B(w)$, we have

[TABLE]

As a consequence, two subposets $B^{\uparrow}(w)$ and $B_{\downarrow}(w)$ in $B(w)$ are isomorphic as posets.

Let $C(w)=\{C_{w}(u)\mid u\in B(w)\}$. Whenever no confusion will arise, define projections with the same symbols $P^{\uparrow}:C(w)\to B^{\uparrow}(w)$ and $P_{\downarrow}:C(w)\to B_{\downarrow}(w)$ by

[TABLE]

Definition 2.11**.**

For $C,D\in C(w)$, define the quotient Bruhat order $C\leq D$ if $P^{\uparrow}(C)\leq P^{\uparrow}(D)$ (equivalently, $P_{\downarrow}(C)\leq P_{\downarrow}(D)$). Call $(C(w),\leq)$ the quotient lower interval of $w$.

As this name suggested, $(C(w),\leq)$ is indeed an interval, i.e., it has both the maximum and minimum. Observe now that

[TABLE]

as posets.

2.4. side, middle length

Definition 2.12**.**

Let $u\in B(w)$. The middle length of $u$ under $w$ is

[TABLE]

The side length of $u$ under $w$ is

[TABLE]

Observe that $\textnormal{mid}_{w}$ is weakly increasing:

[TABLE]

(while $\text{side}_{w}$ is increasing only in the same Bruhat coset of $w$:

[TABLE]

Definition 2.13**.**

The middle length, side length, length of $C\in C(w)$ are

[TABLE]

Consequently, we also have

[TABLE]

(while

[TABLE]

is not always true)

Example 2.14**.**

Let us see the Bruhat cosets of $w=3412=s_{2}s_{1}s_{3}s_{2}$ with $D_{L}(w)=\{s_{2}\}=D_{R}(w)$ in Figure 1; $B(w)$ consists of 14 elements with

[TABLE]

Both of $B^{\uparrow}(w)$ and $B_{\downarrow}(w)$ are isomorphic to the following graded poset of rank 2:

[TABLE]

Thus, $C(w)$ consists of four cosets; three of them are of degree 2 while the rest is of degree 1 (note that $\deg(C)=\text{side}_{w}(C)$). They are all graded posets themselves; however, except the bottom coset, they do not contain the minimum element $e$ of $B(w)$; see Table 1.

To describe such details on subposets in a graded poset, we need to introduce some new terminology as below.

2.5. subposets in a graded poset

Recall that a poset $(P,\leq)$ is graded if there exists some function $r:P\to\{0,1,2,\dots,n\}$ such that

1. [ 1 ]

there exists $\widehat{0}_{P}=\widehat{0}\in P$ such that $\widehat{0}\leq x$ for all $x\in P$, 2. [ 2 ]

$r(\widehat{0})=0$ and $n=\max_{x\in P}r(x)$, 3. [ 3 ]

all maximal chains in $[\widehat{0},x]$ have the same length.

In this case, we say that $(P,\leq,r)$ is a graded poset of rank $n$.

We must be very careful when we talk about subposets of $(P,\leq,r)$. Let $Q$ be a nonempty subset of $P$. Then, naturally $(Q,\leq)$ is a subposet of $(P,\leq)$. It may or may not be graded with the rank function $r$ of $P$, though.

Definition 2.15**.**

Say $(Q,\leq)$ is a faithful subposet of $(P,\leq,r)$ if $(Q,\leq,r)$ is a graded poset.

In this case, it is necessary $\widehat{0}\in Q$. So a little weaker idea is this:

Definition 2.16**.**

Say $(Q,\leq)$ is an almost faithful subposet of $(P,\leq,r)$ if there exists some function $r_{Q}:Q\to\{0,1,2,\dots,n_{Q}\}$ such that

1. [ 1 ]

$(Q,\leq,r_{Q})$ is graded itself, 2. [ 2 ]

$r_{Q}(x)=r_{P}(x)-n_{Q}\,(x\in Q)$ for some nonnegative integer $n_{Q}$.

In this case, there exists a unique $\widehat{0}_{Q}\in Q$ such that $r_{Q}(\widehat{0}_{Q})=0$ and $r_{P}(\widehat{0}_{Q})=n_{Q}$.

2.6. main theorem 1

Now, we summarize all the our results above as a Theorem with remarks.

Theorem 2.17**.**

Every lower interval $B(w)$ is a disjoint union of two-sided cosets parametrized by the poset $(C(w),\leq)$, the quotient lower interval of $w$, as

[TABLE]

with all of the following:

• •

The poset $(C(w),\leq)$ is an interval.

