[theorem] [lemma] [proposition] [corollary]

# Braid variety cluster structures, II: general type

Pavel Galashin  Department of Mathematics, University of California, Los
Angeles, 520 Portola Plaza, Los Angeles, CA 90025, USA<EMAIL_ADDRESS>,  Thomas Lam  Department of Mathematics, University of Michigan, 2074 East
Hall, 530 Church Street, Ann Arbor, MI 48109-1043, USA<EMAIL_ADDRESS>and
Melissa Sherman-Bennett  Massachusetts Institute of Technology, 77
Massachusetts Avenue, Cambridge, MA 02139<EMAIL_ADDRESS>
###### Abstract.

We show that braid varieties for any algebraic group $G$ are cluster
varieties. This includes open Richardson varieties inside the flag variety
$G/B$.

###### Key words and phrases:

Cluster algebra, braid variety, open Richardson variety, local acyclicity,
Deodhar hypersurface, algebraic group.

###### 2020 Mathematics Subject Classification:

Primary: 13F60. Secondary: 14M15.

P.G. was supported by an Alfred P. Sloan Research Fellowship and by the
National Science Foundation under Grants No. DMS-1954121 and No. DMS-2046915.
T.L. was supported by the National Science Foundation under Grant No.
DMS-1953852. M.S.B. was supported by the National Science Foundation under
Award No. DMS-2103282. Any opinions, findings, and conclusions or
recommendations expressed in this material are those of the authors and do not
necessarily reflect the views of the National Science Foundation.

##  1\. Introduction

Let $G$ be a complex simple algebraic group, $B_{\pm}$ opposing Borel
subgroups, $U_{\pm}$ their unipotent radicals, $H:=B_{+}\cap B_{-}$ the torus,
$I$ the vertex set of the Dynkin diagram, $W$ the Weyl group with simple
generators $s_{i},i\in I$, and denote by $\dot{w}$ the lift of $W$ to $G$ as
in (2.1). Let $w_{\circ}\in W$ denote the longest element and $i\mapsto i^{*}$
the action of $w_{\circ}$ on $I$. Let
$\alpha_{i},\alpha_{i}^{\vee},\omega_{i}$ for $i\in I$ denote the simple
roots, simple coroots, fundamental weights, respectively, and let
$A=(a_{ij})_{i,j\in I}$ be the Cartan matrix given by
$a_{ij}:={\langle}\alpha_{i},\alpha^{\vee}_{j}{\rangle}$. Denote
$d_{i}:=2/(\alpha_{i},\alpha_{i})$ so that $d_{i}a_{ij}=d_{j}a_{ji}$.

###  1.1. Double braid varieties

A _double braid word_ $\beta=i_{1}i_{2}\cdots i_{m}$ is a word in the alphabet
$\pm I$. For $i\in I$, we set $(-i)^{*}:=-i^{*}$. For $i\in\pm I$, define

(1.1) $s_{i}^{+}:=\begin{cases}s_{i},&\text{if }i>0,\\\
{\operatorname{id}},&\text{if }i<0,\end{cases}\quad
s_{i}^{-}:=\begin{cases}{\operatorname{id}},&\text{if }i>0,\\\
s_{-i},&\text{if }i<0.\\\ \end{cases}$

A _weighted flag_ is an element $F=gU_{+}\in G/U_{+}$. Two weighted flags
$(F,F^{\prime})$ are called _weakly $w$-related_ (resp., _strictly
$w$-related_) if there exist $g\in G$ and $h\in H$ (resp., $g\in G$) such that
$(gF,gF^{\prime})=(U_{+},h\dot{w}U_{+})$ (resp.,
$(gF,gF^{\prime})=(U_{+},\dot{w}U_{+})$). We write this as
$F\stackrel{{\scriptstyle w\ }}{{\Longrightarrow}}F^{\prime}$ (resp.,
$F\stackrel{{\scriptstyle w\ }}{{\longrightarrow}}F^{\prime}$).

Suppose that the Demazure product of $\beta$ is $w_{\circ}$; see (2.5). We
consider tuples $(X_{\bullet},Y_{\bullet})$ of weighted flags satisfying the
relative position conditions

(1.2)
${X_{0}}$${X_{1}}$${\cdots}$${X_{m}}$${Y_{0}}$${Y_{1}}$${\cdots}$${Y_{m}.}$$\scriptstyle{s_{i_{1}}^{+}}$$\scriptstyle{s_{i_{2}}^{+}}$$\scriptstyle{s_{i_{m}}^{+}}$$\scriptstyle{s_{i_{1}^{\ast}}^{-}}$$\scriptstyle{w_{\circ}}$$\scriptstyle{s_{i_{2}^{\ast}}^{-}}$$\scriptstyle{s_{i_{m}^{\ast}}^{-}}$

The group $G$ acts on such tuples by acting on each individual weighted flag,
and this action is free (Remark 2.15). The _double braid variety_
$\accentset{\circ}{R}_{\beta}$ is a complex, affine, irreducible variety
defined as the quotient modulo the $G$-action of the configuration space of
tuples $(X_{\bullet},Y_{\bullet})$ satisfying (1.2).

Double braid varieties include open Richardson varieties [KLS14, Lec16], open
positroid varieties [KLS13], double Bott-Samelson cells [SW21], the spaces in
[WY07], and the braid varieties of [Mel19, CGGS20]; see [GLSBS22] for further
discussion. For each $\beta$, we construct a cluster seed $\Sigma_{\beta}$.
Our main result is the following.

######  Theorem 1.1.

The coordinate ring of $\accentset{\circ}{R}_{\beta}$ is isomorphic to the
cluster algebra ${\mathcal{A}}_{\beta}={\mathcal{A}}(\Sigma_{\beta})$.

Theorem 1.1 settles conjectures of [Lec16, CGGS21]. In our earlier joint work
[GLSBS22] with Speyer, we constructed cluster structures in the type $A$ case
($G=\operatorname{SL}_{n}$) using Deodhar geometry and the combinatorics of 3D
plabic graphs and associated surfaces. The present work uses a very similar
overall proof strategy, but replaces the combinatorics with Deodhar geometry,
and thus gives an alternative approach even in the type $A$ case. This paper
can be read independently from [GLSBS22].

It would be interesting to understand the relation between our approach and
the cluster-categorical approach of [GLS06, BIRS09, Lec16, Mén22, CK22], as
well as the Legendrian-geometric viewpoint of [CGGS21].

######  Remark 1.2.

Recently, a proof of Theorem 1.1 was independently obtained in [CGG+22]. We
thank the authors of [CGG+22] for updating us on their progress.

One application of Theorem 1.1 is that a _curious Lefschetz theorem_ (see
[HRV08, LS16], [GLSBS22, Theorem 10.1], and [GL20, Theorem 1.5]) holds for
double braid varieties; see Theorem 6.7. In the case of open Richardson
varieties, this implies that the doubly-graded extension group ${\rm
Ext}_{\mathcal{O}}(M_{w},M_{v})$ of two Verma modules in Category
$\mathcal{O}$ satisfies curious Lefschetz; cf. [GL20, Section 1.11].

###  1.2. Constructing the seed

We give an informal summary of our construction; see Section 2 for details.
First, we introduce an open dense algebraic torus
$T_{\beta}\subset\accentset{\circ}{R}_{\beta}$ called the _Deodhar torus_. It
is defined by requiring the weighted flags $X_{c},Y_{c}$ to be weakly
$w_{c}$-related, where $w_{c}\in W$ is maximal possible subject to (1.2) (for
each $c=0,1,\dots,m$). The complement $\accentset{\circ}{R}_{\beta}\setminus
T_{\beta}$ is a union of irreducible _mutable Deodhar hypersurfaces_
$\\{V_{c}\mid c\in J_{\beta}^{{\rm mut}}\\}$. We define a partial
compactification $\tilde{R}_{\beta}$ of $\accentset{\circ}{R}_{\beta}$ so that
the complement $\tilde{R}_{\beta}\setminus T_{\beta}$ also includes _frozen
Deodhar hypersurfaces_ $\\{V_{c}\mid c\in J_{\beta}^{{\rm fro}}\\}$. We let
$J_{\beta}:=J_{\beta}^{{\rm fro}}\sqcup J_{\beta}^{{\rm mut}}$. The following
is the key definition of our approach.

######  Proposition-Definition 1.3.

For $c\in J_{\beta}$, define the _cluster variable_ $x_{c}$ to be the unique
character of $T_{\beta}$ that vanishes to order one on $V_{c}$ and has neither
a pole nor a zero on $V_{e}$ for $e\in J_{\beta}\setminus\\{c\\}$. We denote
the cluster by ${\mathbf{x}}_{\beta}=\\{x_{c}\\}_{c\in J_{\beta}}$.

We show that the cluster variables form a basis of the character lattice of
$T_{\beta}$, and that they extend to regular functions on
$\accentset{\circ}{R}_{\beta}$. In order to define a cluster structure on
$\accentset{\circ}{R}_{\beta}$, we need to also introduce a skew-symmetrizable
matrix ${\tilde{B}}=({\tilde{B}}_{ij})_{i\in J_{\beta},j\in J_{\beta}^{{\rm
mut}}}$. To do so, we consider a $2$-form $\omega_{\beta}$ on
$\accentset{\circ}{R}_{\beta}$. It is defined in terms of generalized minors
of the elements $Z_{c}:=Y_{c}^{-1}X_{c}\in U_{+}\backslash G/U_{+}$, and is a
sum of local contributions for each letter of $\beta$. This is similar in
spirit to [BFZ05, FG06, Ing19, SW21]. We introduce integers
$\mathbf{d}=(d_{i})_{i\in J_{\beta}}$ and expand $\omega_{\beta}$ in the basis
of cluster variables:

(1.3) $\omega_{\beta}=\sum_{c,e\in J_{\beta}:\ c\leq
e}d_{e}{\tilde{B}}_{ce}{\rm dlog}x_{c}\wedge{\rm dlog}x_{e}=\sum_{c,e\in
J_{\beta}:\ c\leq e}d_{c}{\tilde{B}}_{ec}{\rm dlog}x_{e}\wedge{\rm
dlog}x_{c}.$

We show that the coefficients ${\tilde{B}}_{ce}$ are integers for $(c,e)\in
J_{\beta}\times J_{\beta}^{{\rm mut}}$, and define the $J_{\beta}\times
J_{\beta}^{{\rm mut}}$ matrix ${\tilde{B}}:=({\tilde{B}}_{ce})$ . The
principal $J_{\beta}^{{\rm mut}}\times J_{\beta}^{{\rm mut}}$ part of the
matrix ${\tilde{B}}$ is skew-symmetrizable, with symmetrizer ${\rm
diag}(d_{i}\mid i\in J_{\beta}^{{\rm mut}})$. Therefore
$\Sigma_{\beta}:=({\mathbf{x}}_{\beta},{\tilde{B}})$ is a seed of a cluster
algebra $\mathcal{A}(\Sigma_{\beta})$ of geometric type. Our main result
(Theorem 1.1) is an equality
$\mathcal{A}(\Sigma_{\beta})={\mathbb{C}}[\accentset{\circ}{R}_{\beta}]$.

###  1.3. Overview of the proof

The proof that $\Sigma_{\beta}$ provides a cluster structure for
$\accentset{\circ}{R}_{\beta}$ is obtained in two steps. First, we use a
notion of deletion-contraction for cluster seeds developed in Section 3.3. We
apply this in the case when $\beta=ii\beta^{\prime}$ starts with a repeated
letter: the braids $i\beta^{\prime}$ and $\beta^{\prime}$ correspond to
deletion and contraction, respectively. We show in Theorem 4.10 that if
Theorem 1.1 holds for $i\beta^{\prime}$ and $\beta^{\prime}$ then it holds for
$\beta$.

Second, in Section 4, we describe moves $\beta\sim\beta^{\prime}$ on double
braid words that induce natural isomorphisms
$\accentset{\circ}{R}_{\beta}\cong\accentset{\circ}{R}_{\beta^{\prime}}$. We
show that the isomorphisms
$\accentset{\circ}{R}_{\beta}\cong\accentset{\circ}{R}_{\beta^{\prime}}$ give
rise to a sequence of mutations connecting the corresponding seeds
$\Sigma_{\beta}$ and $\Sigma_{\beta^{\prime}}$; see Theorem 4.2. We prove
Theorem 4.2 in the simply-laced case (i.e., for $G$ of type $A$, $D$, $E$) in
Section 4. In the simply-laced case, the seeds $\Sigma_{\beta}$ and
$\Sigma_{\beta^{\prime}}$ either coincide or are related by a single mutation.

