# Acyclic matchings on Bruhat intervals and applications to total positivity

Huanchen Bao  Department of Mathematics, National University of Singapore,
Singapore<EMAIL_ADDRESS>and  Xuhua He  Department of Mathematics and
New Cornerstone Science Laboratory, The University of Hong Kong, Pokfulam,
Hong Kong, Hong Kong SAR, China<EMAIL_ADDRESS>
###### Abstract.

The existence of acyclic complete matchings on the face poset of a regular CW
complex implies that the underlying topological space of the CW complex is
contractible by discrete Morse theory.

In this paper, we construct explicitly acyclic complete matchings on any non-
trivial Bruhat interval $[v,w]$ based on any reflection order on the Coxeter
group $W$. We then apply this combinatorial result to regular CW complexes
arising from the theory of total positivity.

As an application, we show that the totally nonnegative Springer fibers are
contractible. This verifies a conjecture of Lusztig in [Lu94]. As another
application, we show that the totally nonnegative fibers of the natural
projection from full flag varieties to partial flag varieties are
contractible. This leads to a much simplified proof of the regularity property
on totally nonnegative partial flag varieties compared to the proofs in [GKL]
and [BH22].

###### Key words and phrases:

Weyl groups, Bruhat intervals, total positivity

###### 2020 Mathematics Subject Classification:

20F55, 14M15, 15B48

###### Contents

  1. 1 Introduction
  2. 2 Discrete Morse theory
  3. 3 Acyclic matchings on Bruhat intervals
  4. 4 Totally nonnegative Springer fibers
  5. 5 Fibers on totally nonnegative flag varieties

##  1\. Introduction

###  1.1. Acyclic complete matchings

Discrete Morse theory, developed by Forman in [Forman], is an efficient tool
for determining the homotopy type of a regular CW complex. The theory was
later reformulated by Chari [Chari] in purely combinatorial terms of acyclic
matchings on the face poset $P$ of the regular CW complex. Chari’s
reformulation applies to arbitrary posets; see §2.2 for the definition.

The poset we are mainly interested in arises from the Coxeter groups. We state
the main combinatorial result of this paper.

######  Theorem 1 (Theorem 3.1).

A reflection order on a Coxeter group $W$ induces an acyclic complete matching
on any non-trivial Bruhat interval $[v,w]$ for $v<w$ in $W$.

It is known [Bj1] that $[v,w]$ is the augmented face poset of a regular CW
complex homeomorphic to a closed ball. So discrete Morse theory is not very
useful on the poset $[v,w]$ itself. However, there are many regular CW
complexes that are not homeomorphic to closed balls and are not even
equidimensional. The explicit matching from Theorem 3.1 will be a useful tool
to handle some of these complicated situations.

###  1.2. Totally positive spaces

The regular CW complexes we consider in this paper arise from the theory of
total positivity. The theory of total positivity on the reductive groups and
their flag varieties was introduced by Lusztig in the seminal work [Lu94], and
later generalized to arbitrary Kac-Moody types in [Lu19] and [BH21]. The
totally nonnegative flag variety $\mathcal{P}_{K,\geqslant 0}$ is a
“remarkable polyhedral subspace” (cf. [Lu94]). It admits a cellular
decomposition and is a regular CW complex homeomorphic to a closed ball (see
[Rie06], [GKL], [BH22]). The totally nonnegative flag varieties have been used
in many other areas, such as cluster algebras [FZ], Grassmann polytopes
[Lam15], and the physics of scattering amplitudes [AHBC16].

The regular CW complex structure for the totally nonnegative full flag variety
is given by $\mathcal{B}_{\geqslant 0}=\sqcup_{v\leqslant
w}\mathcal{B}_{v,w;>0}$, where $\mathcal{B}_{v,w;>0}$ is the totally
nonnegative Richardson variety of $\mathcal{B}$. In this paper, we will
establish the contractibility of certain subcomplexes inside the totally
nonnegative flag variety $\mathcal{B}_{\geqslant 0}$. This will be established
by constructing explicit acyclic matchings on certain subposets of
$\\{(v,w)\in W^{2}|v\leqslant w\\}$. These matchings will be obtained via
carefully chosen reflection orders on $W$ via Theorem 3.1.

The first subcomplex of $\mathcal{B}_{\geqslant 0}$ we study is the totally
nonnnegative Springer fiber for a reductive group. By definition, a totally
nonnnegative Springer fiber is $\mathcal{B}^{u}_{\geqslant
0}=\\{B\in\mathcal{B}_{\geqslant 0}|u\cdot B=B\\}$ for some unipotent $u\in
G_{\geqslant 0}$. Lusztig [Lu94]*§8.16 conjectured that totally nonnegative
Springer fibers are contractible. In [LuSp], Lusztig showed that
$\mathcal{B}^{u}_{\geqslant 0}$ admits a natural cellular decomposition,
inherited from the cellular decomposition of $\mathcal{B}_{\geqslant 0}$ and
gave an explicit description of the cells inside $\mathcal{B}^{u}_{\geqslant
0}$.

######  Theorem 2 (Theorem 4.2).

The totally nonnegative Springer fiber $\mathcal{B}^{u}_{\geqslant 0}$ is
contractible.

The second subcomplex of $\mathcal{B}_{\geqslant 0}$ we study arise from the
fiber of the projection map from the totally nonnegative full flag variety to
the totally nonnegative partial flag variety for any Kac-Moody group.

The totally nonnegative partial flag variety $\mathcal{P}_{K,\geqslant 0}$
admits the natural cellular decomposition into totally nonnegative projected
Richardson varieties $\mathcal{P}_{K,v,w,\geqslant 0}$ [Rie99, Rie06],
inherited from the cellular the cellular decomposition of
$\mathcal{B}_{\geqslant 0}$. By definition, the closed stratum
$\mathcal{P}_{K,v,w,\geqslant 0}$ is the image of the totally nonnegative
Richardson variety $\mathcal{B}_{v,w,\geqslant 0}$ in the full flag variety.
The natural projection map $\pi_{v,w}:\mathcal{B}_{v,w,\geqslant
0}\rightarrow\mathcal{P}_{K,v,w,\geqslant 0}$ is almost never a homeomorphism.
We show that

######  Theorem 3.

The fibers of the map $\pi_{v,w}:\mathcal{B}_{v,w,\geqslant
0}\rightarrow\mathcal{P}_{K,v,w,\geqslant 0}$ are contractible regular CW
complexes.

We conclude that the map $\pi_{v,w}$ is cell-like in the sense of [Dyd02] or
[DHM]*Definition 2.34. As the name suggests, one should think of cell-like
maps as well-behaved maps between CW complexes. Indeed, Siebenmann’s cell-like
approximation theorem implies that the two spaces $\mathcal{B}_{v,w,\geqslant
0}$ and $\mathcal{P}_{K,v,w,\geqslant 0}$ are abstractly homeomorphic; cf.
[DHM]*Corollary 2.33. Therefore, the regularity of $\mathcal{B}_{v,w,\geqslant
0}$ would readily imply the regularity of $\mathcal{P}_{K,v,w,\geqslant 0}$.

This leads to a new proof of the regularity property for the totally
nonnegative partial flag varieties, assuming the regularity theorem on the
full flag variety. This approach significantly simplifies the proof of the
regularity theorems for totally nonnegative partial flag varieties by
Galashin-Karp-Lam [GKL] for reductive groups and by us [BH22] for arbitrary
Kac-Moody groups, which involves the compatibility of certain total positivity
with the atlas map (see, e.g., [BH22]*§6-8).

Acknowledgement:  XH is partially supported by the New Cornerstone Science
Foundation through the New Cornerstone Investigator Program and the Xplorer
Prize, and by Hong Kong RGC grant 14300023. HB is supported by MOE grants
A-0004586-00-00 and A-0004586-01-00. HB thanks Lauren Williams for introducing
him to discrete Morse theory.

##  2\. Discrete Morse theory

In this section, we recall the discrete Morse theory for regular CW complexes
by Forman [Forman]. We focus on the combinatorial reformulation by Chari in
[Chari] in terms of acyclic matching on the face poset.

###  2.1. Some properties on posets

Let $(P,\leqslant)$ be a poset. For any $x,y\in P$ with $x\leqslant y$, the
interval from $x$ to $y$ is defined to be $[x,y]=\\{z\in P|x\leqslant
z\leqslant y\\}$. The covering relations are indicated by $\gtrdot$ and
$\lessdot$. For any $x,y\in P$ with $x\leqslant y$, a maximal chain from $x$
to $y$ is a finite sequence of elements $y=w_{0}\gtrdot
w_{1}\gtrdot\cdots\gtrdot w_{n}=x$. In general, the maximal chain may not
exist.

The poset $P$ is called pure if for any $x,y\in P$ with $x\leqslant y$, the
maximal chains from $x$ to $y$ always exist and have the same length. A pure
poset $P$ is called thin if every interval of length $2$ has exactly $4$
elements, i.e. has exactly two elements between $x$ and $y$. A pure poset is
called graded if it has a unique minimal and maximal element. We often denote
the unique minimal element (resp. maximal element) of a graded poset by
$\hat{0}$ (resp. $\hat{1}$). The coatoms (resp. atoms) of a graded poset are
the elements covered by $\hat{1}$ (resp. $\hat{0}$).

Assume $P$ is a CW poset, that is, $P$ is the face poset of a regular CW
complex $\Delta(P)$. The topology of $\Delta(P)$ is determined by the poset
$P$. Hence we say that $P$ is the face poset of a ball (resp., sphere) if the
CW complex $\Delta(P)$ is homeomorphic to a ball (resp., a sphere.)

Now we recall the notion of EL-shellability introduced by Bjorner in [Bj1].
Suppose that the poset $P$ is pure. An edge labeling of $P$ is a map $\lambda$
from the set of all the covering relations in $P$ to a poset $\Lambda$.
Labeling $\lambda$ sends any maximal chain of an interval of $P$ to a tuple of
$\Lambda$. A maximal chain is called increasing if the associated tuple of
$\Lambda$ is increasing. An edge labeling of $P$ is called EL-labeling if for
every interval, there exists a unique increasing maximal chain, and all the
other maximal chains of this interval are less than this maximal chain with
respect to the lexicographical order.

###  2.2. Discrete Morse theory

Let $G(P)$ be the Hasse diagram of $P$ with the edges being the covering
relations. A matching of the Hasse diagram of $G(P)$ (or simply a matching of
$P$) is a subset of edges of $G(P)$ such that any vertex $x\in P$ is incident
to at most one edge in $M$. We say that a matching $M$ is complete if there is
no unmatch element. We often equivalently consider $M$ as an involution of the
set $P$ of vertices, such that $M(x)=y$ if the edge $\\{x,y\\}\in M$ and
$M(x)=x$ if $x$ is unmatched.

We can regard the Hasse diagram $G(P)$ of $P$ as a directed graph with edges
directed from large elements to small elements with respect to poset ordering.
By reversing the directed edges in $M$, we obtain a new directed graph
$G_{M}(P)$. We say that $M$ is an acyclic matching (or discrete Morse
matching) if $G_{M}(P)$ has no cycles.

######  Definition 2.1.

Let $M$ be a matching of $P$, considered as an involution of $P$. We say that
a subset $Q\subset P$ is an $M$-subset, if $M$ restricts to an involution of
$Q$. In other words, there is no edge in $M$ connecting any element in $Q$
with any element in $P-Q$.

It is clear that the collection of $M$-subsets is preserved under unions,
intersections, and subtractions. It is also clear that if $M$ is acyclic, then
any $M$ subsets are acyclic.