• •

Each Bruhat coset $C$ of $w$ is a subinterval of $(B(w),\leq)$. Thus, $(C,\leq,\text{side}_{w})$ is an almost faithful subposet of rank  $\text{side}_{w}(C)$. In particular, the bottom coset is a faithful subposet of $(B(w),\leq,\ell)$.

• •

As a directed graph, each $C$ is $\text{side}_{w}(C)$-regular.

• •

There are nonempty subposets $(B_{\downarrow}(w),\leq)$ and $(B^{\uparrow}(w),\leq)$ in $(B(w),\leq,\ell)$ such that

[TABLE]

and $B_{\downarrow}(w)$ are isomorphic to $B^{\uparrow}(w)$ as posets. They both parametrize the partition in the following sense:

[TABLE]

• •

There is a natural partition of $\ell$, the rank function of $(B(w),\leq)$, by two functions both of which take nonnegative integers:

[TABLE]

[TABLE]

Moreover, the function $\textnormal{mid}_{w}:B(w)\to\{0,1,2,\dots,\textnormal{mid}_{w}(w)\}$ is weakly increasing.

Remark 2.18**.**

• •

Note that $C_{w}(u)=W_{D_{L}(w)}uW_{D_{R}(w)}$ and $C_{w}(v)=W_{D_{L}(w)}vW_{D_{R}(w)}$ may or may not be isomorphic as posets even though these two-sided cosets have expressions of the same two-sided indices. This is a phenomenon totally different from one-sided cosets (Fact A.15).

• •

Some of posets $(B_{\downarrow}(w),\leq)$ or $(B^{\uparrow}(w),\leq)$ may or may not be graded.

• •

Let $Q_{1},Q_{2}$ be subsets of a graded poset $P$. We remark that all of the following can happen simultaneously:

• (1)

$(Q_{1},\leq)$ is a faithful subposet of $(P,\leq,r)$ while $(Q_{2},\leq)$ is not.

• (2)

$(Q_{2},\leq)$ is graded with some other rank function.

• (3)

$(Q_{1},\leq)$ and $(Q_{2},\leq)$ are isomorphic as posets.

We will see examples in the next section with $P=B(w)$, $Q_{1}=B_{\downarrow}(w)$ and $Q_{2}=B^{\uparrow}(w)$ for some $w$.

3. Quotient Bruhat graph

3.1. separated element

Next, we study quotient lower intervals in some simplest cases. Let

[TABLE]

Definition 3.1**.**

We say that $w$ is separated if

[TABLE]

In this case, any $u\in[e,P_{\downarrow}(w)]$ satisfies

[TABLE]

and

[TABLE]

As a result, if $w$ is separated, then

[TABLE]

Proposition 3.2**.**

If $w$ is separated, then $(B_{\downarrow}(w),\leq,\ell)$ is a faithful subposet of $(B(w),\leq,\ell)$.

Proof.

$B_{\downarrow}(w)={}^{D_{L}(w)}[e,P_{\downarrow}(w)]^{D_{R}(w)}=[e,P_{\downarrow}(w)]$ is just a lower subinterval of $B(w)$ and hence is graded. ∎

Example 3.3**.**

On the one hand, all of $w=3412,45312,52341$ are separated since

[TABLE]

here $|\,\cdots\,|$ indicates $P_{\downarrow}(w)$ for convenience. On the other hand, $w=456123$ is not separated since

[TABLE]

with $D_{L}(w)\cap S(s_{2}s_{1}s_{4}s_{3}s_{2}s_{5}s_{4})=\{s_{3}\}\neq\emptyset$.

We have already seen that the function $\textnormal{mid}_{w}$ on $B(w)$ is weakly increasing. We can say more with our assumption:

Proposition 3.4**.**

If $w$ is separated and $u\leq v$ in $B(w)$, then

[TABLE]

Proof.

Since $(B(w),\leq,\ell)$ is a graded poset, it is enough to prove this for $u\to v$, $\ell(u,v)=1$. Write

[TABLE]

with $\ell(v)=\ell(x)+\ell(P_{\downarrow}(v))+\ell(y)$.

• •

Case 1: If $u\sim_{w}v$, then $P_{\downarrow}(u)=P_{\downarrow}(v)$ so that

[TABLE]

Thus,

[TABLE]

• •

Case 2: Suppose $u\not\sim_{w}v$. There exists a reduced expression

[TABLE]

for some $a_{i},b_{i},c_{i}\in S$ such that $a_{i}\in D_{L}(w),c_{i}\in D_{R}(w)$ for all $i$ and

[TABLE]

Since $u\to v$ and $u\not\sim_{w}v$, we have

[TABLE]

for some $j$ (Fact A.7). Now, this word must be also reduced since $\ell(u,v)=1$. This implies $P_{\downarrow}(u)=b_{1}\cdots\widehat{b_{j}}\cdots b_{m}$ (reduced) so that $\textnormal{mid}_{w}(u,v)=1$ and

[TABLE]

Thus, $\text{side}_{w}(u)=\text{side}_{w}(v)$.