The proof of Theorem 4.2 in the multiply-laced type is achieved via folding in
Sections 5 and 6. Here, we show that there is always a sequence of mutations
that relates the seeds $\Sigma_{\beta}$ and $\Sigma_{\beta^{\prime}}$. This
generalizes a result of Fock and Goncharov [FG06, Theorem 3.5].

Finally, in Section 7, we give a combinatorial algorithm for computing our
seeds using only root system combinatorics. See [Gal23] for a Sage
implementation and for some examples.

### Acknowledgments

We are indebted to David Speyer for his contributions to this project. We
thank Roger Casals, Eugene Gorsky, and Daping Weng for inspiring
conversations.

##  2\. Deodhar geometry

We discuss the geometry of the double braid variety
$\accentset{\circ}{R}_{\beta}$ with the goal of defining a cluster seed on it.
The ingredients of a cluster seed were outlined in Section 1.2. In Section
2.3, we construct a _Deodhar torus_
$T_{\beta}\subset\accentset{\circ}{R}_{\beta}$. In Section 2.7, we introduce a
family ${\mathbf{x}}_{\beta}=\\{x_{c}\\}_{c\in J_{\beta}}$ of _cluster
variables_ and show that they are regular functions on
$\accentset{\circ}{R}_{\beta}$. Finally, in Section 2.8, we introduce a
$2$-form $\omega_{\beta}$ on $T_{\beta}$ from which the ${\tilde{B}}$-matrix
can be extracted via (1.3).

###  2.1. Background

For each $i\in I$, we fix a group homomorphism

$\phi_{i}:\operatorname{SL}_{2}\to G,\quad\begin{pmatrix}1&t\\\
0&1\end{pmatrix}\mapsto x_{i}(t),\quad\begin{pmatrix}1&0\\\
t&1\end{pmatrix}\mapsto y_{i}(t),$

where $x_{i}(t),y_{i}(t)$ are the exponentiated Chevalley generators. The data
$(H,B_{+},B_{-},x_{i},y_{i};i\in I)$ is called a _pinning_ of $G$; see [Lus94,
Section 1.1].

Let $\Phi$ be the root system of $G$, with positive roots $\Phi^{+}$
corresponding to $B_{+}$. Let
$X^{\ast}(H):=\operatorname{Hom}(H,{\mathbb{C}}^{\times})$ be the _character
lattice_ of $H$ and $X_{\ast}(H):=\operatorname{Hom}({\mathbb{C}}^{\times},H)$
be the _cocharacter lattice_ of $H$. Let $\\{\alpha_{i}\\}_{i\in I}\subset
X^{\ast}(H)$ (resp., $\\{\alpha^{\vee}_{i}\\}_{i\in I}\subset X_{\ast}(H)$,
$\\{\omega_{i}\\}_{i\in I}\subset X^{\ast}(H)$) be the simple roots (resp.,
simple coroots, fundamental weights) of $\Phi^{+}$. We have a natural pairing
${\langle}\cdot,\cdot{\rangle}:X^{\ast}(H)\times X_{\ast}(H)\to{\mathbb{Z}}$
satisfying ${\langle}\omega_{i},\alpha^{\vee}_{j}{\rangle}=\delta_{ij}$ and
${\langle}\alpha_{i},\alpha^{\vee}_{j}{\rangle}=a_{ij}$, where
$A=(a_{ij})_{i,j\in I}$ is the Cartan matrix of $G$.

The Weyl group is given by $W=N_{G}(H)/H$, where $N_{G}(H)$ is the normalizer
of $H$ in $G$. It is a Coxeter group with generators $\\{s_{i}\\}_{i\in I}$
and length function $\ell(\cdot)$. We denote by ${\operatorname{id}}\in W$ the
identity element. For $i\in I$, we set

$\dot{s}_{i}=\overline{s_{i}}:=\phi_{i}\begin{pmatrix}0&-1\\\
1&0\end{pmatrix},\quad\dot{s}_{i}^{-1}=\overline{\overline{s_{i}}}:=\phi_{i}\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}.$

For a reduced expression $w=s_{i_{1}}s_{i_{2}}\cdots s_{i_{l}}$, where
$l=\ell(w)$, we set

(2.1)
$\dot{w}=\overline{w}:=\overline{s_{i_{1}}}\cdot\overline{s_{i_{2}}}\cdots\overline{s_{i_{l}}},\quad\overline{\overline{w}}:=\overline{\overline{s_{i_{1}}}}\cdot\overline{\overline{s_{i_{2}}}}\cdots\overline{\overline{s_{i_{l}}}}.$

The resulting product does not depend on the choice of the reduced expression.
For $u\in W$ and $h\in H$, we set

(2.2) $u\cdot
h:=\dot{u}h\dot{u}^{-1}=\overline{u}h\overline{u}^{-1}=\overline{\overline{u}}h\overline{\overline{u}}^{-1}.$

We also consider elements

(2.3) $z_{i}(t):=\phi_{i}\begin{pmatrix}t&-1\\\
1&0\end{pmatrix}=x_{i}(t)\dot{s}_{i},\quad\bar{z}_{i}(t):=\phi_{i}\begin{pmatrix}t&1\\\
-1&0\end{pmatrix}=x_{i}(-t)\dot{s}^{-1}_{i}.$

For each $w\in W$, it is well known that the multiplication map gives rise to
an isomorphism

(2.4) $(\dot{w}^{-1}U_{+}\dot{w}\cap U_{-})\times(\dot{w}^{-1}U_{+}\dot{w}\cap
U_{+})\xrightarrow{\sim}\dot{w}^{-1}U_{+}\dot{w}.$

###  2.2. Weighted flags

Recall from Section 1.1 that a _weighted flag_ is an element $F=gU_{+}\in
G/U_{+}$. Associated to a weighted flag $F$ is the flag $\pi(F)=gB_{+}$, the
image of $F$ in $G/B_{+}$.

The following elementary facts can be found in e.g. [SW21, Appendix]; see also
[GLSBS22, Section 6.2].

######  Lemma 2.1.

Let $F,F^{\prime},F^{\prime\prime}$ be weighted flags.

  1. (1)

$F\stackrel{{\scriptstyle{\operatorname{id}}\ }}{{\longrightarrow}}F^{\prime}$
if and only if $F=F^{\prime}$.

  2. (2)

If $F\stackrel{{\scriptstyle v\
}}{{\longrightarrow}}F^{\prime}\stackrel{{\scriptstyle w\
}}{{\longrightarrow}}F^{\prime\prime}$ and $\ell(vw)=\ell(v)+\ell(w)$, then
$F\stackrel{{\scriptstyle vw\ }}{{\longrightarrow}}F^{\prime\prime}$.

  3. (3)

Suppose $\ell(vw)=\ell(v)+\ell(w)$. If $F\stackrel{{\scriptstyle vw\
}}{{\longrightarrow}}F^{\prime\prime}$, then there exists a unique
$F^{\prime}$ such that $F\stackrel{{\scriptstyle v\
}}{{\longrightarrow}}F^{\prime}\stackrel{{\scriptstyle w\
}}{{\longrightarrow}}F^{\prime\prime}$. If $F\stackrel{{\scriptstyle vw\
}}{{\Longrightarrow}}F^{\prime\prime}$ then there exist unique
$F^{\prime}_{1},F^{\prime}_{2}$ such that $F\stackrel{{\scriptstyle v\
}}{{\longrightarrow}}F^{\prime}_{1}\stackrel{{\scriptstyle w\
}}{{\Longrightarrow}}F^{\prime\prime}$ and $F\stackrel{{\scriptstyle v\
}}{{\Longrightarrow}}F^{\prime}_{2}\stackrel{{\scriptstyle w\
}}{{\longrightarrow}}F^{\prime\prime}$.

######  Lemma 2.2.

Suppose $F\stackrel{{\scriptstyle s_{i}\ }}{{\longrightarrow}}F^{\prime}$ and
say $F=gU_{+}$. Then there exists a unique $t\in{\mathbb{C}}$ such that
$F^{\prime}=gz_{i}(t)U_{+}$. Similarly, if $F^{\prime}=g^{\prime}U_{+}$, there
exists a unique $t^{\prime}\in{\mathbb{C}}$ such that
$F=g^{\prime}\bar{z}_{i}(t^{\prime})U_{+}$. The maps $(g,F^{\prime})\mapsto t$
and $(g^{\prime},F)\mapsto t^{\prime}$ are regular functions on the
corresponding subvarieties of $G\times G/U_{+}$.

######  Lemma 2.3.

Suppose $F\stackrel{{\scriptstyle v\
}}{{\Longrightarrow}}gU_{+}\stackrel{{\scriptstyle s_{i}\
}}{{\longrightarrow}}gz_{i}(t)U_{+}$ and let $w$ be such that
$F\stackrel{{\scriptstyle w\ }}{{\Longrightarrow}}gz_{i}(t)U_{+}$. If
$vs_{i}>v$, then $w=vs_{i}$ for all $t\in{\mathbb{C}}$. If $vs_{i}<v$, then
there exists $t^{*}\in{\mathbb{C}}$ such that $w=vs_{i}$ for $t=t^{*}$ and
$w=v$ for $t\in{\mathbb{C}}\setminus\\{t^{*}\\}$. In particular,
$w\in\\{v,vs_{i}\\}$.

###  2.3. Deodhar torus

Fix $\beta=i_{1}i_{2}\dots i_{m}\in(\pm I)^{m}$. We write
$[m]:=\\{1,2,\dots,m\\}$ and $[0,m]:=\\{0,1,\dots,m\\}$. Recall the notation
$s_{i}^{+}$ and $s_{i}^{-}$ from (1.1).

Given two elements $u,v\in W$ which differ by a simple transposition (i.e.,
$u=vs_{i}$ or $u=s_{i}v$ for some $i\in I$) and such that, say,
$\ell(u)<\ell(v)$, we write $\max(u,v):=v$ and $\min(u,v):=u$. The _Demazure
product_ of $\beta$ is defined by

(2.5)
$\pi(\beta):=s_{i_{m}}^{-}*s_{i_{m-1}}^{-}*\cdots*s_{i_{1}}^{-}*s_{i_{1}}^{+}*s_{i_{2}}^{+}*\cdots*s_{i_{m}}^{+}\quad\in
W,$

where $*$ denotes the standard Demazure product on $W$, defined so that
$w*s_{i}:=\max(w,ws_{i})$. From now on, we assume that $\pi(\beta)=w_{\circ}$.

######  Definition 2.4.

A _$w_{\circ}$ -subexpression_ of $\beta$ is a sequence
${\mathbf{u}}=(u_{0},u_{1},\dots,u_{m})\in W^{m+1}$ such that
$u_{0}={\operatorname{id}}$, $u_{m}=w_{\circ}$, and such that for each
$c\in[m]$, we have either $u_{c-1}=u_{c}$ or
$u_{c-1}=s_{i_{c}}^{-}u_{c}s_{i_{c}}^{+}$. If $w_{\circ}\leq\beta$ then there
exists a unique “rightmost” $w_{\circ}$-subexpression, called the _$w_{\circ}$
-positive distinguished subexpression ($w_{\circ}$-PDS)_. It is given by
$u_{m}:=w_{\circ}$ and $u_{c-1}:=\min(u_{c},s_{i_{c}}^{-}u_{c}s_{i_{c}}^{+})$
for all $c=m,m-1,\dots,1$.

We set $J_{\beta}:=\\{c\in[m]\mid u_{c}=u_{c-1}\\}$. We refer to the indices
in $J_{\beta}$ as _solid crossings_ and to the indices in $[m]\setminus
J_{\beta}$ as _hollow crossings_. We denote
$d(\beta):=m-\ell(w_{\circ})=|J_{\beta}|$,

$w_{c}:=w_{\circ}u_{c}\quad\text{for
$c\in[0,m]$},\qquad\text{and}\qquad{\mathbf{w}}=w_{\circ}{\mathbf{u}}:=(w_{0},w_{1},\dots,w_{m}).$

Thus, we have $w_{m}={\operatorname{id}}$ and
$w_{c-1}=\max(w_{c},s_{i_{c}^{\ast}}^{-}w_{c}s_{i_{c}}^{+})$ for all
$c\in[m]$, where for $i\in I$, we set $(-i)^{\ast}:=-i^{\ast}$. Because
$\pi(\beta)=w_{\circ}$, we have $u_{0}={\operatorname{id}}$ and
$w_{0}=w_{\circ}$.