The following reformulation of Forman’s discrete Morse theory on regular CW
complexes by Chari [Chari] is crucial for us. Our formulation follows
[RW]*Theorem 7.6.

######  Theorem 2.2.

Let $P$ be the face poset of a regular CW complex $\Delta(P)$ with an acyclic
matching $M$. Let $m_{p}$ be the number of unmatched cells of dimension $p$.
Then $\Delta(P)$ is homotopic to a (not necessarily regular) CW complex of
exactly $m_{p}$ cells in dimension $p$.

In particular, if $m_{0}=1$ and $m_{p}=0$ for all $p>0$, we can conclude that
$\Delta(P)$ is contractible.

If $P$ is the face poset of a regular CW complex $\Delta(P)$, we denote by
$\hat{P}=P\sqcup\\{\hat{0}\\}$ the augmented face poset with an additional
minimal element $\hat{0}$. Assuming that $P$ has an acyclic matching, we
define an extension of the acyclic matching by matching $\hat{0}$ with any
unmatched (if exists) minimal element of $P$. The resultant matching is
clearly acyclic.

######  Remark 2.3.

Our definition of face poset $P$ of a regular CW complex agrees with [Forman]
and differs from [Bj1, Bj2]. In particular, this is no added minimal element
$\\{\hat{0}\\}$ corresponding to the empty set. One should think of the
difference as ordinary homology versus reduced homology.

##  3\. Acyclic matchings on Bruhat intervals

Let $(W,I)$ be a Coxeter group with simple reflections $s_{i}\in I$. We denote
the Bruhat order on $W$ by $v\leqslant w$ and the cover relation by $v\lessdot
w$, respectively. We denote the length function by $\ell(\cdot)$. Let $T$ be
the set of reflections of $W$ and denote an (arbitrary fixed) reflection order
by $\preceq$; cf. [Dyer1]*§2. We shall consider various different reflection
orders, which will be specified in the context.

###  3.1. EL-labelings

For any $v\leqslant w$, let $[v,w]=\\{z\in W|v\leqslant z\leqslant w\\}$ be
the Bruhat interval. By [BB]*Lemma 2.7.3, the poset $[v,w]$ is thin. For any
$w_{1}\lessdot w_{2}$, we label the edge by

$\lambda(w_{1}\lessdot w_{2})=w_{1}w_{2}^{-1}\in T.$ (3.1)

It follows from [Dyer1] that this defines an EL-labeling of $[v,w]$ with
respect to the (any fixed) reflection order $\preceq$ of $T$. Recall that the
opposite order $\preceq^{op}$ on $T$ is again a reflection order. By [Dyer1],
the edge labeling (3.1) is a dual EL-labeling of $[v,w]$ (that is, an EL-
labeling for the dual poset), with respect to the opposite reflection order
$\preceq^{op}$ of $T$.

Let $v\lessdot v_{1}\lessdot\cdots\lessdot w_{1}\lessdot w$ be the unique
increasing maximal chain in $[v,w]$ with respect to the reflection order
$\preceq$ on $T$. We then obtain the following conclusions combining both the
EL-labeling and the dual El-labeling of $[v,w]$.

(a) The maximal chain $w\gtrdot w_{1}\gtrdot\cdots\gtrdot v_{1}\gtrdot v$ is
the unique decreasing maximal chain, which is lexicographically maximal.

(b) Let $\\{v_{1},\dots,v_{n}\\}$ be the atoms of $v$. Then we have
$vv_{1}^{-1}\preceq vv_{i}^{-1}$, for any $1\leqslant i\leqslant n$.

(c) Let $\\{w_{1},\dots,w_{m}\\}$ be the coatoms of $w$. Then we have
$w_{j}w^{-1}\preceq w_{1}w^{-1}$, for any $1\leqslant j\leqslant m$.

###  3.2. Acyclic matchings

Let $G([v,w])$ be the Hasse diagram of $[v,w]$. We define a subset
$M_{\preceq}$ of the edges of $G([v,w])$ with respect to the reflection order
$\preceq$ on $T$ as follows. For any $x\in[v,w]$, let $E(x)$ be the set of
edges in $G([v,w])$ incident to $x$. Recall that the edges in $E(x)$ are
labeled by reflections in $T$ via the labeling function
$\lambda:E(x)\rightarrow T$ in (3.1). Labelings on $E(x)$ are distinct by
definition. Let $E(x,max)$ be the edge such that the labeling is maximal with
respect to the order $\preceq$ in $T$. Let
$M_{\preceq}([v,w])=\cup_{x\in[v,w]}E(x,max)$ as a subset of edges of
$G([v,w])$.

######  Theorem 3.1.

Let $v<w$. Then $M_{\preceq}([v,w])$ is an acyclic complete matching of
$[v,w]$.

The rest of this section is devoted to the proof of Theorem 3.1. We first show
that $M_{\preceq}([v,w])$ is indeed a complete matching of $[v,w]$ in §3.2.1.
We then discuss the relation between the matching $M_{\preceq}([v,w])$ and the
shelling (induced by the EL-labeling) of $[v,w]$ in §3.2.2. Finally, we prove
that the matching $M_{\preceq}([v,w])$ is acyclic in §3.2.3. We write
$M=M_{\preceq}([v,w])$.

#### 3.2.1.

We first show that $M$ is indeed a complete matching of $[v,w]$.

We can clearly consider $M$ as a map from $[v,w]$ to $[v,w]$, such that

$\\{x,M(x)\\}=E(x,max),\quad\text{ for any }x\in[v,w].$

There is no unmatched element by definition. It remains to show that it is an
involution.

Let $x\in[v,w]$. Without loss of generality, we assume $x\lessdot M(x)$. We
assume by contradiction that $M^{2}(x)\neq x$. We divide into two cases.

We first consider the case $M(x)\lessdot M^{2}(x)$. By definition, we have
$\lambda(x\lessdot M(x))\preceq\lambda(M(x)\lessdot M^{2}(x))$. Consider the
length-two interval $[x,M^{2}(x)]$, which consists precisely of $4$ elements,
$x$, $M(x)$, $M^{2}(x)$, $y$. By assumption, $x\lessdot M(x)\lessdot M^{2}(x)$
is the unique increasing maximal chain in $[x,M(x)]$ in the EL-labeling with
respect to the reflection $\preceq$ on $T$. So we must have $\lambda(x\lessdot
M(x))\preceq\lambda(x\lessdot y)$ by §3.1 (b). This contradicts the fact that
$\lambda(x\lessdot M(x))$ is maximal.

We then consider the case $M^{2}(x)\lessdot M(x)$. By definition, we have
$\lambda(x\lessdot M(x))\preceq\lambda(M^{2}(x)\lessdot M(x))$. We consider
the interval $[v,x]$ and let $v\leqslant v_{1}\leqslant\cdots\leqslant
x_{1}\leqslant x$ be the unique increasing maximal chain in the EL-labeling.
Since $\lambda(x_{1}\preceq x)\preceq\lambda(x\lessdot M(x))$ by definition,
the extended chain $v\leqslant v_{1}\leqslant\cdots\leqslant x_{1}\leqslant
x\leqslant M(x)$ is increasing. Hence it is the unique increasing chain in the
interval $[v,w]$. Then by §3.1 (c), we must have $\lambda(M^{2}(x)\lessdot
M(x))\preceq\lambda(x\lessdot M(x))$. This is a contradiction.

#### 3.2.2.

Let $[v^{\prime},w^{\prime}]\subset[v,w]$ be a sub-interval. If
$[v^{\prime},w^{\prime}]$ is an $M_{\preceq}([v,w])$-subset of $[v,w]$, then
the restriction of $M_{\preceq}([v,w])$ is the same as
$M_{\preceq}([v^{\prime},w^{\prime}])$, as involutions/matchings on
$[v^{\prime},w^{\prime}]$.

The next corollary discusses the relation between the matching
$M_{\preceq}([v,w])$ of $[v,w]$ and the shelling of $[v,w]$.

######  Corollary 3.2.

Let $v<w$.

  1. (1)

Let $w_{1},\dots,w_{n}$ be the coatoms of $[v,w]$, ordered such that
$ww_{1}^{-1}\preceq\cdots\preceq ww_{n}^{-1}$ in $T$. Then $\cup_{1\leqslant
i<k}[v,w_{i}]$ are $M_{\preceq}([v,w])$-subsets for any $k\leqslant n$.

  2. (2)

Let $v_{1},\dots,v_{m}$ be the atoms of $[v,w]$, ordered in such a way that
$vv_{1}^{-1}\preceq\cdots\preceq vv_{n}^{-1}$ in $T$. Then $\cup_{1\leqslant
i<k}[v_{i},w]$ are $M_{\preceq}([v,w])$-subsets for any $k\leqslant m$.

###### Proof.

We show part (1). Part (2) is similar and will be omitted.

We write $M=M_{\preceq}([v,w])$ and consider $M$ as an involution on $[v,w]$.
We need to show that for any $x\in\cup_{1\leqslant i<k}[v,w_{i}]$, we have
$M(x)\in\cup_{1\leqslant i<k}[v,w_{i}]$. If $M(x)\lessdot x$, this is trivial.

Assume $x\lessdot M(x)$. By [Dyer1], the interval $[x,w]$ is EL-shellable with
respect to the opposite reflection order $\preceq^{op}$ on $T$. Let
$x\leqslant x_{1}\leqslant\dots\leqslant w^{\prime}\leqslant w$ be the unique
increasing maximal chain with respect to the opposite reflection order
$\preceq^{op}$ on $T$. Applying §3.1 (b) with respect to the order
$\preceq^{op}$, we obtain that $\lambda(x\lessdot x_{1})$ is minimal with
respect to the order $\preceq^{op}$ among atoms of $x$, hence maximal with
respect to the order $\preceq$. So $M(x)=x_{1}$. By §3.1 (c), we see that
$w^{\prime}$ must be $w_{i}$ for some $i<k$. This finishes the proof. ∎

#### 3.2.3.

We show that the matching $M=M_{\preceq}([v,w])$ is acyclic.

We prove this by induction on $m=\ell(w)-\ell(v)$. When $m=1$, this is
trivial. We assume $m\geqslant 2$.

Let $w_{1},\dots,w_{n}$ be the coatoms of $[v,w]$, ordered such that
$ww_{1}^{-1}\preceq\cdots\preceq ww_{n}^{-1}$ in $T$. It follows by Corollary
3.2 that $[v,w_{1}]$ is an $M$-subset of $[v,w]$. Therefore, the involution
$M$ restricts to an involution of $[v,w_{1}]$. It is clear that this is the
same as $M_{\preceq}([v,w_{1}])$. Hence $[v,w_{1}]$ is an acyclic $M$-subset
by the induction hypothesis.

(a) The subset $\cup_{1\leqslant i<n}[v,w_{i}]$ is an acyclic $M$-subset.

Without loss of generality, we assume $n=3$ for notational simplicity. The
general case follows by induction on $n$ via the same argument.

By Corollary 3.2, we know $[v,w_{1}]\cup[v,w_{2}]$ is an $M$-subset. We
consider the partition

$[v,w_{1}]\cup[v,w_{2}]=[v,w_{1}]\sqcup P_{2},\quad\text{where
}P_{2}=[v,w_{2}]-[v,w_{1}].$

The subset $P_{2}=[v,w_{2}]-[v,w_{1}]$ is also an $M$-subset. Then the
restriction of $M$ on $P_{2}$ is the same as the restriction of
$M_{\preceq}([v,w_{2}])$ by the construction in §3.2. Since the matching
$M_{\preceq}([v,w_{2}])$ of $[v,w_{2}]$ is acyclic by the induction
hypothesis, the matching on the subset $P_{2}$ is also acyclic. Now we have
shown that there are no cycles within either $[v,w_{1}]$ or $P_{2}$. On the
other hand, any possible cover relation between $x\in P_{2}$ and
$y\in[v,w_{1}]$ must be of the form $y\lessdot x$ by definition. So, there
cannot be any cycle that traverses both $[v,w_{1}]$ and $P_{2}$. This proves
the claim.