∎

Observation 3.5**.**

Suppose $w$ is separated. Then, $(C(w),\leq,\textnormal{mid}_{w})$ is graded. In particular, if $C\leq D$ in $C(w)$, then

[TABLE]

3.2. quotient Bruhat graph

Now we are ready for introducing quotient Bruhat graphs.

Definition 3.6**.**

The quotient Bruhat graph of $w$ is the directed graph with the vertex set $C(w)$ and edges $C\to D$ if $C\neq D$ and there exists a directed edge $u\to v$ such that $u\in C$ and $v\in D$.

In this way, we can regard $C(w)$ not only a poset but also a directed graph.

If $P_{\downarrow}(C)\to P_{\downarrow}(D)$, then $C\to D$ by definition. If $w$ is separated, the converse also holds:

Proposition 3.7**.**

If $w$ is separated and $C,D\in C(w)$, $C\to D$, then $P_{\downarrow}(C)\to P_{\downarrow}(D)$.

This follows from the following lemma.

Lemma 3.8**.**

Suppose $w$ is separated. If $u\to v$ in $B(w)$ and $u\not\sim_{w}v$, then $P_{\downarrow}(u)\to P_{\downarrow}(v)$.

Proof.

Write

[TABLE]

with $a_{i}\in D_{L}(w),b_{i}\in S,c_{i}\in D_{R}(w)$ and

[TABLE]

Since $u\to v$, we can obtain a word for $u$ by deleting one simple reflection from the reduced word for $v$ above; there are three possible cases:

[TABLE]

[TABLE]

[TABLE]

In the first and third cases, we must have $u\sim_{w}v$, a contradiction. Thus, it is necessary that

[TABLE]

(which may or may not be reduced). Then we have

[TABLE]

where the first inequality is due to Subword Property. Since $w$ is separated, this implies

[TABLE]

and moreover

[TABLE]

With Fact A.7, conclude that $P_{\downarrow}(u)\to P_{\downarrow}(v)$. ∎

Notice that, in general, $u\to v,u\not\sim_{w}v$ does not necessarily imply $P^{\uparrow}(u)\to P^{\uparrow}(v)$ (and $C\to D$ does not necessarily imply $P^{\uparrow}(C)\to P^{\uparrow}(D)$).

3.3. main theorem 2

Now, we again summarize the our results above as a Theorem with remarks.

Theorem 3.9**.**

Suppose $w$ is separated. Then, every lower interval $B(w)$ is a disjoint union of two-sided cosets paramtrized by its quotient lower interval

[TABLE]

with all of the following:

• •

$B_{\downarrow}(w)$ are isomorphic to $B^{\uparrow}(w)$ as posets.

• •

$(B_{\downarrow}(w),\leq)$ is a faithful subposet of $(B(w),\leq,\ell)$.

• •

We have

[TABLE]

• •

There is a natural partition of $\ell$, the rank function of $(B(w),\leq)$, by two functions both of which take nonnegative integers:

[TABLE]

[TABLE]

Moreover, both of $\textnormal{mid}_{w},\text{side}_{w}$ are weakly increasing.

• •

Each Bruhat coset $(C,\leq)$ of $w$ is a subinterval of $(B(w),\leq)$ and hence $(C,\leq,\text{side}_{w})$ is an almost faithful subposet of rank   $\text{side}_{w}(C)$ and, as a directed graph, it is $\text{side}_{w}(C)$-regular. Moreover, the degree of such cosets is weakly increasing in quotient Bruhat order:

[TABLE]

• •

There is an equivalence

[TABLE]

so that $(C(w),\to)\cong(B_{\downarrow}(w),\to)$ as a directed graph.

Remark 3.10**.**

We should not misunderstand this theorem as the direct product of graded posets. Even though $(B(w),\leq,\ell)$ has two kinds of graded poset structures $(B_{\downarrow}(w),\leq,\textnormal{mid}_{w})$ and $(C,\leq,\text{side}_{w})$ inside with $\ell(u)=\textnormal{mid}_{w}(u)+\text{side}_{w}(u)$, $B(w)$ is not necessarily the direct product of them.

3.4. example 1: 45312

Here we observe some examples from type A Coxeter groups.