######  Definition 2.5.

The _Deodhar torus_ $T_{\beta}\subset\accentset{\circ}{R}_{\beta}$ consists of
all tuples $(X_{\bullet},Y_{\bullet})$ satisfying

(2.6) $X_{c}\stackrel{{\scriptstyle\ w_{c}}}{{\Longleftarrow}}Y_{c}\quad\text{
for $c\in[0,m]$.}$

###  2.4. Torus-valued functions

Given $(X_{\bullet},Y_{\bullet})\in\accentset{\circ}{R}_{\beta}$, let
$Z_{c}:=Y_{c}^{-1}X_{c}\in U_{+}\backslash G/U_{+}$. Abusing notation, we
shall use elements $Z_{c}$ of the double coset $U_{+}\backslash G/U_{+}$
interchangeably with their representatives in $G$. For
$(X_{\bullet},Y_{\bullet})\in T_{\beta}$, $Z_{c}$ belongs to the Bruhat cell
$\accentset{\circ}{\mathcal{X}}_{w_{c}}:=B_{+}w_{c}B_{+}$ of $G$. We have
$\accentset{\circ}{\mathcal{X}}_{w_{c}}=U_{+}w_{c}HU_{+}$. There exist unique
elements $h^{+}_{c},h^{-}_{c}\in H$ satisfying

(2.7) $Z_{c}\in U_{+}\dot{w}_{c}h^{+}_{c}U_{+},\quad Z_{c}\in
U_{+}\overline{w_{\circ}}h^{-}_{c}\overline{\overline{u_{c}}}U_{+},\quad\text{thus,}\quad
h^{-}_{c}=u_{c}\cdot h^{+}_{c}.$

The third statement follows from the first two since
$\dot{w}_{c}=\overline{w_{\circ}}\cdot\overline{\overline{u_{c}}}$ and
$u_{c}\cdot
h^{+}_{c}=\overline{\overline{u_{c}}}h^{+}_{c}\overline{\overline{u_{c}}}^{-1}$
by (2.2). The map $(X_{\bullet},Y_{\bullet})\mapsto h^{\pm}_{c}$ is a rational
$H$-valued function on $\accentset{\circ}{R}_{\beta}$, regular on $T_{\beta}$.

######  Lemma 2.6.

There exist rational functions $(t_{c})_{c\in J_{\beta}}$ on
$\accentset{\circ}{R}_{\beta}$ such that for $c\in[m]$,

(2.8) $h^{+}_{c-1}=\begin{cases}s_{i_{c}}\cdot h^{+}_{c},&\text{if $c$ is
hollow, $i_{c}\in I$;}\\\ \alpha^{\vee}_{i_{c}}(t_{c})h^{+}_{c},&\text{if $c$
is solid, $i_{c}\in I$;}\\\ \end{cases}\quad
h^{-}_{c-1}=\begin{cases}s_{|i_{c}|}\cdot h^{-}_{c},&\text{if $c$ is hollow,
$i_{c}\in-I$;}\\\ \alpha^{\vee}_{|i_{c}|}(t_{c})h^{-}_{c},&\text{if $c$ is
solid, $i_{c}\in-I$.}\\\ \end{cases}$

###### Proof.

For $c\in J_{\beta}$, define $t_{c}$ to be such that if
$Z_{c}=\dot{w}_{c}h^{+}_{c}=\overline{w_{\circ}}h^{-}_{c}\overline{\overline{u_{c}}}$
then $Z_{c-1}=Z_{c}z_{i_{c}}(t_{c})$ if $i_{c}\in I$ and
$Z_{c-1}=\bar{z}_{|i_{c}|^{\ast}}(t_{c})^{-1}Z_{c}$ if $i_{c}\in-I$; see Lemma
2.2. The following identities in $G$ can be checked inside
$\operatorname{SL}_{2}$:

(2.9)
$x_{i}(t)\dot{s}_{i}=y_{i}(1/t)\alpha^{\vee}_{i}(t)x_{i}(-1/t)\quad\text{and}\quad\dot{s}_{i}x_{i}(t)=x_{i}(-1/t)\alpha^{\vee}_{i}(1/t)y_{i}(1/t).$

Suppose that $i_{c}\in I$. We have
$Z_{c-1}=\dot{w}_{c}h^{+}_{c}x_{i_{c}}(t_{c})\dot{s}_{i_{c}}$. If $c$ is
hollow then $\dot{w}_{c}h^{+}_{c}x_{i_{c}}(t_{c})\in
U_{+}\dot{w}_{c}h^{+}_{c}$, and thus $Z_{c-1}\in
U_{+}\dot{w}_{c}\dot{s}_{i_{c}}(s_{i_{c}}\cdot h^{+}_{c})$. This implies
$h^{+}_{c-1}=s_{i_{c}}\cdot h^{+}_{c}$. If $c$ is solid then we use the first
identity in (2.9) to write
$Z_{c-1}=\dot{w}_{c}h^{+}_{c}y_{i_{c}}(1/t_{c})\alpha^{\vee}_{i_{c}}(t_{c})x_{i_{c}}(-1/t_{c})$.
Since $\dot{w}_{c}h^{+}_{c}y_{i_{c}}(t)\in U_{+}\dot{w}_{c}h^{+}_{c}$, we see
that $Z_{c-1}\in U_{+}\dot{w}_{c}h^{+}_{c}\alpha^{\vee}_{i_{c}}(t_{c})U_{+}$,
which implies that $h^{+}_{c-1}=h^{+}_{c}\alpha^{\vee}_{i_{c}}(t_{c})$.

The case when $i_{c}\in-I$ is handled similarly. When $c$ is solid, we use the
second identity in (2.9) together with
$\alpha^{\vee}_{|i_{c}|^{\ast}}(1/t_{c})\overline{w_{\circ}}=\overline{w_{\circ}}\alpha^{\vee}_{|i_{c}|}(t_{c})$;
see [FZ99, Equation (1.2)]. ∎

######  Corollary 2.7.

Suppose $c$ is hollow. If $i_{c}\in I$ then $h^{-}_{c-1}=h^{-}_{c}$, and if
$i_{c}\in-I$ then $h^{+}_{c-1}=h^{+}_{c}$.

######  Corollary 2.8.

The Deodhar torus $T_{\beta}\subset\accentset{\circ}{R}_{\beta}$ is isomorphic
to an algebraic torus of dimension $d(\beta)$, and the functions
$(t_{c})_{c\in J_{\beta}}$ form a basis of the character lattice of
$T_{\beta}$.

###### Proof.

For $c\in J_{\beta}$, the function $t_{c}$ is regular on $T_{\beta}$ by Lemma
2.2. By Lemma 2.3, we have $X_{c-1}\stackrel{{\scriptstyle\
w_{c-1}}}{{\Longleftarrow}}Y_{c-1}$ if and only if
$t_{c}\in{\mathbb{C}}\setminus\\{t^{*}\\}$ for some $t^{*}\in{\mathbb{C}}$
depending on the choice of representatives of $G/U_{+}$ in $G$. We claim that
for our choice of representatives, $t^{*}=0$.

Recall that we have $X_{c-1}\stackrel{{\scriptstyle\
w_{c-1}}}{{\Longleftarrow}}Y_{c-1}$ if and only if
$Z_{c-1}\in\accentset{\circ}{\mathcal{X}}_{w_{c-1}}$. Since $c\in J_{\beta}$,
we have $w_{c-1}=w_{c}$. By the definition of $t_{c}$ given in the proof of
Lemma 2.6, we choose the representative
$Z_{c}=\dot{w}_{c}h^{+}_{c}=\overline{w_{\circ}}h^{-}_{c}\overline{\overline{u_{c}}}$.
Suppose that $i_{c}\in I$. Then $Z_{c-1}=Z_{c}z_{i_{c}}(t_{c})$, so for
$t_{c}=0$, we have
$Z_{c-1}=Z_{c}\dot{s}_{i_{c}}=\dot{w}_{c}h^{+}_{c}\dot{s}_{i_{c}}$. Clearly,
this element belongs to $\accentset{\circ}{\mathcal{X}}_{w_{c}s_{i_{c}}}$ and
not to
$\accentset{\circ}{\mathcal{X}}_{w_{c-1}}=\accentset{\circ}{\mathcal{X}}_{w_{c}}$,
a contradiction. The case when $i_{c}\in-I$ is handled similarly. We have
shown that $t^{*}=0$, and thus for $(X_{\bullet},Y_{\bullet})\in T_{\beta}$,
we have $t_{c}\in{\mathbb{C}}^{\times}$ for all $c\in J_{\beta}$.

The fact that the map $T_{\beta}\to({\mathbb{C}}^{\times})^{J_{\beta}}$,
$(X_{\bullet},Y_{\bullet})\mapsto(t_{c})_{c\in J_{\beta}}$ is an isomorphism
follows from (2.8). ∎

###  2.5. Grid and chamber minors

Recall that $A=(a_{ij})_{i,j\in I}$,
$a_{ij}:={\langle}\alpha_{i},\alpha^{\vee}_{j}{\rangle}$ is the Cartan matrix,
and that $d_{i}a_{ij}=d_{j}a_{ji}$. For $i,j\in\pm I$, we define $a_{ij}=0$ if
$i,j$ have different signs, and $a_{ij}=a_{(-i)(-j)}$ otherwise. Also set
$d_{-i}:=d_{i}$ for $i\in I$.

Following [FZ99, Definition 1.4], for $i\in I$ and $v,w\in W$, we have a
_generalized minor_ $\Delta_{v\omega_{i},w\omega_{i}}:G\to{\mathbb{C}}$. It is
a regular function satisfying

(2.10) $\displaystyle\Delta_{\omega_{i},\omega_{i}}(y_{-}xy_{+})$
$\displaystyle=\Delta_{\omega_{i},\omega_{i}}(x)\quad\text{for all
$(y_{-},x,y_{+})\in U_{-}\times G\times U_{+}$;}$ (2.11)
$\displaystyle\Delta_{v\omega_{i},w\omega_{i}}(x)$
$\displaystyle=\Delta_{\omega_{i},\omega_{i}}\left(\overline{\overline{v^{-1}}}x\overline{w}\right)=\Delta_{\omega_{i},\omega_{i}}\left(\dot{v}^{-1}x\dot{w}\right);$

see [FZ99, Section 1.4]. For $h\in H$, we have
$\Delta_{\omega_{i},\omega_{i}}(h)=h^{\omega_{i}}$. For $x\in G$, we also have
[FZ99, Equation (2.14)]

(2.12)
$\Delta_{\omega_{k},\omega_{k}}(xh)=\Delta_{\omega_{k},\omega_{k}}(hx)=h^{\omega_{k}}\Delta_{\omega_{k},\omega_{k}}(x).$

######  Definition 2.9.

For $c\in[0,m]$ and $k\in I$, we define the _grid minors_

(2.13)
$\Delta_{c,k}(X_{\bullet},Y_{\bullet})=\Delta_{w_{c}\omega_{k},\omega_{k}}(Z_{c})\quad\text{and}\quad\Delta_{c,-k}(X_{\bullet},Y_{\bullet})=\Delta_{w_{\circ}\omega_{k},u_{c}^{-1}\omega_{k}}(Z_{c}).$

The _chamber minors_ are defined as $\Delta_{c}:=\Delta_{c-1,i_{c}}$, for
$c\in[m]$.

Recall that $Z_{c}$ is viewed as an element of the double coset
$U_{+}\backslash G/U_{+}$, and that for $(X_{\bullet},Y_{\bullet})\in
T_{\beta}$, we have $Z_{c}\in\accentset{\circ}{\mathcal{X}}_{w_{c}}$ for all
$c\in[0,m]$. Thus, for
$(X_{\bullet},Y_{\bullet})\in\accentset{\circ}{R}_{\beta}$, $Z_{c}$ belongs to
the closure $\mathcal{X}_{w_{c}}$ of $\accentset{\circ}{\mathcal{X}}_{w_{c}}$
inside $G$. The next result implies that the grid minors restricted to
$\accentset{\circ}{\mathcal{X}}_{w_{c}}$ (and therefore to
$\mathcal{X}_{w_{c}}$) are invariant under the $U_{+}\times U_{+}$-action.