(b) We have $[v,w]-\cup_{1\leqslant i<n}[v,w_{i}]=[M(w),w]$.

Since the Bruhat interval $[v,w]$ is thin, any $y\leqslant M(w)$ must be in
$\cup_{1\leqslant i<n}[v,w_{i}]$. This proves (b).

Now we have a partition of $[v,w]=\Big{(}\cup_{1\leqslant
i<n}[v,w_{i}]\Big{)}\sqcup[M(w),w]$. We can prove similarly to the proof of
claim (a) that the matching $M$ on $[v,w]$ is acyclic.

We now finish the proof of Theorem 3.1.

###  3.3. An example

We give an example of our matching. This example also shows that the matching
is not always a special matching in the sense of [BCM].

Let $S_{4}=Perm\\{1,2,3,4\\}$ be the symmetric group, and let $s_{1}=(12)$,
$s_{2}=(23)$, $s_{3}=(34)$ be simple reflections. Fix a reflection order
$\preceq$ on $T$ such that

$s_{1}\preceq s_{1}s_{2}s_{1}\preceq s_{1}s_{2}s_{3}s_{2}s_{1}\preceq
s_{2}\preceq s_{2}s_{3}s_{2}\preceq s_{3}.$

Then the matching $M=M_{\preceq}([s_{1},s_{2}s_{3}s_{1}s_{2}])$ on the
interval $[s_{1},s_{2}s_{3}s_{1}s_{2}]$ consists of the red edges.

$s_{2}s_{3}s_{1}s_{2}$$s_{3}s_{2}s_{3}$$s_{3}s_{1}s_{2}$$s_{2}s_{1}s_{3}$$s_{2}s_{1}s_{2}$$s_{3}s_{2}$$s_{2}s_{3}$$s_{2}s_{1}$$s_{1}s_{2}$$s_{2}$$\scriptstyle
s_{2}s_{1}s_{2}$$\scriptstyle s_{2}$$\scriptstyle
s_{3}s_{2}s_{1}s_{2}s_{3}$$\scriptstyle s_{2}s_{3}s_{2}$$\scriptstyle
s_{2}$$\scriptstyle s_{3}$$\scriptstyle s_{1}$$\scriptstyle
s_{3}$$\scriptstyle s_{2}s_{1}s_{2}$$\scriptstyle
s_{2}s_{3}s_{2}$$\scriptstyle s_{1}$$\scriptstyle s_{2}$$\scriptstyle
s_{3}$$\scriptstyle s_{2}s_{3}s_{2}$$\scriptstyle
s_{1}s_{2}s_{1}$$\scriptstyle s_{1}$

Note that $s_{3}s_{2}s_{3}\lessdot s_{2}s_{3}s_{1}s_{2}$, but
$M(s_{3}s_{2}s_{3})\not\leqslant M(s_{2}s_{3}s_{1}s_{2})$.

##  4\. Totally nonnegative Springer fibers

Our first application is for totally non-negative Springer fibers and the
posets arising from them.

###  4.1. Totally nonnegative flag varieties

We first give a quick review of the totally nonnegative flag varieties. Let
$G$ be a split Kac-Moody group over $\mathbb{R}$. Fix a pinning
$(T,B^{+},B^{-},x_{i},y_{i};i\in I)$ of $G$. Let $(W,I)$ be the Weyl group of
$G$. For any $K\subset I$, we denote by ${}^{K}W$ (resp. $W^{K}$) the set of
left (resp. right) coset minimal-length representatives. Let
$\mathcal{B}=G/B^{+}$ be the (thin) full flag variety of $G$. It admits a
stratification into the (open) Richardson varieties

$\mathcal{B}=\sqcup_{v\leqslant w}\mathcal{B}_{v,w},\text{ where
}\mathcal{B}_{v,w}=B^{+}\dot{w}\cdot B^{+}\cap B^{-}\dot{v}\cdot B^{+}.$

Let $G_{\geqslant 0}$ be the totally nonnegative submonoid defined by Lusztig
in [Lu94, Lu19]. The totally nonnegative full flag variety
$\mathcal{B}_{\geqslant 0}$ is defined to be the (Hausdorff) closure of
$G_{\geqslant 0}\cdot B^{+}$ in $\mathcal{B}$. The totally positive Richardson
varieties are defined by $\mathcal{B}_{v,w,>0}=\mathcal{B}_{\geqslant
0}\cap\mathcal{B}_{v,w}$. Let $\mathcal{B}_{v,w,\geqslant 0}$ be the Hausdorff
closure of $\mathcal{B}_{v,w,>0}$. Then we have $\mathcal{B}_{\geqslant
0}=\sqcup_{v\leqslant w}\mathcal{B}_{v,w,>0}$.

It is known that for any $v\leqslant w$,

  * •

$\mathcal{B}_{v,w,>0}$ is a semi-algebraic cell isomorphic to
$\mathbb{R}_{>0}^{\ell(w)-\ell(v)}$ (see [Rie99], [BH21]);

  * •

$\mathcal{B}_{v,w,\geqslant 0}$ equals $\sqcup_{v\leqslant v^{\prime}\leqslant
w^{\prime}\leqslant w}\mathcal{B}_{v^{\prime},w^{\prime},>0}$ (see [Rie06],
[BH22]);

  * •

$\mathcal{B}_{v,w,\geqslant 0}$ is a regular CW complex homeomorphic to a
closed ball (see [GKL], [BH22]).

###  4.2. Total positivity on Springer fibers

In the rest of this section, we assume furthermore that $G$ is a reductive
group, i.e. a Kac-Moody group of finite type. Then $W$ is a finite Coxeter
group. We denote the longest element by $w_{0}$. For any $J\subset I$, we
denote the longest element of the parabolic subgroup $W_{J}$ by $w_{J}$.

Let $u\in G_{\geqslant 0}$ be a unipotent element. By [Lu94], one may
associate with $u$ two subsets $J$ and $J^{\prime}$ of $I$ with $J\cap
J^{\prime}=\emptyset$. Let $\mathcal{B}^{u}_{\geqslant
0}=\\{B\in\mathcal{B}_{\geqslant 0}|u\cdot B=B\\}$ be the totally nonnegative
Springer fiber of $u$. Lusztig proved in [LuSp] that

$\mathcal{B}^{u}_{\geqslant 0}=\sqcup_{z\in
Z_{J,J^{\prime}}}\mathcal{B}_{v,w,>0},$

where

$\begin{split}Z_{J,J^{\prime}}=\\{(v,w)\in W\times W|&v\leqslant
w;s_{i}w\leqslant w,v\not\leqslant s_{i}w,\text{ for any }i\in J;\\\
&v\leqslant s_{j}v,s_{j}v\not\leqslant w,\text{ for any }j\in
J^{\prime}\\}.\end{split}$ (4.1)

Then $Z_{J,J^{\prime}}$ is naturally a poset by with
$(v,w)\leqslant(v^{\prime},w^{\prime})$ if and only if $v^{\prime}\leqslant
v^{\prime}\leqslant w^{\prime}\leqslant w$. It follows from the regularity of
$\mathcal{B}_{\geqslant 0}$ that $\mathcal{B}^{u}_{\geqslant 0}$ is also a
regular CW complex. However, $\mathcal{B}^{u}_{\geqslant 0}$ is not
equidimensional, in general.

Now we state the main result of this section.

######  Theorem 4.1.

Suppose that $W$ is a finite Coxeter group. Then the poset $Z_{J,J^{\prime}}$
admits an acyclic matching with a unique unmatched element
$(w_{J^{\prime}}w_{0},w_{J^{\prime}}w_{0})$.

Combining Theorem 4.1 with Theorem 2.2, we establish the contractibility of
the totally nonnegative Springer fiber, conjectured by Lusztig in
[Lu94]*§8.16.

######  Theorem 4.2.

The totally nonnegative Springer fiber $\mathcal{B}^{u}_{\geqslant 0}$ is
contractible.

######  Remark 4.3.

Lusztig also conjectured that the totally nonnegative Grothendieck fiber
$\mathcal{B}^{g}_{\geqslant 0}$ for any $g\in G_{\geqslant 0}$ is
contractible. This conjecture would follow from Theorem 4.2 together with the
conjecture on the Jordan decomposition of $G_{\geqslant 0}$ [HL1]*Conjecture
3.1.

###  4.3. Proof of Theorem 4.1

This section is devoted to a proof of Theorem 4.1. We keep the notations from
§4.2.

####  4.3.1. Some subposets

By definition, if $(v,w)\in Z_{J,J^{\prime}}$, then $v\in{}^{J^{\prime}}W$.
For $j\in J^{\prime}$, since $v\leqslant w$ and $s_{j}v\neq w$, we have
$w\leqslant s_{j}w$. Thus $w\in{}^{J^{\prime}}W$. In particular, if $(v,w)\in
Z_{J,J^{\prime}}$ and $v=w_{J^{\prime}}w_{0}$, then $w=w_{J^{\prime}}w_{0}$.

Let $v\in{}^{J^{\prime}}W$. We define

$\displaystyle Z_{v}$ $\displaystyle=\\{w\in W|(v,w)\in Z_{J,J^{\prime}}\\};$
$\displaystyle P_{v}$ $\displaystyle=\\{w\in W|v\leqslant
w,s_{j}v\not\leqslant w\,\,\forall j\in J^{\prime}\\}\supset Z_{v};$
$\displaystyle Q_{v}$ $\displaystyle=\\{w\in W|v\leqslant w,v\not\leqslant
s_{i}w\,\,\forall i\in J\\}\supset Z_{v}.$

Then it follows that $Z_{v}=P_{v}\cap Q_{v}$ and
$Z_{J,J^{\prime}}=\sqcup_{v\in{}^{J^{\prime}}W}Z_{v}$.

By [Dyer1]*Proposition 2.3, there exists a reflection order $\preceq$ on $T$
such that

$\begin{split}t_{1}\preceq t_{2},\quad\text{for any }t_{1}\in T\cap
W_{J^{\prime}},t_{2}\in T-W_{J^{\prime}};\\\ t_{1}\preceq t_{2},\quad\text{for
any }t_{1}\in T-W_{J},t_{2}\in T\cap W_{J}.\end{split}$ (4.2)

This reflection order induces an EL-labeling on $[v,w_{0}]$. Let
$M=M_{\preceq}([v,w_{0}])$ be the acyclic complete matching of $G([v,w_{0}])$
defined in §3.2.

####  4.3.2. The $M$-subset $P_{v}$

Suppose that $v\neq w_{J^{\prime}}w_{0}$. By Theorem 3.1, $M$ gives an acyclic
complete matching on $[v,w_{0}]$. Let $\\{v_{1},\dots,v_{n}\\}$ be the set of
atoms of $[v,w_{0}]$. Recall that $v\in{}^{J^{\prime}}W$. So if
$vv_{a}^{-1}\in W_{J^{\prime}}$, then $v_{a}=s_{i}v$ for some $i\in
J^{\prime}$. By the choice of the reflection order $\preceq$ on $T$ and
Corollary 3.2, the set $\cup_{i\in J^{\prime}}[s_{i}v,w_{0}]$ is an
$M$-subset. Thus $P_{v}=[v,w_{0}]-\cup_{i\in J^{\prime}}[s_{i}v,w_{0}]$ is
again an $M$-subset. Therefore, $M$ gives an acyclic complete matching on
$P_{v}$.