Let $w=45312=s_{2}s_{3}s_{2}|s_{1}s_{4}|s_{2}s_{3}s_{2}$ (separated). We have

[TABLE]

On one hand, $B_{\downarrow}(w)=$

\textstyle{\CIRCLE}$$\textstyle{1}$$\textstyle{\CIRCLE}$$\textstyle{0}$$\textstyle{\CIRCLE}$$\textstyle{2}$$\textstyle{\CIRCLE}$$\textstyle{1}

is a faithful subposet of $B(w)$; Those numbers indicate length in $B(w)$.

On the other hand, $B^{\uparrow}(w)=$

\textstyle{\CIRCLE}$$\textstyle{\footnotesize 6}$$\textstyle{\CIRCLE}$$\textstyle{\footnotesize 3}$$\textstyle{\CIRCLE}$$\textstyle{\footnotesize 8}$$\textstyle{\CIRCLE}$$\textstyle{\footnotesize 6}

is not an almost faithful subposet of $(B(w),\leq,\ell)$.

The quotient Bruhat graph $C(w)$ consists of four cosets; degree of those is 3, 5, 5 and 6, respectively.

$C(w)\cong[e,s_{1}s_{4}]=$

\textstyle{\CIRCLE}$$\textstyle{\CIRCLE}$$\textstyle{\CIRCLE}$$\textstyle{\CIRCLE}

.

3.5. example 2: 52341

Let $w=52341=s_{1}s_{4}|s_{2}s_{3}s_{2}|s_{1}s_{4}$ (Table 3).

$B_{\downarrow}(w)=$

\textstyle{\CIRCLE}$$\textstyle{0}$$\textstyle{\CIRCLE}$$\textstyle{1}$$\textstyle{\CIRCLE}$$\textstyle{1}$$\textstyle{\CIRCLE}$$\textstyle{2}$$\textstyle{\CIRCLE}$$\textstyle{2}$$\textstyle{\CIRCLE}$$\textstyle{3}

$B^{\uparrow}(w)=$

\textstyle{\CIRCLE}$$\textstyle{2}$$\textstyle{\CIRCLE}$$\textstyle{4}$$\textstyle{\CIRCLE}$$\textstyle{4}$$\textstyle{\CIRCLE}$$\textstyle{6}$$\textstyle{\CIRCLE}$$\textstyle{6}$$\textstyle{\CIRCLE}$$\textstyle{7}

$C(w)\cong[e,s_{2}s_{3}s_{2}]$=

\textstyle{\CIRCLE}$$\textstyle{\CIRCLE}$$\textstyle{\CIRCLE}$$\textstyle{\CIRCLE}$$\textstyle{\CIRCLE}$$\textstyle{\CIRCLE}

4. Future research

We end with recording some ideas for our future research.

1. [ 1 ]

Let $\deg_{w}(u)$ be the degree of $u$ in $B(w)$ (graph-theoretic degree = the number of edges incident to $u$). We can show that if $u\sim_{w}v$, then $\deg_{w}(u)=\deg_{w}(v)$. Moreover, we always have $\deg_{w}(u)\geq\ell(w)$ (known as Deodhar inequality [2]). Can we prove

[TABLE] 2. [ 2 ]

What if $w$ is not separated? What more can you say about (poset, graded poset, directed graph) structures of $B^{\uparrow}(w),B_{\downarrow}(w),C(w)$ in $(B(w),\leq,\ell)$? 3. [ 3 ]

Study the Poincare polynomial of $B(w)$

[TABLE]

from an aspect of quotient lower intervals: for example, find a relation between $\mathcal{P}_{w}(q)$ and $\mathcal{P}_{P_{\downarrow}(w)}(q)$.

Appendix A Basics on Coxeter groups

A.1. reduced words, weak orders, subword property

Let $W=(W,S)$ be a Coxeter system. By $T$, we mean the set of its reflections:

[TABLE]

Since $S$ is a group-theoretic generator of $W$, for each $w\in W$, we have

[TABLE]

for some $s_{1},\dots,s_{n}\in S$.

Definition A.1**.**

The length of $w$ is

[TABLE]

We say the word $w=s_{1}\cdots s_{l}$ is reduced if $l=\ell(w)$.

Definition A.2**.**

[TABLE]

[TABLE]

Fact A.3**.**

[1, Lifting Property, Proposition 2.2.7] Let $u<w$. If $s\in D_{L}(w)\setminus D_{L}(u)$, then $su\leq w.$ If $s\in D_{R}(w)\setminus D_{R}(u)$, then $us\leq w.$

Definition A.4**.**

Write $u\to v$ if $v=tu$ for some $t\in T$ and $\ell(u)<\ell(v)$.