######  Lemma 2.10.

For $c\in[0,m]$, $k\in I$, and $(X_{\bullet},Y_{\bullet})\in T_{\beta}$, we
have

(2.14)
$\Delta_{c,k}(X_{\bullet},Y_{\bullet})=(h^{+}_{c})^{\omega_{k}}\quad\text{and}\quad\Delta_{c,-k}(X_{\bullet},Y_{\bullet})=(h^{-}_{c})^{\omega_{k}}.$

###### Proof.

Write $Z_{c}=y_{+}^{\prime}\dot{w}_{c}h^{+}_{c}y_{+}^{\prime\prime}$ for
$y_{+}^{\prime},y_{+}^{\prime\prime}\in U_{+}$. We have

$\Delta_{c,k}(X_{\bullet},Y_{\bullet})=\Delta_{w_{c}\omega_{k},\omega_{k}}(Z_{c})=\Delta_{\omega_{k},\omega_{k}}(\overline{\overline{w_{c}^{-1}}}y_{+}^{\prime}\dot{w}_{c}h^{+}_{c}y_{+}^{\prime\prime})=\Delta_{\omega_{k},\omega_{k}}(\dot{w}_{c}^{-1}y_{+}^{\prime}\dot{w}_{c}h^{+}_{c}).$

Factorizing $\dot{w}_{c}^{-1}y_{+}^{\prime}\dot{w}_{c}=b_{-}b_{+}$ for
$(b_{-},b_{+})\in U_{-}\times U_{+}$ using (2.4), and using
$(h^{+}_{c})^{-1}b_{+}h^{+}_{c}\in U_{+}$, we get the first identity in
(2.14). The proof of the second identity is similar. ∎

Combining Lemma 2.10 with (2.8), we get the following.

######  Corollary 2.11.

If $c$ is solid and $k\in\pm I$ has the same sign as $i_{c}$ then

(2.15) $\Delta_{c-1,k}=\begin{cases}t_{c}\Delta_{c,k},&\text{if $k=i_{c}$;}\\\
\Delta_{c,k},&\text{if $k\neq i_{c}$.}\\\ \end{cases}$

######  Proposition 2.12.

  1. (1)

The grid minors are characters of $T_{\beta}$.

  2. (2)

The solid chamber minors $(\Delta_{c})_{c\in J_{\beta}}$ form a basis of the
character lattice of $T_{\beta}$.

###### Proof.

We relate the parameters $(t_{c})_{c\in J_{\beta}}$ from Corollary 2.8 to the
grid and chamber minors by combining (2.8) with (2.14). Suppose that $i_{c}\in
I$ and let $k\in I$. If $k\neq i_{c}$ then $\Delta_{c-1,k}=\Delta_{c,k}$ by
Corollary 2.11. If $c$ is hollow then

$\Delta_{c-1,i_{c}}=(h^{+}_{c-1})^{\omega_{i_{c}}}=(s_{i_{c}}\cdot
h^{+}_{c})^{\omega_{i_{c}}}=(h^{+}_{c})^{s_{i_{c}}\omega_{i_{c}}}.$

We expand $s_{i_{c}}\omega_{i_{c}}=\omega_{i_{c}}-\alpha_{i_{c}}$ in the basis
of fundamental weights using $\alpha_{i_{c}}=\sum_{j\in
I}a_{i_{c}j}\omega_{j}$. This implies

(2.16) $\Delta_{c-1,i}\Delta_{c,i}\prod_{j\neq
i}\Delta_{c,j}^{a_{ij}}=1,\quad\text{if $c$ is hollow and $i:=i_{c}$}.$

If $c$ is solid, Corollary 2.11 yields
$\Delta_{c-1,i_{c}}=t_{c}\Delta_{c,i_{c}}$. We apply a similar argument to the
case when $i_{c}\in-I$. In particular, (2.16) holds for $i_{c}\in\pm I$. We
obtain the following statements; cf. [GLSBS22, Lemma 7.3].

  1. (i)

For each solid $c\in J_{\beta}$, $t_{c}=\Delta_{c-1,i_{c}}/\Delta_{c,i_{c}}$
is a ratio of two grid minors.

  2. (ii)

For each solid $c\in J_{\beta}$, the grid minors $(\Delta_{c-1,j})_{j\in\pm
I}$ are Laurent monomials in the grid minors $(\Delta_{c,k})_{k\in\pm I}$ and
the chamber minor $\Delta_{c}=\Delta_{c-1,i_{c}}$.

  3. (iii)

For each hollow $c\in[m]\setminus J_{\beta}$, the grid minors
$(\Delta_{c-1,j})_{j\in\pm I}$ are Laurent monomials in the grid minors
$(\Delta_{c,k})_{k\in\pm I}$.

  4. (iv)

Every grid minor $\Delta_{c,j}$ is a Laurent monomial in the solid chamber
minors $(\Delta_{e})_{e\in J_{\beta}}$.

We have already shown (i)–(iii), and (iv) follows from (ii)–(iii). This
implies the result. ∎

###  2.6. Almost positive sequences and Deodhar hypersurfaces

Recall that we have $u_{c-1}=\min(u_{c},s_{i_{c}}^{-}u_{c}s_{i_{c}}^{+})$ and
$w_{c-1}=\max(w_{c},s_{i_{c}^{\ast}}^{-}w_{c}s_{i_{c}}^{+})$ for all
$c\in[m]$.

######  Definition 2.13 ([GLSBS22, Definition 3.15]).

Let $e\in J_{\beta}$. Let $u_{m}^{{\langle}e{\rangle}}:=w_{\circ}$, and for
$c=m,m-1,\dots,1$, define

$u_{c-1}^{{\langle}e{\rangle}}:=\begin{cases}\max(u_{e}^{{\langle}e{\rangle}},s_{i_{e}}^{-}u_{e}^{{\langle}e{\rangle}}s_{i_{e}}^{+}),&\text{if
$c=e$,}\\\
\min(u_{c}^{{\langle}e{\rangle}},s_{i_{c}}^{-}u_{c}^{{\langle}e{\rangle}}s_{i_{c}}^{+}),&\text{otherwise}.\end{cases}$

We call the sequence
${\mathbf{u}}^{{\langle}e{\rangle}}:=(u_{0}^{{\langle}e{\rangle}},\dots,u_{m}^{{\langle}e{\rangle}})$
the _$(w_{\circ},e)$ -almost positive sequence_ (_$(w_{\circ},e)$ -APS_). We
set $w_{c}^{{\langle}e{\rangle}}:=w_{\circ}u_{c}^{{\langle}e{\rangle}}$ for
all $c\in[0,m]$, and write
${\mathbf{w}}^{{\langle}e{\rangle}}=w_{\circ}{\mathbf{u}}^{{\langle}e{\rangle}}:=(w_{\circ}u_{0}^{{\langle}e{\rangle}},\dots,w_{\circ}u_{m}^{{\langle}e{\rangle}})$.

######  Definition 2.14.

We say that $e\in J_{\beta}$ is _mutable_ if
$u_{0}^{{\langle}e{\rangle}}={\operatorname{id}}$. Otherwise, $e$ is called
_frozen_. We let $J_{\beta}^{{\rm mut}}$ (resp., $J_{\beta}^{{\rm fro}}$)
denote the set of mutable (resp., frozen) indices.

We define

$\accentset{\circ}{\mathcal{Y}}_{\beta}:=\\{(X_{\bullet},Y_{\bullet})\text{
satisfying
}\eqref{eq:braidDiagram}\\}\subset(G/U_{+})^{[0,m]}\times(G/U_{+})^{[0,m]}.$

Removing the condition $X_{0}\stackrel{{\scriptstyle\
w_{\circ}}}{{\Longleftarrow}}Y_{0}$ yields a partial compactification
$\mathcal{Y}_{\beta}$ of $\accentset{\circ}{\mathcal{Y}}_{\beta}$.

######  Remark 2.15.

It follows that $\accentset{\circ}{R}_{\beta}$ is the quotient of
$\accentset{\circ}{\mathcal{Y}}_{\beta}$ by the $G$-action. This action is
free since the $G$-action on pairs $(X_{0},Y_{0})$ which are weakly
$w_{\circ}$-related is free; cf. [GLSBS22, Proposition 6.8]. The $G$-action on
$\mathcal{Y}_{\beta}$ is no longer free.

######  Remark 2.16.

We can parameterize the variety $\mathcal{Y}_{\beta}$ as follows. We choose an
arbitrary weighted flag $X_{m}=Y_{m}$, and then for $c=m,m-1,\dots,1$,
assuming $(X_{c},Y_{c})=(g_{c}U_{+},g^{\prime}_{c}U_{+})$, we set (cf. Lemma
2.2)

(2.17)
$(X_{c-1},Y_{c-1}):=\begin{cases}(g_{c}z_{i_{c}}(t^{\prime}_{c})U_{+},g^{\prime}_{c}U_{+}),&\text{if
$i_{c}>0$,}\\\
(g_{c}U_{+},g^{\prime}_{c}\bar{z}_{|i_{c}|^{*}}(t^{\prime}_{c})U_{+}),&\text{if
$i_{c}<0$,}\\\ \end{cases}$

for arbitrary parameters
${\mathbf{t}}^{\prime}:=(t^{\prime}_{1},t^{\prime}_{2},\dots,t^{\prime}_{m})\in{\mathbb{C}}^{m}$.
For $(X_{\bullet},Y_{\bullet})$ to be a point in
$\accentset{\circ}{\mathcal{Y}}_{u,\beta}$, we require further that
$X_{0}\stackrel{{\scriptstyle\ w_{\circ}}}{{\Longleftarrow}}Y_{0}$, which is
an extra open condition on the parameters
$({\mathbf{t}}^{\prime},X_{m}=Y_{m})$.

We have the following basic result; cf. [GLSBS22, Lemma 6.19].

######  Lemma 2.17.

For each $c\in[0,m]$ and $k\in\pm I$, the grid minor $\Delta_{c,k}$ gives rise
to a $G$-invariant regular function on $\mathcal{Y}_{\beta}$ and
$\accentset{\circ}{\mathcal{Y}}_{\beta}$. These regular functions are
compatible with the quotient map
$\accentset{\circ}{\mathcal{Y}}_{u,\beta}\to\accentset{\circ}{R}_{u,\beta}$
and the inclusion map
$\accentset{\circ}{\mathcal{Y}}_{u,\beta}\hookrightarrow\mathcal{Y}_{u,\beta}$.

######  Definition 2.18.

Let $e\in J_{\beta}$. Define the _Deodhar hypersurface_
$\tilde{V}_{e}\subset\mathcal{Y}_{\beta}$ to be the closure of the locus
satisfying

(2.18) $X_{c}\stackrel{{\scriptstyle\
w_{c}^{{\langle}e{\rangle}}}}{{\Longleftarrow}}Y_{c}\quad\text{for all
$c\in[0,m]$.}$

It follows that an index $e\in J_{\beta}$ is mutable (resp., frozen) if and
only if $\tilde{V}_{e}\subset\accentset{\circ}{\mathcal{Y}}_{\beta}$ (resp.,
$\tilde{V}_{e}\cap\accentset{\circ}{\mathcal{Y}}_{\beta}=\emptyset$). If $e$
is mutable, then the $G$-action on $\tilde{V}_{e}$ is free and
$V_{e}=\tilde{V}_{e}/G$ is a subvariety of $\accentset{\circ}{R}_{\beta}$. We
let $\tilde{T}_{\beta}$ denote the preimage of $T_{\beta}$ under the quotient
map $\accentset{\circ}{\mathcal{Y}}_{\beta}\to\accentset{\circ}{R}_{\beta}$.

######  Proposition 2.19.

The closed subset $\mathcal{Y}_{\beta}\setminus\tilde{T}_{\beta}$ is a union
of the Deodhar hypersurfaces $\tilde{V}_{e}$ for $e\in J_{\beta}$. Each
$\tilde{V}_{e}$ is irreducible and has codimension $1$ in
$\mathcal{Y}_{\beta}$.