####  4.3.3. The $M$-subset $Q_{v}$

######  Lemma 4.4.

Let $v,w\in W$ with $v\leqslant w$. Then $\\{a\in W_{J}|v\leqslant
a\,{}^{J}w\\}=[x,w_{J}]$ for some $x\in W_{J}$.

###### Proof.

We can assume $w=w_{J}{}^{J}w$, where $w_{J}$ is the longest element in
$W_{J}$. Let $Q=\\{a\in W_{J}|v\leqslant a\,{}^{J}w\\}$. It is easy to see
that $w_{J}$ is the unique maximal element in $Q$. It remains to show that the
minimal element of $Q$ is unique. Let $x\in Q$ be a minimal element. Recall
$[v,w]$ is EL-shellable with respect to the reflection order $\preceq$ on $T$.
We consider the unique increasing maximal chain in $[v,x\,{}^{J}w]$, denoted
by $c(v,x\,{}^{J}w)=v\leqslant v_{1}\leqslant v_{2}\leqslant\cdots\leqslant
v_{l}\leqslant x\,{}^{J}w$. Let $c(x\,{}^{J}w,w)=x\,{}^{J}w\leqslant
x_{1}\leqslant\cdots\leqslant w$ be the unique increasing maximal chain in
$[x\,{}^{J}w,w]$. Then we must have $xv^{-1}_{l}\not\in W_{J}$ by the
minimality of $x$. We also have $x_{1}x^{-1}\in W_{J}$, since we can construct
the chain in $[x,w_{J}]$ by the convention (3.1). So the combined chain

$v\leqslant v_{1}\leqslant v_{2}\leqslant\cdots\leqslant v_{l}\leqslant
x\,{}^{J}w\leqslant x_{1}\leqslant\cdots\leqslant w$

is an increasing maximal chain in $[v,w]$. Since this chain is unique, the
element $x$ must be unique. ∎

For any $w\in{}^{J}W$, we define $Q_{v,w}=[v,w_{0}]\cap W_{J}w$. Then by
definition, $Q_{v}=[v,w_{0}]-\sqcup_{w}Q_{v,w}$, where $w$ runs over the
elements in ${}^{J}W$ with $Q_{v,w}\neq\\{w_{J}w\\}$.

By the definition of the EL-labeling in (3.1), any edge in $Q_{v,w}$ is
labeled by a reflection in $t\in W_{J}\cap T$. On the other hand, if an edge
$\\{x,y\\}$ on the Hasse diagram $G([v,w_{0}])$ connects some $Q_{v,w}\neq
Q_{v,w^{\prime}}$, it will be labeled by some $t^{\prime}\not\in W_{J}$. We
have $t^{\prime}\preceq t$ by the definition of the reflection order. By Lemma
4.4 and the isomorphism of posets $Q_{v,w}\cong Q_{v,w}w^{-1}$, the Hasse
diagram of $Q_{v,w}$ is a connected graph. Hence by the definition in §3.2 and
choice of the reflection order $\preceq$ in (4.2), $Q_{v,w}$ is an $M$-subset
if $Q_{v,w}\neq\\{w_{J}w\\}$. Hence $Q_{v}$ is again an $M$-subset. Since $M$
gives an acyclic complete matching on $[v,w_{0}]$, $M$ also gives an acyclic
complete matching on $Q_{v}$.

####  4.3.4. The matching on $Z_{J,J^{\prime}}$

Recall that $Z_{J,J^{\prime}}=\sqcup_{v\in{}^{J^{\prime}}W}Z_{v}$ and
$Z_{v}=P_{v}\cap Q_{v}$. For $v\neq w_{J^{\prime}}w_{0}$, $M$ gives an acyclic
complete matching on $P_{v}$ and $Q_{v}$. Thus $M$ also gives acyclic complete
matching on $Z_{v}$. On the other hand, if $v=w_{J^{\prime}}w_{0}$, then
$Z_{v}=\\{(w_{J^{\prime}}w_{0},w_{J^{\prime}}w_{0})\\}$ is a singleton.
Therefore $M$ gives a matching on $Z_{J,J^{\prime}}$ with the only unmatched
element of $(w_{J^{\prime}}w_{0},w_{J^{\prime}}w_{0})$.

It remains to show that this matching on $Z_{J,J^{\prime}}$ is acyclic. Note
that any arrow from $(v,w)\in Z_{v}$ to $(v^{\prime},w^{\prime})\in
Z_{v^{\prime}}$ for $v\neq v^{\prime}$ must be of the form

$(v,w)\rightarrow(v^{\prime},w^{\prime}),\quad\text{with }v\lessdot
v^{\prime},w=w^{\prime}.$

Such arrows always increase the length of $v$, i.e.,
$\ell(v^{\prime})=\ell(v)+1$. So we cannot have any cycle traversing $Z_{v}$
for different $v\in W$. Since we do not have any cycles within each $Z_{v}$
either, the matching has to be acyclic.

The proof of Theorem 4.1 is complete.

##  5\. Fibers on totally nonnegative flag varieties

###  5.1. Partial flag varieties

We keep the notations in §4.1. For any $K\subset I$, we denote by
$P_{K}^{+}\supset B^{+}$ the corresponding standard parabolic subgroup of $G$
and $\mathcal{P}_{K}=G/P^{+}_{K}$ the partial flag variety. Let
$\pi:\mathcal{B}\to\mathcal{P}_{K}$, $g\cdot B^{+}\mapsto g\cdot P^{+}_{K}$ be
the natural projection map. The totally nonnegative partial flag variety
$\mathcal{P}_{K,\geqslant 0}$ is defined to be the image
$\pi(\mathcal{B}_{\geqslant 0})$. Since $\pi$ is proper,
$\mathcal{P}_{K,\geqslant 0}$ equals the Hausdorff closure of $G_{\geqslant
0}\cdot P^{+}_{K}$ in $\mathcal{P}_{K}$.

The cellular decomposition $\mathcal{B}_{\geqslant 0}=\sqcup_{v\leqslant
w}\mathcal{B}_{v,w,>0}$ induces a cellular decomposition on
$\mathcal{P}_{K,\geqslant 0}$ (see [Rie06]):

$\mathcal{P}_{K,\geqslant 0}=\sqcup_{(v,w)\in
Q_{K}}\mathcal{P}_{K,v,w,>0},\quad\text{ where
}\mathcal{P}_{K,v,w,>0}=\pi(\mathcal{B}_{v,w,>0})\text{ by definition}.$

The face poset $Q_{K}$ is defined as $Q_{K}=\\{(v,w)\in W\times W|w\in
W^{K},v\leqslant w\\}$, where $(v^{\prime},w^{\prime})\leqslant(v,w)$ if and
only if there exists $u\in W_{K}$ with $v\leqslant v^{\prime}u\leqslant
w^{\prime}u\leqslant w$.

Let $(v,w)\in Q_{K}$. Recall that $\mathcal{B}_{v,w,\geqslant 0}$ is the
Hausdorff closure of $\mathcal{B}_{v,w,>0}$. Let $\mathcal{P}_{K,v,w,\geqslant
0}=\pi(\mathcal{B}_{v,w,\geqslant 0})$ be the Hausdorff closure of
$\mathcal{P}_{K,v,w,>0}$. We denote by $\pi_{v,w}:\mathcal{B}_{v,w,\geqslant
0}\rightarrow\mathcal{P}_{K,v,w,\geqslant 0}$ the projection obtained by
restricting $\pi$.

###  5.2. The fibers

We first recall some results on the Coxeter group $W$. By [He]*Lemma 1111We
use the notation $\circ_{l}$ and $\circ_{r}$ for $\triangleright$ and
$\triangleleft$ in loc.cit., respectively., for any $x,y\in W$,

  * •

there exists a unique maximal element (with respect to $\leqslant$) in
$\\{x^{\prime}y^{\prime};x^{\prime}\leqslant x,y^{\prime}\leqslant y\\}$. We
denote this element by $x\ast y$. Moreover, $x\ast y=x^{\prime}y=xy^{\prime}$
for some $x^{\prime}\leqslant x$ and $y^{\prime}\leqslant y$ with $\ell(x\ast
y)=\ell(x^{\prime})+\ell(y)=\ell(x)+\ell(y^{\prime})$;

  * •

there exists a unique minimal element (with respect to $\leqslant$) in
$\\{x^{\prime}y;x^{\prime}\leqslant x\\}$. We denote this element by
$x\circ_{l}y$. Moreover, $x\circ_{l}y=x^{\prime}y$ for some
$x^{\prime}\leqslant x$ with $\ell(x\circ_{l}y)=\ell(y)-\ell(x^{\prime})$;

  * •

there exists a unique minimal element (with respect to $\leqslant$) in
$\\{xy^{\prime};y^{\prime}\leqslant y\\}$. We denote this element by
$x\circ_{r}y$. Moreover, $x\circ_{r}y=xy^{\prime}$ for some
$y^{\prime}\leqslant y$ with $\ell(x\circ_{r}y)=\ell(x)-\ell(y^{\prime})$.

Let $(v^{\prime},w^{\prime})\leqslant(v,w)$ in $Q_{K}$. We define

$\displaystyle F^{K}_{(v^{\prime},w^{\prime}),(v,w)}=\\{(a,b)\in W_{K}\times
W_{K}|$ $\displaystyle a\leqslant b;v\leqslant v^{\prime}a\leqslant
w^{\prime}b\leqslant w,v^{\prime}a\circ_{r}b^{-1}=v^{\prime},$
$\displaystyle\ell(v^{\prime}a)=\ell(v^{\prime})+\ell(a)\\}.$

In particular, if $(v^{\prime},w^{\prime})=(v,w)$, then $F=\\{(e,e)\\}$ is a
singleton. The set $F^{K}_{(v^{\prime},w^{\prime}),(v,w)}$ is naturally a
poset with $(a,b)\leqslant(a^{\prime},b^{\prime})$ if and only if $a\leqslant
a^{\prime}\leqslant b^{\prime}\leqslant b$ in the usual Bruhat order of
$W_{K}$.

######  Proposition 5.1.

Let $(v^{\prime},w^{\prime})\leqslant(v,w)$ in $Q_{K}$. For any
$x\in\mathcal{P}_{K,v^{\prime},w^{\prime},>0}$, we have

$\pi_{v,w}^{-1}(x)\cong\bigsqcup_{(a,b)\in
F^{K}_{(v^{\prime},w^{\prime}),(v,w)}}\mathcal{B}_{a,b,>0}.$

In particular, the fiber $\pi_{v,w}^{-1}(x)$ is a regular CW complex with the
face poset $F^{K}_{(v^{\prime},w^{\prime}),(v,w)}$.