Definition A.5**.**

We say that $t\in T$ is a left inversion for $w$ if $tw\to w$. Similarly, $t$ is a right inversion for $w$ if $wt\to w$. Define

[TABLE]

Left, right weak orders on $W$ are simply defined as follows:

[TABLE]

[TABLE]

From each reduced word for $w$, we can construct its inversions as follows: Let $w=s_{1}s_{2}\cdots s_{l}$ be any reduced word. Set $t_{i}=s_{1}s_{2}\cdots s_{i-1}s_{i}s_{i-1}\cdots s_{2}s_{1}$ for each $i=1,2,\cdots,l$. Then:

Fact A.6**.**

We have $T_{L}(w)=\{t_{1},t_{2},\dots,t_{l}\}$ and $t_{i}$ are all distinct (independent from the choice of a reduced word for $w$). Consequently, $l=|T_{L}(w)|=\ell(w)$.

Fact A.7**.**

Let $w=s_{1}\cdots s_{l}$ be a reduced word and $t\in T$. Then, the following are equivalent:

1. [ 1 ]

$t\in T_{L}(w)$. 2. [ 2 ]

$t=s_{1}s_{2}\cdots s_{i-1}s_{i}s_{i-1}\cdots s_{2}s_{1}$ for some $i$. 3. [ 3 ]

$tw=s_{1}\cdots\widehat{s_{i}}\cdots s_{l}$ for some $i$.

Moreover, the number $i$ the above is unique.

Notice that

[TABLE]

is a subword of the reduced word $s_{1}s_{2}\cdots s_{l}$ for $w$. This may or may not be reduced, though.

Fact A.8** (Subword Property).**

Let $w=s_{1}\cdots s_{l}$ be an arbitrary reduced word for $w$. Then, the following are equivalent:

1. [ 1 ]

$u\leq w$ in Bruhat order. 2. [ 2 ]

There exists a subword of $s_{1}\cdots s_{l}$ for $u$; that is,

[TABLE]

(this word need not be reduced)

A.2. one-sided cosets and quotients

Here, we review some well-known facts.

Definition A.9**.**

The standard parabolic subgroup $W_{I}$ is the group-theoretic subgroup of $W$ generated by $I$.

The subgroup $W_{I}$ is a Coxeter system itself with the canonical generators $I$, reflections $T(I)\overset{\text{def}}{=}T\cap W_{I}$, Bruhat suborder $\leq$ and length function $\ell_{I}$ with respect to $I$.

Definition A.10**.**

Let $V\subseteq W$. Say $V$ is a left coset if

[TABLE]

for some $I\subseteq S$ and $u\in W$. Similarly, it is a right coset if

[TABLE]

for some $I\subseteq S$ and $u\in W$. It is a one-sided coset if it is a left or right coset.

Remark A.11**.**

Note that the choice of index $I$ for each one-sided coset is unique:

[TABLE]

[TABLE]

Fact A.12**.**

For any $W_{I}u$, there exist $v_{0},v_{1}\in W_{I}u$, its minimal and maximal length coset representatives, such that

[TABLE]

for all $v\in W_{I}u$. Indeed, $v_{0}=\min W_{I}u$ and $v_{1}=\max W_{I}u$ (here min and max are taken in Bruhat order). The same is true for right cosets.

Next, let us review classical results on one-sided quotients. Define subsets of $W$ for $I,J\subseteq S$ as

[TABLE]

Call these left and right quotients.

Fact A.13**.**

Let $J\subseteq S$. For each $w\in W$, there exists a unique pair $(w_{J},w^{J})\in W_{J}\times W^{J}$ such that

[TABLE]

Fact A.14** (Chain property).**

Every one-sided quotient is graded with the rank function $\ell$: If $u<w$ in ${\vphantom{W}}^{I}{W}$ ($W^{J}$), there exists a chain in ${\vphantom{W}}^{I}{W}$ ($W^{J}$)

[TABLE]

such that $\ell(v_{i},v_{i+1})=1$ for all $i$ (thus $k=\ell(u,w)$). Consequently, ${}^{I}[u,w]\overset{\text{def}}{=}[u,w]\cap{}^{I}W$ $\left({[u,w]^{J}\overset{\text{def}}{=}[u,w]\cap W^{J}}\right)$ is graded.

Fact A.15**.**

Every lower interval $B(w)$ is a disjoint union of left cosets with the same index $D_{L}(w)$:

[TABLE]

Moreover, all cosets in

[TABLE]

are isomorphic to each other as posets (and even as graded posets and as directed graphs). Moreover, each $W_{D_{L}(w)}u$ is indeed an (left weak) interval and a regular graph of the same degree.