######  Remark 2.20.

Alternatively, our results may be phrased in terms of the stack
$\tilde{R}_{\beta}:=[\mathcal{Y}_{\beta}/G]$, which gives a partial
compactification of $\accentset{\circ}{R}_{\beta}$. For instance, the above
proposition states that the complement $\tilde{R}_{\beta}\setminus T_{\beta}$
is a union of irreducible codimension-$1$ Deodhar hypersurfaces
$V_{e}=[\tilde{V}_{e}/G]$ (viewed also as stacks) for $e\in J_{\beta}$.

###### Proof.

We first show that the closed subset
$\mathcal{Y}_{\beta}\setminus\tilde{T}_{\beta}$ is a union of the Deodhar
hypersurfaces $\tilde{V}_{e}$ for $e\in J_{\beta}$. Applying (2.17), we see
that the conditions (2.18) cut out an iterated fiber bundle over
$G/U_{+}\times G/U_{+}$, where each fiber is either ${\mathbb{C}}$,
${{\mathbb{C}}^{\times}}$, or (in the case of the crossing $c=e$) a point. It
follows that $\tilde{V}_{e}$ is an irreducible subvariety of
$\mathcal{Y}_{\beta}$ of codimension one.

Let
$(X_{\bullet},Y_{\bullet})\in\mathcal{Y}_{\beta}\setminus\tilde{T}_{\beta}$.
Then (2.6) must fail, and let $e\in[0,m]$ be the largest index such that we do
not have $X_{e}\stackrel{{\scriptstyle\ w_{e}}}{{\Longleftarrow}}Y_{e}$. Let
$\tilde{V}^{\prime}_{e}\subset\mathcal{Y}_{\beta}$ be the locus of points
where (2.6) holds for $c>e$ and fails for $c=e$. By Lemma 2.3 and Definition
2.13, an open dense subset $\tilde{V}^{\prime\prime}_{e}$ of
$\tilde{V}^{\prime}_{e}$ consists of points satisfying (2.18). Thus,
$\tilde{V}^{\prime}_{e}\subset\tilde{V}_{e}$, and therefore
$(X_{\bullet},Y_{\bullet})\in\tilde{V}_{e}$. We have shown that
$\mathcal{Y}_{\beta}\setminus\tilde{T}_{\beta}=\bigcup_{e\in
J_{\beta}}\tilde{V}_{e}$. ∎

###  2.7. Cluster variables

The irreducible components of $\mathcal{Y}_{\beta}\setminus\tilde{T}_{\beta}$
are the Deodhar hypersurfaces $\tilde{V}_{e}$, $e\in J_{\beta}$. For a grid
minor $\Delta_{c,k}$ and $e\in J_{\beta}$, we denote by
${\mathrm{ord}}_{V_{e}}\Delta_{c,k}\in{\mathbb{Z}}$ the order of vanishing of
$\Delta_{c,k}$ on the hypersurface $\tilde{V}_{e}$; cf. Lemma 2.17. Since
$\Delta_{c,k}$ is regular on $\mathcal{Y}_{\beta}$, we have

(2.19) ${\mathrm{ord}}_{V_{e}}\Delta_{c,k}\geq 0.$

We have the basic triangularity property.

######  Proposition 2.21.

For $e\in J_{\beta}$ solid, $c\in[0,m]$, and $k\in\pm I$, we have

(2.20) ${\mathrm{ord}}_{V_{e}}\Delta_{c,k}=\begin{cases}0,&\text{if $e\leq
c$;}\\\ 1,&\text{if $(c,k)=(e-1,i_{e})$, i.e., $\Delta_{c,k}=\Delta_{e}$.}\\\
\end{cases}$

###### Proof.

Suppose that $e\leq c$. Let $(X_{\bullet},Y_{\bullet})$ be a generic point in
$\tilde{V}_{e}$. Then we have $X_{c}\stackrel{{\scriptstyle\
w_{c}}}{{\Longleftarrow}}Y_{c}$, and thus
$\Delta_{c,k}(X_{\bullet},Y_{\bullet})\neq 0$. It follows that
${\mathrm{ord}}_{V_{e}}\Delta_{c,k}=0$ when $e\leq c$.

Suppose now that $(c,k)=(e-1,i_{e})$. We may parameterize
$\mathcal{Y}_{\beta}$ using parameters $({\mathbf{t}}^{\prime},X_{m}=Y_{m})$
as in (2.17). The union $\tilde{T}_{\beta}\cup\tilde{V}_{e}$ contains an open
dense subset of tuples $(X_{\bullet},Y_{\bullet})$ satisfying
$X_{{e^{\prime}}}\stackrel{{\scriptstyle\
w_{{e^{\prime}}}}}{{\Longleftarrow}}Y_{{e^{\prime}}}$ for all
${e^{\prime}}\geq e$. Suppose that
$(X_{\bullet},Y_{\bullet})\in\tilde{T}_{\beta}\cup\tilde{V}_{e}$ is _generic_
, i.e., satisfies these conditions. In particular,
$X_{e}\stackrel{{\scriptstyle\ w_{e}}}{{\Longleftarrow}}Y_{e}$.

Assume that $i_{e}\in I$. Thus, we have $Z_{e}\in U_{+}\dot{w}_{e}hU_{+}$ for
some $h\in H$. The proof of Lemma 2.10 implies that $\dot{w}_{e}^{-1}Z_{e}\in
U_{-}hU_{+}$. Let us write $\dot{w}_{e}^{-1}Z_{e}=y_{-}hy_{+}$ for
$(y_{-},y_{+})\in U_{-}\times U_{+}$. Since $e\in J_{\beta}$ is solid, we have
$w_{e-1}=w_{e}$. Setting $t:=t^{\prime}_{e}$, we find

$\Delta_{e-1,k}(X_{\bullet},Y_{\bullet})=\Delta_{w_{e-1}\omega_{k},\omega_{k}}(Z_{e}z_{k}(t))=\Delta_{\omega_{k},\omega_{k}}(\dot{w}_{e}^{-1}Z_{e}z_{k}(t))=\Delta_{\omega_{k},\omega_{k}}(y_{-}hy_{+}z_{k}(t)).$

Recall that $z_{k}(t)=x_{k}(t)\dot{s}_{k}$. Let
$\Psi:=\Phi^{+}\setminus\\{\alpha_{k}\\}$ and let
$U_{+}(\Psi):=(\dot{s}_{k}^{-1}U_{+}\dot{s}_{k})\cap U_{+}$ be the
corresponding _root subgroup_ ; see [Hum75, Theorem 26.3]. We have
$x_{k}(-t)U_{+}(\Psi)x_{k}(t)\subset U_{+}(\Psi)$ by [Hum75, Lemma 32.5].
Next, we have $\dot{s}_{k}^{-1}U_{+}(\Psi)\dot{s}_{k}\subset U_{+}(\Psi)$,
since $s_{k}$ permutes $\Psi$. Using (2.4), we can factorize
$y_{+}=x_{k}(p)y_{+}^{\prime}$ for some $p\in{\mathbb{C}}$ and
$y_{+}^{\prime}\in U_{+}(\Psi)$. We therefore get
$y_{+}^{\prime}x_{k}(t)\dot{s}_{k}\in x_{k}(t)\dot{s}_{k}U_{+}$. Using (2.10),
we get

$\Delta_{e-1,k}(X_{\bullet},Y_{\bullet})=\Delta_{\omega_{k},\omega_{k}}(y_{-}hx_{k}(p)y_{+}^{\prime}x_{k}(t)\dot{s}_{k})=\Delta_{\omega_{k},\omega_{k}}(hx_{k}(p+t)\dot{s}_{k}).$

It is clear that if $p+t=0$ then $\Delta_{e-1,k}(X_{\bullet},Y_{\bullet})=0$.
If $p+t\neq 0$, applying the first identity in (2.9) to
$x_{k}(p+t)\dot{s}_{k}$ and using (2.12), we find

(2.21)
$\Delta_{e-1,k}(X_{\bullet},Y_{\bullet})=(p+t)\Delta_{\omega_{k},\omega_{k}}(h).$

It follows that (2.21) holds regardless of whether $p+t=0$, and that we have
$p+t=0$ if and only if the condition $X_{e-1}\stackrel{{\scriptstyle\
w_{e-1}}}{{\Longleftarrow}}Y_{e-1}$ fails, i.e.,
$(X_{\bullet},Y_{\bullet})\in\tilde{V}_{e}$. By (2.21), since
$\Delta_{\omega_{k},\omega_{k}}(h)\neq 0$, we have $p+t=0$ if and only if
$\Delta_{e-1,k}(X_{\bullet},Y_{\bullet})=0$. Since $\Delta_{e-1,k}$ is of
degree $1$ in $t$, we find that ${\mathrm{ord}}_{V_{e}}\Delta_{e-1,k}\leq 1$.
On the other hand, we have shown that $\Delta_{e-1,k}$ vanishes on
$\tilde{V}_{e}$, so ${\mathrm{ord}}_{V_{e}}\Delta_{e-1,k}\geq 1$, which shows
the result. ∎

Recall from (2.19) that the integers ${\mathrm{ord}}_{V_{e}}\Delta_{c,k}$ are
nonnegative. Our next result shows that whether
${\mathrm{ord}}_{V_{e}}\Delta_{c,k}$ is zero or positive is determined by the
$w_{\circ}$-APS ${\mathbf{u}}^{{\langle}e{\rangle}}$ similarly to [GLSBS22,
Proposition 7.10]. However, determining the precise value of
${\mathrm{ord}}_{V_{e}}\Delta_{c,k}$ is more subtle for $G$ of arbitrary type;
cf. Section 7.

######  Proposition 2.22.

For all $c\in[0,m]$, $e\in J_{\beta}$, and $k\in I$, we have

${\mathrm{ord}}_{V_{e}}\Delta_{c,k}=0\quad\Longleftrightarrow\quad
u_{c}\omega_{k}=u_{c}^{{\langle}e{\rangle}}\omega_{k}\quad\text{and}\quad{\mathrm{ord}}_{V_{e}}\Delta_{c,-k}=0\quad\Longleftrightarrow\quad
u_{c}^{-1}\omega_{k}=(u_{c}^{{\langle}e{\rangle}})^{-1}\omega_{k}.$

###### Proof.

Recall that the _Bruhat order_ $(W,\leq)$ is given by $u\leq w$ if some
(equivalently, any) reduced word for $w$ contains a subword which is a reduced
word for $u$. It follows that $u\leq w$ is equivalent to $u^{-1}\leq w^{-1}$
and $w_{\circ}u\geq w_{\circ}w$. For each $k\in I$, we have a parabolic
quotient order $(W\omega_{k},\leq)$ such that $u\leq w$ implies that
$u\omega_{k}\leq w\omega_{k}$. Comparing Definitions 2.4 and 2.13 and
proceeding by induction on $c=m,m-1,\dots,1$ (for a fixed $e\in J_{\beta}$),
we see that $u_{c}\leq u_{c}^{{\langle}e{\rangle}}$ for all $c\in[0,m]$.
Consequently, $w_{c}\geq w_{c}^{{\langle}e{\rangle}}$ for all $c\in[0,m]$. We
find

$w_{c}\omega_{k}\geq w_{c}^{{\langle}e{\rangle}}\omega_{k}\quad\text{for all
$c\in[0,m]$ and $k\in I$.}$

Let $\tilde{V}_{e}^{\prime}\subset\tilde{V}_{e}$ be the open dense subset of
points satisfying (2.18). For
$(X_{\bullet},Y_{\bullet})\in\tilde{V}_{e}^{\prime}$, we have
$Z_{c}\in\accentset{\circ}{\mathcal{X}}_{w_{c}^{{\langle}e{\rangle}}}\subset\mathcal{X}_{w_{c}}$
for all $c\in[0,m]$, because $w_{c}^{{\langle}e{\rangle}}\leq w_{c}$. Recall
that
$\Delta_{c,k}(X_{\bullet},Y_{\bullet})=\Delta_{w_{c}\omega_{k},\omega_{k}}(Z_{c})$.
The fact that the function $\Delta_{w_{c}\omega_{k},\omega_{k}}(Z_{c})$ when
restricted to $Z_{c}\in\mathcal{X}_{w_{c}}$ does not vanish at
$Z_{c}\in\accentset{\circ}{\mathcal{X}}_{w_{c}^{{\langle}e{\rangle}}}$ if and
only if $w_{c}^{{\langle}e{\rangle}}\omega_{k}=w_{c}\omega_{k}$ is well known,
and can be shown by an argument analogous to the proof of [FZ99, Proposition
2.4]. Similarly, we consider
$\Delta_{c,-k}(X_{\bullet},Y_{\bullet})=\Delta_{w_{\circ}\omega_{k},u_{c}^{-1}\omega_{k}}(Z_{c})$
and observe that this function does not vanish at
$Z_{c}\in\accentset{\circ}{\mathcal{X}}_{w_{\circ}u_{c}^{{\langle}e{\rangle}}}\subset\mathcal{X}_{w_{\circ}u_{c}}$
if and only if
$u_{c}^{-1}\omega_{k}=(u_{c}^{{\langle}e{\rangle}})^{-1}\omega_{k}$. ∎

######  Corollary 2.23.

The $J_{\beta}\times J_{\beta}$ matrix
$M_{\beta}=({\mathrm{ord}}_{V_{e}}\Delta_{c})_{c,e\in J_{\beta}}$ is upper
unitriangular.