###### Proof.

We fix a reduced expression ${\bf w^{\prime}}=s_{i_{1}}s_{i_{2}}\cdots
s_{i_{n}}$ of $w^{\prime}$ and let ${\bf
v^{\prime}_{+}}=t_{i_{1}}t_{i_{2}}\cdots t_{i_{n}}$ be the unique positive
subexpression of ${\bf w^{\prime}}$ in the sense of [MR]*Lemma 3.5. Here
$t_{i_{j}}=1$ or $s_{i_{j}}$ for any $1\leqslant j\leqslant n$. Following
[MR]*Definition 5.1, we define

$G_{\bf v^{\prime}_{+},\bf w^{\prime},>0}=\\{g_{1}g_{2}\cdots
g_{n}|g_{j}=\dot{s}_{i_{j}},\text{ if }t_{i_{j}}=1;\text{ and }g_{j}\in
y_{i_{j}}(\mathbb{R}_{>0}),\text{ if }t_{i_{j}}=s_{i_{j}}.\\}.$

Then the natural map $G_{\bf v^{\prime}_{+},\bf
w^{\prime},>0}\rightarrow\mathcal{P}_{K,v^{\prime},w^{\prime},>0}$, $g\mapsto
g\cdot\mathcal{P}_{K}^{+}$, is an isomorphism. We write

$x=g\cdot\mathcal{P}_{K}^{+},\qquad\text{ for the unique }g\in G_{\bf
v^{\prime}_{+},\bf w^{\prime},>0}.$

We then define $\phi:\pi_{v,w}^{-1}(x)\rightarrow\mathcal{B}$, $y\mapsto
g^{-1}y$.

Let $y\in\pi_{v,w}^{-1}(x)$ such that $y\in\mathcal{B}_{p,q,>0}$ for some
$v\leqslant p\leqslant q\leqslant w$ in $W$. We have $q=wb$ for some $b\in
W_{J}$. We can fix a reduced expression ${\bf q}={\bf w^{\prime}}{\bf b}$ of
$q$ starting with the expression ${\bf w^{\prime}}$.

Let ${\bf p_{+}}$ be the unique positive subexpression of $p$ in ${\bf q}$. We
write ${\bf p_{+}}=({\mathbf{p}_{1}},{\bf p_{2}})$ as a subexpression of
$({\bf w^{\prime}},{\bf b})$. Since $\pi_{v,w}(y)=x$, we must have ${\bf
p_{1}}={\bf v^{\prime}_{+}}$ and ${\bf p_{2}}={\bf a_{+}}$ as positive
expressions of some $a\in W_{J}$ in ${\bf b}$. In particular, we have

$G_{\bf p_{+},\bf q,>0}=G_{\bf v^{\prime}_{+},\bf w^{\prime},>0}\cdot G_{\bf
a_{+},\bf b,>0}.$ (5.1)

We hence can write $y=g^{\prime}g^{\prime\prime}\cdot\mathcal{B}^{+}$ for some
$g^{\prime}\in G_{\bf v^{\prime}_{+},\bf w^{\prime},>0}$ and
$g^{\prime\prime}\in G_{\bf a_{+},\bf b,>0}$. We also see that
$v^{\prime}a\circ_{r}b^{-1}=v^{\prime}$ and
$\ell(p)=\ell(v^{\prime})+\ell(a)$, since this subexpression ${\bf p_{+}}$
must be rightmost and reduced/non-decreasing in the sense of [MR]*Definition
3.4.

Since $\pi_{v,w}(y)=x$, we obtain $g^{\prime}=g$. Then we conclude that
$g^{-1}y=g^{\prime\prime}\cdot\mathcal{B}^{+}$. This shows
$\phi(y)\in\mathcal{B}_{\geqslant 0}$. More precisely, we have shown that

$\phi(\pi_{v,w}^{-1}(x))\subset\bigsqcup_{(a,b)\in
F^{K}_{(v^{\prime},w^{\prime}),(v,w)}}\mathcal{B}_{a,b,>0}.$

On the other hand, it is easy to show that the reverse inclusion using (5.1).
This shows that $\pi_{v,w}^{-1}(x)$ is a regular CW complex with the face
poset $F^{K}_{(v^{\prime},w^{\prime}),(v,w)}$. ∎

Now we state the main combinatorial result of this section.

######  Theorem 5.2.

For any $(v^{\prime},w^{\prime})\leqslant(v,w)$ in $Q_{K}$, the poset
$F^{K}_{(v^{\prime},w^{\prime}),(v,w)}$ admits an acyclic matching with a
unique unmatched element $(\tilde{z},\tilde{z})\in W_{K}\times W_{K}$.

Theorem 5.2 will be proved in §5.4. Combining Theorem 5.2 with Theorem 2.2, we
establish the following contractibility result.

######  Theorem 5.3.

The fibers of the projection map $\pi_{v,w}:\mathcal{B}_{v,w,\geqslant
0}\rightarrow\mathcal{P}_{K,v,w,\geqslant 0}$ are contractible.

In §5.5, we will use Theorem 5.3 to give a new and simple proof of the
regularity property of $\mathcal{P}_{K,\geqslant 0}$. This is our main
motivation to study the fibers of $\pi_{v,w}:\mathcal{B}_{v,w,\geqslant
0}\rightarrow\mathcal{P}_{K,v,w,\geqslant 0}$. It is also interesting to study
the fiber of $\pi:\mathcal{B}_{\geqslant 0}\to\mathcal{P}_{\geqslant 0}$,
which we will not pursue here.

###  5.3. Reformulation of $F^{K}_{(v^{\prime},w^{\prime}),(v,w)}$

The result in this subsection holds for any Coxeter group. We first prove a
counterpart of Lemma 4.4.

######  Lemma 5.4.

Let $w^{\prime}\leqslant w$ in $W$. Then $\\{a\in W_{K}|w^{\prime}a\leqslant
w\\}=[e,z^{\prime}]$ for some $z^{\prime}\in W_{K}$.

###### Proof.

The proof is similar to the proof of Lemma 4.4. In order to match the
convention used in the edge labeling in (3.1), we prove the equivalent
statement that $Q=\\{a\in W_{K}|a(w{{}^{\prime}})^{-1}\leqslant
w^{-1}\\}=[e,x]$ for some $x\in W_{K}$.

It is clear that $e$ is the unique minimal element of $Q$. Let $x$ be any
maximal element of $Q$. It is clear that $[e,x]\subset Q$.

We fix a reflection order $\preceq$ on $W$ such that $t_{1}\preceq t_{2}$, for
any $t_{1}\in T\cap W_{K}$, $t_{2}\in T-W_{K}$. Recall that
$[(w{{}^{\prime}})^{-1},w^{-1}]$ is EL-shellable with respect to the
reflection order $\preceq$ on $T$.

We consider the unique increasing maximal chain in
$[x(w{{}^{\prime}})^{-1},w^{-1}]$, denoted by
$c(x(w{{}^{\prime}})^{-1},w^{-1})=x(w{{}^{\prime}})^{-1}\leqslant
x_{1}\leqslant\cdots\leqslant w$. We must have
$x(w{{}^{\prime}})^{-1}x_{1}^{-1}\not\in W_{K}$ by the maximality of $x$. Let
$c((w{{}^{\prime}})^{-1},x(w{{}^{\prime}})^{-1})=(w{{}^{\prime}})^{-1}\leqslant
y_{1}\leqslant\cdots\leqslant y_{l}\leqslant x(w{{}^{\prime}})^{-1}$ be the
unique increasing maximal chain in
$[(w{{}^{\prime}})^{-1},x(w{{}^{\prime}})^{-1}]$. We then have
$y_{l}w^{\prime}x^{-1}\in W_{K}$, since we can construct the increasing
maximal chain in $[e,x]\subset W_{K}$ by the labeling convention (3.1).

Hence the combined chain

$(w{{}^{\prime}})^{-1}\leqslant y_{1}\leqslant\cdots\leqslant y_{l}\leqslant
x(w{{}^{\prime}})^{-1}\leqslant x_{1}\leqslant\cdots\leqslant w^{-1}$

is increasing and maximal. By the definition of the EL-labeling on
$[(w{{}^{\prime}})^{-1},w^{-1}]$, this chain must be unique. Therefore the
element $x$ is unique. ∎

######  Proposition 5.5.

Let $(v^{\prime},w^{\prime})\leqslant(v,w)$ in $Q_{K}$. Let
$z=(v^{\prime})^{-1}\circ_{l}v$ and $z^{\prime}$ be the element defined in
Lemma 5.4. Set $N_{R}(v^{\prime})=\\{t\in T|v^{\prime}t<v^{\prime}\\}$. Then

$\displaystyle F^{K}_{(v^{\prime},w^{\prime}),(v,w)}$
$\displaystyle=\\{(a,b)\in W_{K}\times W_{K}|z\leqslant a\leqslant b\leqslant
z^{\prime},v^{\prime}a\circ_{r}b^{-1}=v^{\prime}\\}$
$\displaystyle=\\{(a,b)\in W_{K}\times W_{K}|z\leqslant a\leqslant b\leqslant
z^{\prime},ta\not\leqslant b\,\,\forall t\in N_{R}(v^{\prime})\\}$
$\displaystyle=\\{(a,b)\in W_{K}\times W_{K}|z\leqslant a\leqslant b\leqslant
z^{\prime},ta\not\leqslant b\,\,\forall t\in N_{R}(v^{\prime})\text{ with
}a\lessdot ta\\}.$

###### Proof.

Note that $z\in W_{K}$ by the definition of the partial ordering on $Q_{K}$.
By [HL]*Lemma 4.3, we have $v\leqslant v^{\prime}a$ if and only if
$(v^{\prime})^{-1}\circ_{l}v\leqslant a$ and $w^{\prime}b\leqslant w$ if and
only if $b\leqslant(w^{\prime})^{-1}\circ_{l}w$. Moreover, if $a\leqslant b$
and $v^{\prime}a\circ_{r}b^{-1}=v^{\prime}$, then $v^{\prime}\geqslant
v^{\prime}a\circ_{r}a^{-1}\geqslant v^{\prime}a\circ_{r}b^{-1}=v^{\prime}$.
Therefore $v^{\prime}a\circ_{r}a^{-1}=v^{\prime}$. By definition, there exists
$a^{\prime}\leqslant a$ such that $v^{\prime}=v^{\prime}a(a^{\prime})^{-1}$
and $\ell(v^{\prime}a)-\ell(a^{\prime})=\ell(v^{\prime})$. Hence
$a^{\prime}=a$ and $\ell(v^{\prime}a)=\ell(v^{\prime})+\ell(a)$. Therefore we
conclude that

$F^{K}_{(v^{\prime},w^{\prime}),(v,w)}=\\{(a,b)\in W_{K}\times
W_{K}|z\leqslant a\leqslant b\leqslant
z^{\prime};v^{\prime}a\circ_{r}b^{-1}=v^{\prime}\\}.$

Note that if $ta\leqslant b$ for some $t\in N_{R}(v^{\prime})$, then
$v^{\prime}a\circ_{r}b^{-1}\leqslant
v^{\prime}aa^{-1}t=v^{\prime}t<v^{\prime}$. Thus we have

$F^{K}_{(v^{\prime},w^{\prime}),(v,w)}\subset\\{(a,b)\in W_{K}\times
W_{K}|z\leqslant a\leqslant b\leqslant z^{\prime},ta\not\leqslant b\,\,\forall
t\in N_{R}(v^{\prime})\\}.$

It remains to prove that if $(a,b)\in W_{K}\times W_{K}$ with $z\leqslant
a\leqslant b\leqslant z^{\prime}$ and $ta\not\leqslant b$ for all $t\in
N_{R}(v^{\prime})$ with $a\lessdot ta$, we have
$F^{K}_{(v^{\prime},w^{\prime}),(v,w)}$.