Inverting the matrix $M_{\beta}$, we arrive at the following definition, which
is crucial for our analysis; cf. 1.3. Recall from Proposition 2.12 that a
character $T_{\beta}$ is just a Laurent monomial in the solid chamber minors
$\\{\Delta_{c}\\}_{c\in J_{\beta}}$.

######  Definition 2.24.

For $c\in J_{\beta}$, the _cluster variable_ $x_{c}$ is the unique character
of $T_{\beta}$ satisfying

${\mathrm{ord}}_{V_{e}}x_{c}=\begin{cases}1,&\text{if $c=e$,}\\\
0,&\text{otherwise,}\end{cases}\quad\text{for all $e\in J_{\beta}$.}$

We denote the _cluster_ by ${\mathbf{x}}_{\beta}=\\{x_{c}\\}_{c\in
J_{\beta}}$.

######  Corollary 2.25.

  1. (1)

For $c\in J_{\beta}$, the character $x_{c}\in X^{\ast}(T_{\beta})$ extends to
a regular function on $\accentset{\circ}{R}_{\beta}$,
$\accentset{\circ}{\mathcal{Y}}_{\beta}$, $\mathcal{Y}_{\beta}$.

  2. (2)

For $c\in J_{\beta}^{{\rm fro}}$, the frozen cluster variable $x_{c}$ is
invertible in ${\mathbb{C}}[\accentset{\circ}{R}_{\beta}]$ and
${\mathbb{C}}[\accentset{\circ}{\mathcal{Y}}_{\beta}]$.

  3. (3)

The cluster variables in ${\mathbf{x}}_{\beta}$ are irreducible and
algebraically independent.

###### Proof.

For $c\in J_{\beta}$, $x_{c}$ is a rational function on $\mathcal{Y}_{\beta}$
which does not have a pole on $\tilde{T}_{\beta}$ and on $\tilde{V}_{e}$ for
all $e\in J_{\beta}$. This implies that $x_{c}$ is regular. By Proposition
2.19, $x_{c}$ is irreducible. Since ${\mathbf{x}}_{\beta}=\\{x_{c}\\}_{c\in
J_{\beta}}$ is a basis of the character lattice of $T_{\beta}$, we see that
the cluster variables are algebraically independent. Finally, for $c\in
J_{\beta}^{{\rm fro}}$, the function $1/x_{c}$ is regular on
$\mathcal{Y}_{\beta}\setminus\tilde{V}_{c}$, and as we mentioned after
Definition 2.18, we have
$\tilde{V}_{c}\cap\accentset{\circ}{\mathcal{Y}}_{\beta}=\emptyset$. ∎

Let us denote
${\mathrm{ord}}(\beta;e,c,k):={\mathrm{ord}}_{V_{e}}\Delta_{c,k}$. Recall from
Proposition 2.21 that ${\mathrm{ord}}(\beta;e,c,k)$ can only be nonzero when
$e>c$. The following result will be used later in the proof; cf. Section 4.8.

######  Lemma 2.26.

The integer ${\mathrm{ord}}_{V_{e}}\Delta_{c,k}$ only depends on
$i_{c+1},\dots,i_{m}$. That is, suppose that $\beta=i_{1}i_{2}\cdots i_{m}$
and $\beta^{\prime}=i^{\prime}_{1}i^{\prime}_{2}\cdots
i^{\prime}_{m^{\prime}}$ are two double braid words in the alphabet $\pm I$
such that for some $c\in[m]$ and $c^{\prime}\in[m^{\prime}]$ (with
$m-c=m^{\prime}-c^{\prime}$), we have $i_{c+1}\cdots
i_{m}=i^{\prime}_{c^{\prime}+1}\cdots i^{\prime}_{m^{\prime}}$. Then we have

${\mathrm{ord}}(\beta;e,c,k)={\mathrm{ord}}(\beta^{\prime};e^{\prime},c^{\prime},k)$

for all $k\in\pm I$, $e>c$, and $e^{\prime}>c^{\prime}$ such that
$m-e=m^{\prime}-e^{\prime}$.

###### Proof.

Parameterize $\mathcal{Y}_{\beta}$ (resp., $\mathcal{Y}_{\beta^{\prime}}$)
using parameters $({\mathbf{t}}^{\prime},X_{m}=Y_{m})$ (resp.,
$({\mathbf{t}}^{\prime\prime},X^{\prime}_{m}=Y^{\prime}_{m})$) as in (2.17).
We may assume that $X_{m}=Y_{m}=X^{\prime}_{m}=Y^{\prime}_{m}$, and that for
each $j\in[m-c]$, we have
$t^{\prime}_{m+1-j}=t^{\prime\prime}_{m^{\prime}+1-j}$. Then it is clear that
the integers ${\mathrm{ord}}(\beta;e,c,k)$ and
${\mathrm{ord}}(\beta^{\prime};e^{\prime},c^{\prime},k)$ are computed in the
same way as the degree with which the polynomial
$\Delta_{c,k}(X_{\bullet},Y_{\bullet})$ (resp.,
$\Delta_{c^{\prime},k}(X_{\bullet}^{\prime},Y_{\bullet}^{\prime})$) contains
the defining equation of the hypersurface $\tilde{V}_{e}$ (resp.,
$\tilde{V}^{\prime}_{e^{\prime}}$). ∎

###  2.8. A two-form on the braid variety

We start by introducing a family of $1$-forms on $T_{\beta}$. For $i,j\in\pm
I$, recall that $a_{ij}=0$ if $i,j$ have different signs, and
$a_{ij}=a_{(-i)(-j)}$ otherwise, and that $d_{i}:=d_{|i|}$. For each
$c\in[0,m]$ and $i\in\pm I$, we set

(2.22) $L_{c,i}:=\frac{1}{2}\sum_{k\in\pm I}a_{ik}{\rm dlog}\Delta_{c,k}.$

Consider the following $2$-forms on $T_{\beta}$:

(2.23) $\omega_{\beta,c}:={\rm sign}(i)d_{i}L_{c-1,i}\wedge
L_{c,i}\quad\text{for $c\in[m]$,
$i:=i_{c}$,}\qquad\text{and}\qquad\omega_{\beta}:=\sum_{c\in[m]}\omega_{\beta,c}.$

Since $T_{\beta}$ is open dense in $\accentset{\circ}{R}_{\beta}$, the forms
$\omega_{\beta}$ and $\omega_{\beta,c}$ are rational $2$-forms on
$\accentset{\circ}{R}_{\beta}$. Though it is not apparent from the above
formula, it will follow from our main result that $\omega_{\beta}$ is actually
a regular $2$-form on $\accentset{\circ}{R}_{\beta}$.

Recall that (2.16) holds for $i_{c}\in\pm I$. Taking ${\rm dlog}$ of both
sides of (2.16), we get

(2.24) $L_{c-1,i_{c}}+L_{c,i_{c}}=0\quad\text{if $c$ is hollow.}$

Thus, $\omega_{\beta,c}=0$ for all $c\in[m]\setminus J_{\beta}$, which implies
the following result.

######  Corollary 2.27.

We have $\omega_{\beta}=\sum_{c\in J_{\beta}}\omega_{\beta,c}$.

##  3\. Cluster algebras

Cluster algebras were discovered by Fomin and Zelevinsky [FZ02]. We shall
consider skew-symmetrizable cluster algebras of geometric type, and rely on
formalism similar to that of Fock and Goncharov [FG09].

###  3.1. Background

######  Definition 3.1.

A rank $n$ and dimension $n+m$ _(abstract) seed_ is a quadruple
$\Sigma=(T,{\mathbf{x}},\mathbf{d},\omega)$, where

  1. (1)

$T$ is a complex algebraic torus of dimension ${n+m}$,

  2. (2)

${\mathbf{x}}=(x_{1},\ldots,x_{{n+m}})$ is an ordered basis of characters of
$T$, where we have designated $x_{1},\ldots,x_{n}$ (resp.,
$x_{n+1},\ldots,x_{{n+m}}$) as mutable (resp., frozen) variables,

  3. (3)

$\mathbf{d}=(d_{1},\ldots,d_{{n+m}})$ is a collection of positive integers,

  4. (4)

$\omega$ is a 2-form on $T$ of the form

(3.1) $\omega=\sum_{i\leq j}d_{j}{\tilde{B}}_{ij}{\rm dlog}x_{i}\wedge{\rm
dlog}x_{j}=\sum_{i\leq j}d_{i}{\tilde{B}}_{ji}{\rm dlog}x_{j}\wedge{\rm
dlog}x_{i},$

where ${\tilde{B}}_{ii}=0$ for $i\in[n+m]$ and ${\tilde{B}}_{ij}\in\mathbb{Q}$
for all $i,j\in[{n+m}]$.

We say that $\Sigma$ is _integral_ if ${\tilde{B}}_{ij}\in{\mathbb{Z}}$ for
all $i\in[{n+m}]$ and $j\in[n]$.

The matrix ${\tilde{B}}=({\tilde{B}}_{ij})_{(i,j)\in[{n+m}]\times[n]}$ is the
usual $({n+m})\times n$ _extended exchange matrix_ in the theory of cluster
algebras. We have $d_{j}{\tilde{B}}_{ij}=-d_{i}{\tilde{B}}_{ji}$ for
$i,j\in[{n+m}]$; in particular, the top $n\times n$ _principal part_ $B$ of
${\tilde{B}}$ is skew-symmetrizable.

######  Definition 3.2.

Let $\Sigma=(T,{\mathbf{x}},\mathbf{d},\omega)$ be a seed and $k$ a mutable
index. We say that $\Sigma$ is _integral at $k$_ if
${\tilde{B}}_{jk}\in{\mathbb{Z}}$ for all $j\in[{n+m}]$. In this case, we
define

(3.2)
$x_{k}^{\prime}:=\frac{\prod_{{\tilde{B}}_{jk}>0}x_{j}^{{\tilde{B}}_{jk}}+\prod_{{\tilde{B}}_{jk}<0}x_{j}^{-{\tilde{B}}_{jk}}}{x_{k}}.$

The _mutation_ of $\Sigma$ in the direction $k$ is the seed
$\mu_{k}(\Sigma)=(T^{\prime},{\mathbf{x}}^{\prime},\mathbf{d},\omega^{\prime})$
where $T^{\prime}$ is the algebraic torus with basis of characters
${\mathbf{x}}^{\prime}=(x_{1},\ldots,x^{\prime}_{k},\ldots,x_{{n+m}})$ and the
2-form $\omega^{\prime}$ on $T^{\prime}$ is the pullback of $\omega$ via the
natural rational map $T^{\prime}\to T$; see [FG09]. We will sometimes denote
$\mu_{k}({\mathbf{x}}):={\mathbf{x}}^{\prime}$ and $\mu_{k}(T):=T^{\prime}$.