Assume the contrary that $v^{\prime}a\circ_{r}b^{-1}<v^{\prime}$. Let
$b=s_{i_{1}}\cdots s_{i_{n}}$ be a reduced expression of $b$. We write
$v_{n}=v^{\prime}a\circ_{r}s_{i_{n}}$ and $v_{i}=v_{i+1}\circ_{r}s_{i_{i}}$
inductively. Fix a reduced expression $v^{\prime}a=s_{k_{1}}\cdots
s_{k_{l}}s_{k_{l+1}}\cdots s_{k_{l+l^{\prime}}}$, such that
$v^{\prime}=s_{k_{1}}\cdots s_{k_{l}}$ and $a=s_{k_{l+1}}\cdots
s_{k_{l+l^{\prime}}}$. Then by the deletion property of Coxeter groups, we
have either $v_{i}=v_{i+1}$ or $v_{i}$ is a subexpression of $v_{i+1}$ by
deleting a unique simple reflection in the (fixed) reduced expression. The
resulting expression is always reduced by the definition of
$-\circ_{r}s_{i_{i}}$. Since $a\leqslant b$, the resulting expression of
$v_{1}=v^{\prime}a\circ_{r}b^{-1}$ must be a subexpression of $v^{\prime}$. By
the assumption, $v^{\prime}a\circ_{r}b^{-1}<v^{\prime}$. Suppose that
$v_{j-1}$ is the first element obtained from $v_{j}$ by deleting a simple
reflection in $s_{k_{1}}\cdots s_{k_{l}}$. In other words, $j$ is the largest
integer such that $v_{j}=v^{\prime}a_{j}$ with $a_{j}\leqslant a$ and
$v_{j}\circ_{r}s_{i_{j}}=v^{\prime}_{j}a_{j}$ for $v^{\prime}_{j}\lessdot
v^{\prime}$.

By the definition of $\circ_{r}$, we have
$\ell(a)=\ell(a_{j})+\ell(a_{j}^{-1}a)$. We also have that
$v^{\prime}_{j}a_{j}=v^{\prime}a_{j}s_{i_{j}}=v^{\prime}(a_{j}s_{i_{j}}a_{j}^{-1})a_{j}$
and $v^{\prime}_{j}=v^{\prime}(a_{j}s_{i_{j}}a_{j}^{-1})$. Thus
$a_{j}s_{i_{j}}a_{j}^{-1}\in N_{R}(v^{\prime})$. Note that
$a_{j}s_{i_{j}}a_{j}^{-1}$ is a reflection and we have either
$a<a_{j}s_{i_{j}}a_{j}^{-1}a$ or $a>a_{j}s_{i_{j}}a_{j}^{-1}a$. If
$a>a_{j}s_{i_{j}}a_{j}^{-1}a$, then
$v^{\prime}a=(v^{\prime}a_{j}s_{i_{j}}a_{j}^{-1})(a_{j}s_{i_{j}}a_{j}^{-1}a)$
and
$\ell(v^{\prime})+\ell(a)=\ell(v^{\prime}a)\leqslant\ell(v^{\prime}a_{j}s_{i_{j}}a_{j}^{-1})+\ell(a_{j}s_{i_{j}}a_{j}^{-1}a)<\ell(v^{\prime})+\ell(a)$.
That is a contradiction. Hence
$a<a_{j}s_{i_{j}}a_{j}^{-1}a=a_{j}s_{i_{j}}(a_{j}^{-1}a)$. We have
$\ell(a)<\ell(a_{j}s_{i_{j}}(a_{j}^{-1}a))\leqslant\ell(a_{j})+1+\ell(a_{j}^{-1}a)=\ell(a)+1$.
Hence we conclude that $\ell(a_{j}s_{i_{j}}(a_{j}^{-1}a))=\ell(a)+1$ and
$a\lessdot a_{j}s_{i_{j}}(a_{j}^{-1}a)$.

Since $v^{\prime}a\circ_{r}b^{-1}=v^{\prime}a_{j}\circ_{r}(s_{i_{1}}\cdots
s_{i_{j}})^{-1}<v^{\prime}$. By definition, we have $a_{j}<s_{i_{1}}\cdots
s_{i_{j}}$. Also $a_{j}s_{i_{j}}>a_{j}$. Thus $a_{j}s_{i_{j}}\leqslant
s_{i_{1}}\cdots s_{i_{j}}$. So $a_{j}s_{i_{j}}(a_{j}^{-1}a)\leqslant
s_{i_{1}}\cdots s_{i_{j}}s_{i_{j+1}}\cdots s_{i_{n}}=b$. This contradicts the
assumption that $ta\not\leqslant b$ for all $t\in N_{R}(v^{\prime})$ with
$a\lessdot ta$. Hence $v^{\prime}a\circ_{r}b^{-1}=v^{\prime}$.

The proposition is proved. ∎

######  Remark 5.6.

Thanks to the reformulation in Proposition 5.5, one may regard the poset
$F^{K}_{(v^{\prime},w^{\prime}),(v,w)}$ as a generalization of
$Z_{\emptyset,J^{\prime}}$ considered in (4.1).

######  Corollary 5.7.

Let $(a,b)\in F^{K}_{(v^{\prime},w^{\prime}),(v,w)}$. Then for any
$(a^{\prime},b^{\prime})\in W_{K}\times W_{K}$ with $a\leqslant
a^{\prime}\leqslant b^{\prime}\leqslant b$, we have
$(a^{\prime},b^{\prime})\in F^{K}_{(v^{\prime},w^{\prime}),(v,w)}$.

###### Proof.

In the case where $W$ is the Weyl group of a Kac-Moody group, this statement
follows from Proposition 5.1 and the closeness of the fibers of $\pi_{v,w}$.
We give a combinatorial proof here, which applies to arbitrary Coxeter groups.

By Proposition 5.5, it suffices to show that
$v^{\prime}a^{\prime}\circ_{r}(b^{\prime})^{-1}=v^{\prime}$ for any
$a\leqslant a^{\prime}\leqslant b^{\prime}\leqslant b$.

Since $a^{\prime}\leqslant b^{\prime}$, by definition we have
$v^{\prime}a^{\prime}\circ_{r}(b^{\prime})^{-1}\leqslant v^{\prime}$. We prove
the other direction. By [He]*Lemma 2, we have

$v^{\prime}=v^{\prime}a\circ_{r}(b)^{-1}\leqslant(v^{\prime}\ast
a^{\prime})\circ_{r}(b^{\prime})^{-1}\leqslant(v^{\prime}\ast
a^{\prime})\circ_{r}(a^{\prime})^{-1}\leqslant v^{\prime}.$

In particular, we have $(v^{\prime}\ast
a^{\prime})\circ_{r}(b^{\prime})^{-1}=(v^{\prime}\ast
a^{\prime})\circ_{r}(a^{\prime})^{-1}=v^{\prime}$. We deduce from
$(v^{\prime}\ast a^{\prime})\circ_{r}(a^{\prime})^{-1}=v^{\prime}$ that
$v^{\prime}a^{\prime}=v^{\prime}\ast a^{\prime}$. Thus
$(v^{\prime}a^{\prime})\circ_{r}(b^{\prime})^{-1}=v^{\prime}$. The statement
is proved. ∎

###  5.4. Proof of Theorem 5.2

By [Dyer1]*Lemma 2.7, there exists an reflection order $\preceq$ on $T$ such
that

$t_{1}\preceq t_{2}\text{ for any }t_{1}\in N_{R}(v^{\prime})\text{ and
}t_{2}\in T-N_{R}(v^{\prime}).$ (5.2)

Let $M=M_{\preceq}([a,z^{\prime}])$ be the acyclic complete matching
constructed in §3. We show $M$ gives the desired matching on
$F^{K}_{(v^{\prime},w^{\prime}),(v,w)}$. The strategy of the proof is similar
to Theorem 4.1 thanks to the reformulation of the poset
$F^{K}_{(v^{\prime},w^{\prime}),(v,w)}$ in Proposition 5.5. However, we need a
detailed analysis on the generalized quotients of intervals, which we will
discuss below.

Let $z,z^{\prime}$ be as in Proposition 5.5. Let
$\\{v^{\prime}\\}\backslash[z,z^{\prime}]=\\{a\in[z,z^{\prime}]|\ell(v^{\prime}a)=\ell(v^{\prime})+\ell(a)\\}$
be the (left) generalised quotient of the interval $[z,z^{\prime}]$ in the
sense of [BW].222Note that we consider the left quotient here, while the right
quotient was considered in [BW]. There is clearly no essential difference.
Note that $a\in\\{v^{\prime}\\}\backslash[z,z^{\prime}]$ if and only if
$(a,a)\in F^{K}_{(v^{\prime},w^{\prime}),(v,w)}$.

Let $a,a^{\prime}\in\\{v^{\prime}\\}\backslash[z,z^{\prime}]$. Let $u$ be a
minimal upper bound of $a,a^{\prime}$ in $[z,z^{\prime}]$ (hence also in
$W_{K}$). Then by [BW]*Theorem 3.7, we see that
$\ell(v^{\prime}u)=\ell(v^{\prime})+\ell(u)$. Since the interval
$[z,z^{\prime}]$ is finite, we obtain a unique maximal element in
$\\{v^{\prime}\\}\backslash[z,z^{\prime}]$ by iterating the process. We denote
this element by $\tilde{z}$. We define

$\displaystyle P_{a}$ $\displaystyle=\\{b\in W_{K}|a\leqslant b\leqslant
z^{\prime},v^{\prime}a\circ_{r}b^{-1}=v^{\prime}\\}$ $\displaystyle=\\{b\in
W_{K}|a\leqslant b\leqslant z^{\prime},ta\not\leqslant b\,\,\forall t\in
N_{R}(v^{\prime})\\}.$

(a) We have $P_{a}=\\{a\\}$ if and only if $a=\tilde{z}$.

It is easy to see that $P_{\tilde{z}}=\\{\tilde{z}\\}$. Suppose that
$a<\tilde{z}$. By [BW]*Theorem 3.4, the generalized quotient
$\\{v^{\prime}\\}\backslash[z,z^{\prime}]$ is a graded poset with grading
given by the length function of $W$. Let
$a^{\prime}\in\\{v^{\prime}\\}\backslash[z,z^{\prime}]$ be such that
$a\lessdot a^{\prime}\leqslant\tilde{z}$.

We show $v^{\prime}a\circ_{r}(a^{\prime})^{-1}=v^{\prime}$. Assume the
contrary that $v^{\prime}a\circ_{r}(a^{\prime})^{-1}=u<v^{\prime}$. Since
$\ell(u)\geqslant\ell(v^{\prime}a)-\ell(a^{\prime})=\ell(v)-1$, we have
$v^{\prime}a(a^{\prime})^{-1}=u=v^{\prime}t$ for some $t\in
N_{R}(v^{\prime})$. Therefore $ta=a^{\prime}$ and
$\ell(v^{\prime}a^{\prime})=\ell(v^{\prime}ta)<\ell(v^{\prime})+\ell(a)$. This
contradicts the fact that
$a^{\prime}\in\\{v^{\prime}\\}\backslash[z,z^{\prime}]$. Thus
$P_{a}\supset\\{a,a^{\prime}\\}$. (a) is proved.

(b) The poset $P_{a}=\\{b\in W_{K}|a\leqslant b\leqslant
z^{\prime},ta\not\leqslant b\,\,\forall t\in N_{R}(v^{\prime})\\}$ has a
complete acyclic matching for any
$a\in\\{v^{\prime}\\}\backslash[z,z^{\prime}]$ unless $a=\tilde{z}$.

Assume $a<\tilde{z}\leqslant z^{\prime}$. Let $a_{1},a_{2},\dots,a_{s}$ be all
the atoms of $[a,z^{\prime}]$ such that $a_{i}a^{-1}\in N_{R}(v^{\prime})$.
Then, similarly to §4.3.2, we see that $[a,z^{\prime}]-\cup_{1\leqslant
i\leqslant s}[a_{i},z^{\prime}]$ is an $M$-subset due to the choice of the
reflection order $\preceq$ in (5.2). Recall if $b\in W_{K}$ with
$v^{\prime}a\circ_{r}b^{-1}<v^{\prime}$, then $a\lessdot ta\leqslant b$ for
some $t\in N_{R}(v^{\prime})$ by Proposition 5.5. So $b\in\cup_{1\leqslant
i\leqslant s}[a_{i},z^{\prime}]$. Therefore
$P_{a}=[a,z^{\prime}]-\cup_{1\leqslant i\leqslant s}[a_{i},z^{\prime}]$ has a
complete acyclic matching unless $a=\tilde{z}$.

The rest of the proof is entirely similar to that of Theorem 4.1.

###  5.5. Regularity theorem for partial flag varieties

The regularity theorem for totally nonnegative partial flag varieties was
first established for reductive groups in [GKL] and later for arbitrary Kac-
Moody groups in [BH22]. In both cases, the proofs involve the compatibility of
the totally positivity on partial flag varieties with certain atlas maps,
which are rather difficult to establish; see, for example, [BH22]*§6-§8. On
the other hand, the regularity theorem for totally nonnegative full flag
varieties is dramatically easier to establish based on the product structure
introduced in [BH22]*§5.1.

In this section, we give a new proof of the regularity theorem for totally
nonnegative partial flag varieties, assuming the regularity theorem for the
full flag varieties. We refer to [DHM]*§2.1 for the definition of CW
complexes.

We first recall the following topological result.

######  Proposition 5.8.

[DHM]*Corollary 2.33 Let $B^{n}$ be a $n$-dimensional closed ball and
$S^{n-1}$ be the boundary sphere of $B^{n}$. Let $\sim$ be an equivalent
relation on the close ball $B^{n}$ such that

  1. (1)

all equivalently classes are contractible ;

  2. (2)

$S^{n-1}/\sim$ is homeomorphisc to $S^{n-1}$;

  3. (3)

if $x\sim y$ with $x\in S^{n-1}$, then $y\in S^{n-1}$;

  4. (4)

if $x\sim y$ with $x\not\in S^{n-1}$, then $y=x$.

Then $B^{n}$ is homeomorphic to $B^{n}/\sim$.

Let $Q=\\{(v,w)|v\leqslant w\text{ in }W\\}$. We equip $Q$ with a partial
order $\leqslant$ such that $(a,b)\leqslant(v,w)$ if and only if $v\leqslant
a\leqslant b\leqslant w$ in the Bruhat order of $W$.

Recall §4.1 that $\mathcal{B}_{v,w,\geqslant 0}$ is a regular CW complex
homeomorphic to a closed ball $B^{\ell(w)-\ell(v)}$ for any $(v,w)\in Q$. By
[BH22]*Theorem 5.2, the boundary sphere of
$B^{n}\cong\mathcal{B}_{v,w,\geqslant 0}$ is precisely
$S^{n-1}\cong\sqcup_{(v^{\prime},w^{\prime})<(v,w)}\mathcal{B}_{v^{\prime},w^{\prime},>0}$.
We explain how to deduce that $\mathcal{P}_{K,v,w,\geqslant 0}$ is a regular
CW complex homeomorphic to a closed ball.

The natural projection $\pi_{v,w}:\mathcal{B}_{v,w,\geqslant
0}\rightarrow\mathcal{P}_{K,v,w,\geqslant 0}$ restricts to a homeomorphism
$\mathcal{B}_{v,w,>0}\cong\mathcal{P}_{K,v,w,>0}$ for any $(v,w)\in Q_{K}$.
Moreover, $\pi_{v,w}$ maps $S^{\ell(w)-\ell(v)-1}\cong\sqcup_{(a,b)<(v,w)\in
Q}\mathcal{B}_{a,b,>0}$ to $\sqcup_{(v^{\prime},w^{\prime})<(v,w)\in
Q_{K}}\mathcal{P}_{K,v^{\prime},w^{\prime},>0}$.

Since the choice of $(v,w)\in Q_{K}$ is arbitrary, this shows that
$\mathcal{P}_{K,v,w,\geqslant 0}$ is a CW complex with cells
$\mathcal{P}_{K,v^{\prime},w^{\prime},>0}$ for
$(v^{\prime},w^{\prime})\leqslant(v,w)\in Q_{K}$ with characteristic maps
$\pi_{v^{\prime},w^{\prime}}:\mathcal{B}_{v^{\prime},w^{\prime},\geqslant
0}\rightarrow\mathcal{P}_{K,v^{\prime},w^{\prime},\geqslant 0}$.

It remains to show that $\mathcal{P}_{K,v^{\prime},w^{\prime},\geqslant 0}$ is
homeomorphic to a closed ball for any $(v^{\prime},w^{\prime})\in Q_{K}$.
Without loss of generality, we show that $\mathcal{P}_{K,v,w,\geqslant 0}$ is
homeomorphic to a closed ball by induction on $n=\ell(w)-\ell(v)$. The base
case $n=0$ is trivial.

The projection $\pi_{v,w}$ factors through the natural quotient

${\mathcal{B}_{v,w,\geqslant 0}}$${\mathcal{P}_{K,v,w,\geqslant
0}}$${\mathcal{B}_{v,w,\geqslant
0}/\sim}$$\scriptstyle{\pi_{v,w}}$$\scriptstyle{\tilde{\pi}_{v,w}}$

Since $\pi_{v,w}$ is proper, hence close, $\tilde{\pi}_{v,w}$ is also a closed
map. Then we see that $\tilde{\pi}_{v,w}$ is a homeomorphism, since it is a
bijective continuous closed map.

We next show $(\mathcal{B}_{v,w,\geqslant
0}/\sim)\cong\mathcal{B}_{v,w,\geqslant 0}\cong B^{\ell(w)-\ell(v)}$ following
Lemma 5.8 and the induction hypothesis.

The equivalent classes of $\sim$ are precisely the fibers of $\pi_{v,w}$,
whose contractibility follows from Theorem 5.3. This verifies the condition
(1) in Proposition 5.8.

The boundary sphere of $B^{n}\cong\mathcal{B}_{v,w,\geqslant 0}$ is precisely
$S^{n-1}\cong\sqcup_{(v^{\prime},w^{\prime})<(v,w)}\mathcal{B}_{v^{\prime},w^{\prime},>0}$
by [BH22]*Theorem 5.2. Therefore, we have

$(S^{n-1}/\sim)\cong\pi_{v,w}(S^{n-1})=\bigsqcup_{(v^{\prime},w^{\prime})<(v,w)\in
Q_{K}}\mathcal{P}_{K,v^{\prime},w^{\prime},>0}.$

By the induction hypothesis, each closure
$\overline{\mathcal{P}_{K,v^{\prime},w^{\prime},>0}}=\mathcal{P}_{K,v^{\prime},w^{\prime},\geqslant
0}$ is a regular CW complex homeomorphic to a closed ball. Hence
$\sqcup_{(v^{\prime},w^{\prime})<(v,w)\in
Q_{K}}\mathcal{P}_{K,v^{\prime},w^{\prime},>0}$ is a regular CW complex. We
then see that it is a sphere of dimension $n-1$ by the combinatorics of the
face poset following [Bj2]. Hence $(S^{n-1}/\sim)\cong S^{n-1}$ verifying
condition (2) in Proposition 5.8.

Since $\pi_{v,w}:\mathcal{B}_{v,w,\geqslant
0}\rightarrow\mathcal{P}_{K,v,w,\geqslant 0}$ is the characteristic map of a
CW complex. Both conditions (3) and (4) in Proposition 5.8 follow immediately.

We have now verified all the conditions in Proposition 5.8. We conclude that
$\mathcal{P}_{v,w,\geqslant 0}$ is a regular CW complex homeomorphic to
$B^{\ell(w)-\ell(v)}$.

## References

  * [1]
Arkani-
HamedNimaBourjailyJacobCachazoFreddyGoncharovAlexanderPostnikovAlexanderTrnkaJaroslavGrassmannian
geometry of scattering amplitudesCambridge University Press,
Cambridge2016ix+194ISBN 978-1-107-08658-6Review
MathReviewsDocument@book{AHBC16, author = {Arkani-Hamed, Nima}, author =
{Bourjaily, Jacob}, author = {Cachazo, Freddy}, author = {Goncharov,
Alexander}, author = {Postnikov, Alexander}, author = {Trnka, Jaroslav}, title
= {Grassmannian geometry of scattering amplitudes}, publisher = {Cambridge
University Press, Cambridge}, date = {2016}, pages = {ix+194}, isbn =
{978-1-107-08658-6}, review = {\MR{3467729}}, doi =
{10.1017/CBO9781316091548}}

  * [3]
BjörnerAndersShellable and cohen-macaulay partially ordered setsTrans. Amer.
Math. Soc.26019801159–183ISSN 0002-9947Review MathReviewsDocument@article{Bj1,
author = {Bj\"{o}rner, Anders}, title = {Shellable and Cohen-Macaulay
partially ordered sets}, journal = {Trans. Amer. Math. Soc.}, volume = {260},
date = {1980}, number = {1}, pages = {159–183}, issn = {0002-9947}, review =
{\MR{0570784}}, doi = {10.2307/1999881}}

  * [5]
BjörnerAndersPosets, regular cw complexes and bruhat orderEuropean J.
Combin.5198417–16ISSN 0195-6698Review MathReviewsDocument@article{Bj2, author
= {Bj\"{o}rner, Anders}, title = {Posets, regular CW complexes and Bruhat
order}, journal = {European J. Combin.}, volume = {5}, date = {1984}, number =
{1}, pages = {7–16}, issn = {0195-6698}, review = {\MR{0746039}}, doi =
{10.1016/S0195-6698(84)80012-8}}

  * [7]
BjörnerAndersBrentiFrancescoCombinatorics of coxeter groupsGraduate Texts in
Mathematics231Springer, New York2005xiv+363ISBN 978-3540-442387ISBN
3-540-44238-3Review MathReviews@book{BB, author = {Bj\"{o}rner, Anders},
author = {Brenti, Francesco}, title = {Combinatorics of Coxeter groups},
series = {Graduate Texts in Mathematics}, volume = {231}, publisher =
{Springer, New York}, date = {2005}, pages = {xiv+363}, isbn =
{978-3540-442387}, isbn = {3-540-44238-3}, review = {\MR{2133266}}}

  * [9]
BrentiFrancescoThe intersection cohomology of schubert varieties is a
combinatorial invariantEuropean J. Combin.25200481151–1167ISSN 0195-6698Review
MathReviewsDocument@article{Brenti, author = {Brenti, Francesco}, title = {The
intersection cohomology of Schubert varieties is a combinatorial invariant},
journal = {European J. Combin.}, volume = {25}, date = {2004}, number = {8},
pages = {1151–1167}, issn = {0195-6698}, review = {\MR{2095476}}, doi =
{10.1016/j.ejc.2003.10.011}}

  * [11]
BrentiFrancescoCaselliFabrizioMariettiMarioSpecial matchings and kazhdan-
lusztig polynomialsAdv. Math.20220062555–601ISSN 0001-8708Review
MathReviewsDocument@article{BCM, author = {Brenti, Francesco}, author =
{Caselli, Fabrizio}, author = {Marietti, Mario}, title = {Special matchings
and Kazhdan-Lusztig polynomials}, journal = {Adv. Math.}, volume = {202}, date
= {2006}, number = {2}, pages = {555–601}, issn = {0001-8708}, review =
{\MR{2222360}}, doi = {10.1016/j.aim.2005.01.011}}