For the rest of this subsection, we assume that all seeds are integral. Let
$X$ be an irreducible complex algebraic variety of dimension ${n+m}$. A _seed
on $X$_ is an abstract seed $\Sigma=(T,{\mathbf{x}},\mathbf{d},\omega)$
together with an identification $T\subset X$ of $T$ with an open dense subset
of $X$. The inclusion $T\hookrightarrow X$ induces an identification of the
field ${\mathbb{C}}(X)$ of rational functions on $X$ with the field of
rational functions
${\mathbb{C}}({\mathbf{x}}):={\mathbb{C}}(x_{1},\ldots,x_{n+m})$ in the
initial cluster variables ${\mathbf{x}}$. In practice, we abuse notation and
write ${\mathbf{x}}$ as a tuple of elements of ${\mathbb{C}}(X)$.

The _cluster algebra_ ${\mathcal{A}}(\Sigma)$ is defined as a subring of
${\mathbb{C}}({\mathbf{x}})$ generated by all cluster variables together with
inverses of frozen variables. We let
$\mathcal{X}(\Sigma):=\operatorname{Spec}({\mathcal{A}}(\Sigma))$ denote the
_cluster variety_. We say that _$(X,\Sigma)$ is a cluster variety_ if $X$ is
an affine variety and the coordinate ring ${\mathbb{C}}[\mathcal{X}]$ is
identified with ${\mathcal{A}}(\Sigma)$ under the identification
${\mathbb{C}}(X)\cong{\mathbb{C}}({\mathbf{x}})$.

We will need the following property of cluster variables.

######  Proposition 3.3 ([GLS13, Theorem 3.1]).

Each cluster variable is an irreducible element of $\mathcal{A}(\Sigma)$.

######  Definition 3.4.

Let $\Sigma$ be an abstract seed of rank $n$ and dimension ${n+m}$, and let
$I\subset[n]$. The _freezing of $\Sigma$ at $I$_ , denoted $\Sigma^{\backslash
I}$, is an abstract seed obtained from $\Sigma$ by declaring the variables
$\\{x_{c}\\}_{c\in I}$ to be frozen. It is a seed of rank $n-|I|$ and
dimension ${n+m}$. Abusing notation, for $k\in[n]$, we denote
$\Sigma^{\backslash k}:=\Sigma^{\backslash\\{k\\}}$.

To a seed $\Sigma$ we associate a directed graph $\tilde{\Gamma}(\Sigma)$ with
vertex set $[n+m]$ and an arrow $i\to j$ whenever ${\tilde{B}}_{ij}>0$. We let
$\Gamma:=\Gamma(\Sigma)$ be the _mutable part_ of $\tilde{\Gamma}(\Sigma)$,
i.e., the induced subgraph of $\tilde{\Gamma}(\Sigma)$ with vertex set $[n]$.
We say that a mutable index $s\in[n]$ is a _sink_ if it has no outgoing arrows
in $\Gamma$. Let $N^{\operatorname{in}}_{s}(\Gamma)$ denote the set of
vertices of $\Gamma$ having an arrow to $s$, and denote
$\widehat{N}^{\operatorname{in}}_{s}(\Gamma):=N^{\operatorname{in}}_{s}(\Gamma)\cup\\{s\\}$.
We have the following extension of [GLSBS22, Definition 5.4]; see also [GL22,
Section 5.4 and Remark 5.14].

######  Definition 3.5.

The class of _sink-recurrent seeds_ is defined recursively as follows.

  * •

Any seed $\Sigma$ such that $\Gamma(\Sigma)$ has no arrows is sink-recurrent.

  * •

Any seed that is mutation equivalent to a sink-recurrent seed is sink-
recurrent.

  * •

Suppose that $\Sigma$ is a seed with a sink $s\in[n]$ such that the seeds
$\Sigma^{\backslash s}$ and
$\Sigma^{\backslash\widehat{N}^{\operatorname{in}}_{s}(\Gamma(\Sigma))}$ are
sink-recurrent. Then $\Sigma$ is sink-recurrent.

The _upper cluster algebra_
$\mathcal{U}(\Sigma)\subset{\mathbb{C}}[{\mathbf{x}}^{\pm 1}]$ is the
intersection ${\mathbb{C}}[{\mathbf{x}}^{\pm
1}]\cap\bigcap_{k\in[n]}{\mathbb{C}}[\mu_{k}({\mathbf{x}})^{\pm 1}]$; see
[BFZ05].

######  Proposition 3.6.

Suppose that $\Sigma$ is a sink-recurrent seed. Then
$\mathcal{A}(\Sigma)=\mathcal{U}(\Sigma)$.

###### Proof.

We will show that $\mathcal{A}(\Sigma)$ is _locally acyclic_ in the sense of
[Mul13, Mul14]. We assume familiarity with Muller’s results and terminology.
Our argument is identical to the one given in [GLSBS22, Proposition 5.5].

Let $\Gamma:=\Gamma(\Sigma)$. We proceed by induction. The statement is clear
if $\Gamma$ has no arrows. Suppose that $s$ is a sink such that
$\Sigma^{\backslash s}$ and
$\Sigma^{\backslash\widehat{N}^{\operatorname{in}}_{s}(\Gamma)}$ are sink-
recurrent. Let $\mathcal{A}_{1}:=\mathcal{A}(\Sigma^{\backslash s})$ and
$\mathcal{A}_{2}:=\Sigma^{\backslash\widehat{N}^{\operatorname{in}}_{s}(\Gamma)}$.
By the induction hypothesis, $\mathcal{A}_{1},\mathcal{A}_{2}$ are locally
acyclic, and therefore they are cluster localizations.

Choosing $t$ to be any neighbor of $s$ in $\Gamma$, one can check analogously
to [Mul13, Lemma 5.3] (where it was done for quivers) that $x_{s}$ and $x_{t}$
cannot simultaneously vanish on $\mathcal{A}(\Sigma)$. Thus,
$\mathcal{A}_{1},\mathcal{A}_{2}$ form a cover of $\mathcal{A}(\Sigma)$, which
implies that $\mathcal{A}(\Sigma)$ is locally acyclic. By [Mul14, Theorem 2],
we have $\mathcal{A}(\Sigma)=\mathcal{U}(\Sigma)$. ∎

The following notion is needed for applications (such as Theorem 6.7). We will
prove it for the seeds $\Sigma_{\beta}$ from Section 2 in Corollary 6.6.

######  Definition 3.7 ([LS16, Section 5.1]).

A seed $\Sigma$ is called _really full rank_ if the columns of ${\tilde{B}}$
span ${\mathbb{Z}}^{n}$ over ${\mathbb{Z}}$.

###  3.2. Quasi-equivalence

######  Definition 3.8.

Two seeds $\Sigma=(T,{\mathbf{x}},\mathbf{d},\omega)$ and
$\tilde{\Sigma}=(\tilde{T},\tilde{\mathbf{x}},\tilde{\mathbf{d}},\tilde{\omega})$
of rank $n$ and dimension ${n+m}$ are _quasi-equivalent_ , denoted
$\Sigma\sim\tilde{\Sigma}$, if the following conditions are satisfied:

  * •

$T=\tilde{T}$, $\mathbf{d}=\tilde{\mathbf{d}}$, $\omega=\tilde{\omega}$;

  * •

the sublattice of $X^{\ast}(T)$ spanned by the frozen variables
$x_{n+1},\dots,x_{{n+m}}$ coincides with the sublattice spanned by
$\tilde{x}_{n+1},\dots,\tilde{x}_{{n+m}}$;

  * •

for each $k\in[n]$, we have $\tilde{x}_{k}=x_{k}M_{k}$, where $M_{k}$ is a
Laurent monomial in $x_{n+1},\dots,x_{{n+m}}$.

It is easy to see that if $\Sigma$ is integral and $\Sigma\sim\tilde{\Sigma}$
then $\tilde{\Sigma}$ is integral. The following is also straightforward to
check.

######  Lemma 3.9.

If $\Sigma$ and $\tilde{\Sigma}$ are quasi-equivalent seeds then
$\mu_{k}(\Sigma)\sim\mu_{k}(\tilde{\Sigma})$ for all mutable $k$.

######  Corollary 3.10.

Suppose two seeds $\Sigma,\tilde{\Sigma}$ are quasi-equivalent. Then they
define the same cluster algebra
${\mathcal{A}}(\Sigma)={\mathcal{A}}(\Sigma^{\prime})\subset{\mathbb{C}}(T)$.

###### Proof.

It follows from Lemma 3.9 that each cluster variable in
${\mathcal{A}}(\Sigma)$ differs from the corresponding cluster variable in
${\mathcal{A}}(\tilde{\Sigma})$ by a factor equal to a Laurent monomial in the
frozen variables. ∎

###  3.3. Deletion-contraction

We give an inductive criterion for a pair $(X,\Sigma)$ to be a sink-recurrent
cluster variety. In Section 4.9, we will apply this criterion to the seeds
$\Sigma_{\beta}$ constructed in Section 2.

######  Assumption 3.11.

Throughout this section, we let $\Sigma=(T,{\mathbf{x}},\mathbf{d},\omega)$ be
an abstract seed of rank $n$ and dimension ${n+m}$. Let
$\Gamma:=\Gamma(\Sigma)$. We assume that $\Sigma$ is sink-recurrent, with sink
$s$ in $\Gamma$ such that $\Sigma$ is integral at $s$. Further, we assume that
there exists a frozen index $f$ such that ${\tilde{B}}_{fs}=\pm 1$ and
${\tilde{B}}_{fj}=0$ for $j\in[n]\setminus\\{s\\}$. Suppose that the exchange
relation for $x_{s}$ in $\Sigma$ is given by
$x_{s}x_{s}^{\prime}=M_{1}+x_{f}M_{2}$ for some monomials $M_{1},M_{2}$ in
$\\{x_{j}\\}_{j\in[n+m]\setminus\\{s,f\\}}$.

######  Definition 3.12.

Suppose that $s$ has $q:=|N^{\operatorname{in}}_{s}(\Gamma)|$ mutable
neighbors. The _contraction_
$\Sigma^{/s}=(T^{/s},{\mathbf{x}}^{/s},\mathbf{d}^{/s},\omega^{/s})$ is a seed
of rank $n-q-1$ and dimension ${n+m}-2$ defined as follows.

  1. (1)

${\mathbf{x}}^{/s}$ is obtained from ${\mathbf{x}}$ by omitting $x_{s}$ and
$x_{f}$ and declaring the indices in $N^{\operatorname{in}}_{s}(\Gamma)$ to be
frozen.

  2. (2)

$T^{/s}$ is an algebraic torus with character lattice generated by
${\mathbf{x}}^{/s}$.

  3. (3)

$\mathbf{d}^{/s}$ is obtained by restricting the sequence $\mathbf{d}$ to the
set $[{n+m}]\setminus\\{s,f\\}$.

  4. (4)

$\omega^{/s}$ is obtained from $\omega$ by writing it in the form (3.1) and
substituting ${\rm dlog}x_{s}:=0$ and ${\rm dlog}x_{f}:={\rm dlog}M_{1}-{\rm
dlog}M_{2}$.

The _deletion_ $\Sigma^{\backslash s}$ is the seed of rank $n-1$ and dimension
${n+m}$ obtained by declaring $x_{s}$ to be frozen (cf. Definition 3.4).

[GL22, Corollary 5.15] gives some evidence that the nomenclature of deletion
and contraction is appropriate.

######  Theorem 3.13 (Deletion-contraction recurrence).

Let $X$ be an affine, normal, irreducible, complex algebraic variety, and let
$\Sigma=(T,{\mathbf{x}},\mathbf{d},\omega)$ be a seed on $X$ with a sink $s$
satisfying 3.11. Assume that all cluster variables in
${\mathbf{x}}\sqcup\\{x_{s}^{\prime}\\}$ are regular on $X$. Define
subvarieties $W:=\\{x_{s}\neq 0\\}$ and $V:=\\{x_{s}=0\\}$ of $X$. Suppose
that the seeds $\Sigma^{\backslash s},\Sigma^{/s}$ are integral, and that we
have isomorphisms $W\cong\mathcal{X}(\Sigma^{\backslash s})$ and
$V\cong\operatorname{Spec}({\mathbb{C}}[x^{\prime}_{s}])\times\mathcal{X}(\Sigma^{/s})\cong{\mathbb{C}}\times\mathcal{X}(\Sigma^{/s})$
satisfying:

  1. (1)

Under the inclusion $\iota_{W}:W\hookrightarrow X$, the pullbacks of the
cluster variables $x_{1},\ldots,x_{n+m}$ of $\Sigma$ are the same-named
cluster variables of $\Sigma^{\backslash s}$.