  * [13]
BaoHuanchenHeXuhuaFlag manifolds over semifieldsAlgebra Number
Theory15202182037–2069ISSN 1937-0652Document@article{BH21, author = {Bao,
Huanchen}, author = {He, Xuhua}, title = {Flag manifolds over semifields},
journal = {Algebra Number Theory}, volume = {15}, year = {2021}, number = {8},
pages = {2037–2069}, issn = {1937-0652}, doi = {10.2140/ant.2021.15.2037}}

  * [15]
BaoHuanchenHeXuhuaProduct structure and regularity theorem for totally
nonnegative flag
varietiesarXiv2022https://arxiv.org/abs/2203.02137@article{BH22, author =
{Bao, Huanchen}, author = {He, Xuhua}, title = {Product structure and
regularity theorem for totally nonnegative flag varieties}, publisher =
{arXiv}, year = {2022}, eprint = {https://arxiv.org/abs/2203.02137}}

  * [17]
BjörnerAndersWachsMichelle L.Generalized quotients in coxeter groupsTrans.
Amer. Math. Soc.308198811–37ISSN 0002-9947Review
MathReviewsDocument@article{BW, author = {Bj\"{o}rner, Anders}, author =
{Wachs, Michelle L.}, title = {Generalized quotients in Coxeter groups},
journal = {Trans. Amer. Math. Soc.}, volume = {308}, date = {1988}, number =
{1}, pages = {1–37}, issn = {0002-9947}, review = {\MR{0946427}}, doi =
{10.2307/2000946}}

  * [19]
ChariManoj K.On discrete morse functions and combinatorial
decompositionsEnglish, with English and French summariesFormal power series
and algebraic combinatorics (Vienna, 1997)Discrete Math.21720001-3101–113ISSN
0012-365XReview MathReviewsDocument@article{Chari, author = {Chari, Manoj K.},
title = {On discrete Morse functions and combinatorial decompositions},
language = {English, with English and French summaries}, note = {Formal power
series and algebraic combinatorics (Vienna, 1997)}, journal = {Discrete
Math.}, volume = {217}, date = {2000}, number = {1-3}, pages = {101–113}, issn
= {0012-365X}, review = {\MR{1766262}}, doi = {10.1016/S0012-365X(99)00258-7}}

  * [21]
DydakJerzyCohomological dimension theorytitle={Handbook of geometric
topology}, publisher={North-Holland, Amsterdam}, ISBN
0-444-82432-42002423–470Review MathReviews@article{Dyd02, author = {Dydak,
Jerzy}, title = {Cohomological dimension theory}, conference =
{title={Handbook of geometric topology}, }, book = {publisher={North-Holland,
Amsterdam}, }, isbn = {0-444-82432-4}, date = {2002}, pages = {423–470},
review = {\MR{1886675}}}

  * [23]
DyerMatthewHecke algebras and shellings of bruhat intervalsCompositio
Math.891993191–115ISSN 0010-437XReview MathReviews@article{Dyer1, author =
{Dyer, Matthew}, title = {Hecke algebras and shellings of Bruhat intervals},
journal = {Compositio Math.}, volume = {89}, date = {1993}, number = {1},
pages = {91–115}, issn = {0010-437X}, review = {\MR{1248893}}}

  * [25]
DvisJames F.HershPatriciaMillerEzraFibers of maps to totally nonnegative
spacesarXiv2019https://arxiv.org/abs/1903.01420@article{DHM, author = {Dvis,
James F.}, author = {Hersh, Patricia}, author = {Miller, Ezra}, title =
{Fibers of maps to totally nonnegative spaces}, publisher = {arXiv}, year =
{2019}, eprint = {https://arxiv.org/abs/1903.01420}}

  * [27]
FormanRobinMorse theory for cell complexesAdv. Math.1341998190–145ISSN
0001-8708Review MathReviewsDocument@article{Forman, author = {Forman, Robin},
title = {Morse theory for cell complexes}, journal = {Adv. Math.}, volume =
{134}, date = {1998}, number = {1}, pages = {90–145}, issn = {0001-8708},
review = {\MR{1612391}}, doi = {10.1006/aima.1997.1650}}

  * [29]
FominSergeyZelevinskyAndreiCluster algebras. i. foundationsJ. Amer. Math.
Soc.1520022497–529ISSN 0894-0347Review MathReviewsDocument@article{FZ, author
= {Fomin, Sergey}, author = {Zelevinsky, Andrei}, title = {Cluster algebras.
I. Foundations}, journal = {J. Amer. Math. Soc.}, volume = {15}, date =
{2002}, number = {2}, pages = {497–529}, issn = {0894-0347}, review =
{\MR{1887642}}, doi = {10.1090/S0894-0347-01-00385-X}} GalashinPavelKarpSteven
N.LamThomasRegularity theorem for totally nonnegative flag varietiesJ. Amer.
Math. Soc.3520222513–579ISSN 0894-0347Review MathReviewsDocument@article{GKL,
author = {Galashin, Pavel}, author = {Karp, Steven N.}, author = {Lam,
Thomas}, title = {Regularity theorem for totally nonnegative flag varieties},
journal = {J. Amer. Math. Soc.}, volume = {35}, date = {2022}, number = {2},
pages = {513–579}, issn = {0894-0347}, review = {\MR{4374956}}, doi =
{10.1090/jams/983}}

  * [32]
HeXuhuaA subalgebra of 0-hecke algebraJ. Algebra3222009114030–4039ISSN
0021-8693Review MathReviewsDocument@article{He, author = {He, Xuhua}, title =
{A subalgebra of 0-Hecke algebra}, journal = {J. Algebra}, volume = {322},
date = {2009}, number = {11}, pages = {4030–4039}, issn = {0021-8693}, review
= {\MR{2556136}}, doi = {10.1016/j.jalgebra.2009.04.003}}

  * [34]
HeXuhuaLuJiang-HuaOn intersections of certain partitions of a group
compactificationInt. Math. Res. Not. IMRN2011112534–2564ISSN 1073-7928Review
MathReviewsDocument@article{HL, author = {He, Xuhua}, author = {Lu, Jiang-
Hua}, title = {On intersections of certain partitions of a group
compactification}, journal = {Int. Math. Res. Not. IMRN}, date = {2011},
number = {11}, pages = {2534–2564}, issn = {1073-7928}, review =
{\MR{2806588}}, doi = {10.1093/imrn/rnq169}}

  * [36]
HeXuhuaLusztigGeorgeTotal positivity and conjugacy
classesarXiv2022https://arxiv.org/abs/2201.12479@article{HL1, author = {He,
Xuhua}, author = {Lusztig, George}, title = {Total positivity and conjugacy
classes}, publisher = {arXiv}, year = {2022}, eprint =
{https://arxiv.org/abs/2201.12479}}

  * [38]
LamThomasTotally nonnegative grassmannian and grassmann
polytopestitle={Current developments in mathematics 2014}, publisher={Int.
Press, Somerville, MA}, ISBN 978-1-57146-313-5ISBN
1-57146-313-5201651–152Review MathReviews@article{Lam15, author = {Lam,
Thomas}, title = {Totally nonnegative Grassmannian and Grassmann polytopes},
conference = {title={Current developments in mathematics 2014}, }, book =
{publisher={Int. Press, Somerville, MA}, }, isbn = {978-1-57146-313-5}, isbn =
{1-57146-313-5}, date = {2016}, pages = {51–152}, review = {\MR{3468251}}}

  * [40]
MarshR. J.RietschK.Parametrizations of flag varietiesRepresent.
Theory82004212–242Review MathReviewsDocument@article{MR, author = {Marsh, R.
J.}, author = {Rietsch, K.}, title = {Parametrizations of flag varieties},
journal = {Represent. Theory}, volume = {8}, date = {2004}, pages = {212–242},
review = {\MR{2058727}}, doi = {10.1090/S1088-4165-04-00230-4}}

  * [42]
JonesBrant C.An explicit derivation of the möbius function for bruhat
orderOrder2620094319–330ISSN 0167-8094Review
MathReviewsDocument@article{Jones, author = {Jones, Brant C.}, title = {An
explicit derivation of the M\"{o}bius function for Bruhat order}, journal =
{Order}, volume = {26}, date = {2009}, number = {4}, pages = {319–330}, issn =
{0167-8094}, review = {\MR{2591858}}, doi = {10.1007/s11083-009-9128-6}}

  * [44]
LusztigG.Total positivity in reductive groupstitle={Lie theory and geometry},
series={Progr. Math.}, volume={123}, publisher={Birkh\"{a}user Boston, Boston,
MA}, ISBN 0-8176-3761-31994531–568Review MathReviewsDocument@article{Lu94,
author = {Lusztig, G.}, title = {Total positivity in reductive groups},
conference = {title={Lie theory and geometry}, }, book = {series={Progr.
Math.}, volume={123}, publisher={Birkh\"{a}user Boston, Boston, MA}, }, isbn =
{0-8176-3761-3}, date = {1994}, pages = {531–568}, review = {\MR{1327548}},
doi = {10.1007/978-1-4612-0261-5\\_20}}

  * [46]
LusztigG.Total positivity in reductive groups, IIBull. Inst. Math. Acad. Sin.
(N.S.)1420194403–459ISSN 2304-7909DocumentLink@article{Lu19, author =
{Lusztig, G.}, title = {Total positivity in reductive groups, {II}}, journal =
{Bull. Inst. Math. Acad. Sin. (N.S.)}, volume = {14}, year = {2019}, number =
{4}, pages = {403–459}, issn = {2304-7909}, doi = {10.21915/bimas.2019402},
url = {https://doi.org/10.21915/bimas.2019402}}

  * [48]
LusztigG.Total positivity in springer fibresQ. J. Math.7220211-231–49ISSN
0033-5606Review MathReviewsDocument@article{LuSp, author = {Lusztig, G.},
title = {Total positivity in Springer fibres}, journal = {Q. J. Math.}, volume
= {72}, date = {2021}, number = {1-2}, pages = {31–49}, issn = {0033-5606},
review = {\MR{4271379}}, doi = {10.1093/qmathj/haaa021}}

  * [50]
RietschKonstanzeAn algebraic cell decomposition of the nonnegative part of a
flag varietyJ. Algebra21319991144–154ISSN 0021-8693Review
MathReviewsDocument@article{Rie99, author = {Rietsch, Konstanze}, title = {An
algebraic cell decomposition of the nonnegative part of a flag variety},
journal = {J. Algebra}, volume = {213}, date = {1999}, number = {1}, pages =
{144–154}, issn = {0021-8693}, review = {\MR{1674668}}, doi =
{10.1006/jabr.1998.7665}}

  * [52]
RietschKonstanzeClosure relations for totally nonnegative cells in $G/P$Math.
Res. Lett.1320065-6775–786ISSN 1073-2780Review
MathReviewsDocument@article{Rie06, author = {Rietsch, Konstanze}, title =
{Closure relations for totally nonnegative cells in $G/P$}, journal = {Math.
Res. Lett.}, volume = {13}, date = {2006}, number = {5-6}, pages = {775–786},
issn = {1073-2780}, review = {\MR{2280774}}, doi =
{10.4310/MRL.2006.v13.n5.a8}}

  * [54]
RietschKonstanzeWilliamsLaurenDiscrete morse theory for totally non-negative
flag varietiesAdv. Math.223201061855–1884ISSN 0001-8708Review
MathReviewsDocument@article{RW, author = {Rietsch, Konstanze}, author =
{Williams, Lauren}, title = {Discrete Morse theory for totally non-negative
flag varieties}, journal = {Adv. Math.}, volume = {223}, date = {2010}, number
= {6}, pages = {1855–1884}, issn = {0001-8708}, review = {\MR{2601003}}, doi =
{10.1016/j.aim.2009.10.011}}

  * [56]