  2. (2)

Under the inclusion $\iota_{V}:V\hookrightarrow X$, the pullbacks of the
cluster variables $x_{j}$ for $j\in[n+m]\setminus\\{s,f\\}$ are the same-named
cluster variables of $\Sigma^{/s}$, and the pullback of the mutated cluster
variable $x^{\prime}_{s}$ is the same-named function on the first factor
$\operatorname{Spec}({\mathbb{C}}[x^{\prime}_{s}])$ under the isomorphism
$V\cong\operatorname{Spec}({\mathbb{C}}[x^{\prime}_{s}])\times\mathcal{X}(\Sigma^{/s})$.

Then $(X,\Sigma)$ is a cluster variety.

###### Proof.

First, it is clear that since $\Sigma$ is integral at $s$ and
$\Sigma^{\backslash s}$ is integral, $\Sigma$ is also integral.

Let $j\in[n]\setminus\\{s\\}$ be a mutable index. Clearly, the (pullback under
$\iota_{W}$ of the) exchange relation for $x_{j}$ in $\Sigma$ coincides with
the exchange relation for $x_{j}$ in $\Sigma^{\backslash s}$. Thus, the
mutated variable $x_{j}^{\prime}$ is regular on $W$. Next, assume that
$j\notin N^{\operatorname{in}}_{s}(\Gamma)$. By assumption, $j$ is not
connected to $s,f$ in $\Gamma$, and thus the terms involving ${\rm dlog}x_{j}$
are unchanged when passing from $\omega$ to $\omega^{/s}$. Thus, the pullback
of the exchange relation for $x_{j}$ under $\iota_{V}$ is still the exchange
relation for $x_{j}$ in $\Sigma^{/s}$, and therefore the mutated variable
$x_{j}^{\prime}$ is regular on $V$. For $j\in
N^{\operatorname{in}}_{s}(\Gamma)$, $x_{j}^{\prime}$ must also be regular on
$V$ since the pullback of $x_{j}$ is a frozen variable. It follows that for
all $j\in[n]\setminus\\{s\\}$, the mutated variable $x_{j}^{\prime}$ is a
regular function on $X$ since it is regular on both $V$ and $W$. For $j=s$,
$x_{s}^{\prime}$ is regular on $X$ by assumption.

Next, we show that ${\mathbb{C}}[X]\subset\mathcal{U}(\Sigma)$. This is
equivalent to constructing inclusions $T\hookrightarrow X$ and
$\mu_{j}(T)\hookrightarrow X$ for all $j\in[n]$. Here, constructing an
inclusion $\mu_{j}(T)\hookrightarrow X$ means showing that the subset
$X_{j}\subset X$ where the regular functions in $\mu_{j}({\mathbf{x}})$ are
all non-vanishing is isomorphic to an algebraic torus
$\mu_{j}(T)\cong({\mathbb{C}}^{\times})^{{n+m}}$ via a map
$\varphi_{j}:X_{j}\to\mu_{j}(T)$ sending $y\in X_{j}$ to
$z:=(x_{1}(y),\dots,x_{j}^{\prime}(y),\dots,x_{{n+m}}(y))$. Since $\Sigma$ is
a seed on $X$, we have $T\subset X$. Next, if $j\in[n]\setminus s$ then we
have $X_{j}\subset W$, and thus the statement follows since $W$ is a cluster
variety. Finally, assume that $j=s$. Consider the torus
$\mu_{s}(T)\cong({\mathbb{C}}^{\times})^{{n+m}}$. Let
$p:=M_{1}+x_{f}M_{2}\in{\mathbb{C}}[X]$ be the exchange binomial for $x_{s}$
(cf. 3.11). Since $p$ does not involve $x_{s}$ and $x_{s}^{\prime}$, we can
also view $p$ as a regular function on $\mu_{s}(T)$ compatible with pullback
under $\varphi_{s}$. Let $z=(z_{1},\dots,z_{{n+m}})\in\mu_{s}(T)$. Our goal is
to show that $z$ has a unique preimage under $\varphi_{s}$. Suppose first that
$p(z)\neq 0$. Then $\varphi_{s}^{-1}(z)\subset T$, and the result follows.
Suppose now that $p(z)=0$. Then $\varphi_{s}^{-1}(z)\subset V$. Recall that
$V\cong\operatorname{Spec}({\mathbb{C}}[x^{\prime}_{s}])\times\mathcal{X}(\Sigma^{/s})$.
Since $x^{\prime}_{s}=z_{s}$, the
$\operatorname{Spec}({\mathbb{C}}[x^{\prime}_{s}])$-coordinate of the preimage
is uniquely determined by $z$. The $\mathcal{X}(\Sigma^{/s})$-coordinate of
the preimage is uniquely determined by
$(z_{i})_{i\in[{n+m}]\setminus\\{s,f\\}}$. We have shown that $z$ has a unique
preimage under $\varphi_{s}$, which completes the proof of the inclusion
${\mathbb{C}}[X]\subset\mathcal{U}(\Sigma)$. The statement of the theorem now
follows from Proposition 3.14 below. ∎

######  Proposition 3.14.

Let $X$ be an affine, normal, irreducible, complex algebraic variety, and let
$\Sigma=(T,{\mathbf{x}},\mathbf{d},\omega)$ be an integral sink-recurrent seed
on $X$. Suppose that ${\mathbb{C}}[X]\subset\mathcal{U}(\Sigma)$. Then
$(X,\Sigma)$ is a cluster variety.

###### Proof.

The inclusions ${\mathbb{C}}[X]\subset{\mathbb{C}}[x_{1}^{\pm
1},\ldots,(x^{\prime}_{j})^{\pm 1},\ldots,x_{n+m}^{\pm 1}]$ give tori
$\mu_{j}(T)\cong X_{j}\subset X$ as in the proof of Theorem 3.13. By a
standard argument, this implies that the complement of
$T\cup\bigcup_{j\in[n]}X_{j}$ has codimension greater than or equal to two in
$X$; see [Zel00, Section 3], [BFZ05, Proof of Theorem 2.10], or [GLSBS22,
Lemmas 9.5–9.8]. Since $X$ is normal, we have
${\mathbb{C}}[X]={\mathbb{C}}[T\cup\bigcup_{j\in[n]}X_{j}]=\mathcal{U}(\Sigma)$.
By assumption, $\Sigma$ is sink-recurrent, and we are done by Proposition 3.6.
∎

######  Remark 3.15.

If the seeds $\Sigma^{\backslash s},\Sigma^{/s}$ are both really full rank
then it follows from 3.11 that $\Sigma$ is really full rank. Indeed, row $f$
of the exchange matrix of $\Sigma$ contains a single nonzero entry equal to
$\pm 1$ in column $s$. The exchange matrix of $\Sigma^{\backslash s}$ is
obtained from that of $\Sigma$ by removing column $s$. This implies that if
$\Sigma^{\backslash s}$ is really full rank then so is $\Sigma$.

######  Remark 3.16.

The statement of Theorem 3.13 remains true if “sink-recurrent” is replaced
with “locally acyclic.”

##  4\. Double braid moves

In this section, we study natural isomorphisms between braid varieties
corresponding to _double braid moves_ , and determine the effect of these
isomorphisms on seeds. Double braid moves are defined as follows:

  1. (B1)

$ij\leftrightarrow ji$ if $i,j\in\pm I$ have different signs;

  2. (B2)

$ij\leftrightarrow ji$ if $i,j\in\pm I$ have the same sign and
$(s_{|i|}s_{|j|})^{2}=1$;

  3. (B3)

$\underbrace{iji\dots}_{\text{$m_{ij}$
letters}}\leftrightarrow\underbrace{jij\dots}_{\text{$m_{ij}$ letters}}$ if
$i,j\in\pm I$ have the same sign and $(s_{|i|}s_{|j|})^{m_{ij}}=1$ with
$m_{ij}\geq 3$;

  4. (B4)

$\beta_{0}i\leftrightarrow\beta_{0}(-i^{\ast})$ for $i\in\pm I$ and
$\beta_{0}\in(\pm I)^{m-1}$;

  5. (B5)

$i\beta_{0}\leftrightarrow(-i)\beta_{0}$ for $i\in\pm I$ and $\beta_{0}\in(\pm
I)^{m-1}$.

If double braid words $\beta$ and $\beta^{\prime}$ are related by one of the
moves (B1)–(B5), there is a natural isomorphism
$\phi:\accentset{\circ}{R}_{\beta}\xrightarrow{\sim}\accentset{\circ}{R}_{\beta^{\prime}}$.

######  Definition 4.1.

Suppose that $\beta$ and $\beta^{\prime}$ are related by one of the moves
(B1)–(B3). If this move involves indices $l,l+1,\dots,r$, the isomorphism
$\phi$ sends $(X_{\bullet},Y_{\bullet})\in\accentset{\circ}{R}_{\beta}$ to the
unique tuple
$(X_{\bullet}^{\prime},Y_{\bullet}^{\prime})\in\accentset{\circ}{R}_{\beta^{\prime}}$
such that $X_{c}^{\prime}=X_{c}$ and $Y_{c}^{\prime}=Y_{c}$ for $0\leq c<l$ or
$r\leq c\leq m$. The remaining weighted flags
$X_{l}^{\prime},\dots,X_{r}^{\prime},Y_{l}^{\prime},\dots,Y_{r}^{\prime}$ are
uniquely determined using Lemma 2.1.

For the moves (B4) and (B5), the isomorphism $\phi$ is described in Sections
4.6 and 4.7, respectively.

The main result of this section is the following.

######  Theorem 4.2.

Suppose that $\beta$ and $\beta^{\prime}$ are related by one of the moves
(B1)–(B5). If $(\accentset{\circ}{R}_{\beta},\Sigma_{\beta})$ is a cluster
variety then so is
$(\accentset{\circ}{R}_{\beta^{\prime}},\Sigma_{\beta^{\prime}})$. In this
case, the seeds $\Sigma_{\beta}$ and $\phi^{*}\Sigma_{\beta^{\prime}}$ are
mutation equivalent (up to a relabeling of the cluster variables) for the
moves (B1)–(B4) and are quasi-equivalent for the move (B5).

Here, for a seed
$\Sigma_{\beta^{\prime}}=(T_{\beta^{\prime}},{\mathbf{x}}_{\beta^{\prime}},\mathbf{d}_{\beta^{\prime}},\omega_{\beta^{\prime}})$,
$\phi^{*}\Sigma_{\beta^{\prime}}=(T^{\ast},{\mathbf{x}}^{\ast},\mathbf{d}^{\ast},\omega^{\ast})$
is an abstract seed on $\accentset{\circ}{R}_{\beta}$ defined by
$T^{\ast}:=\phi^{-1}(T_{\beta^{\prime}})$,
${\mathbf{x}}^{\ast}:=\phi^{*}{\mathbf{x}}_{\beta^{\prime}}$,
$\mathbf{d}^{\ast}:=\mathbf{d}_{\beta^{\prime}}$, and
$\omega^{\ast}:=\phi^{\ast}\omega_{\beta^{\prime}}$.

The proof will occupy Sections 4–6. Along the way, we will construct a seed
$\Sigma^{\prime}=(T^{\prime},{\mathbf{x}}^{\prime},\mathbf{d}^{\prime},\omega^{\prime})$
obtained from $\Sigma_{\beta}=(T,{\mathbf{x}},\mathbf{d},\omega)$ by one or
several mutations, followed by a relabeling. We will then show the following
intermediate statements for moves (B1)–(B5).

  1. (F)[IS0]

The $2$-form is invariant: $\phi^{*}\omega_{\beta^{\prime}}=\omega_{\beta}$.

  2. (Q)[IS2]

Suppose that $(\accentset{\circ}{R}_{\beta},\Sigma_{\beta})$ is a cluster
variety. Then the seeds $\Sigma^{\prime}$ and
$\phi^{*}\Sigma_{\beta^{\prime}}$ are quasi-equivalent.

  3. (E)[IS3]

Suppose that $(\accentset{\circ}{R}_{\beta},\Sigma_{\beta})$ is a cluster
variety. Then for the moves (B1)–(B4), the seeds $\Sigma^{\prime}$ and
$\phi^{*}\Sigma_{\beta^{\prime}}$ are equal.

It is clear that LABEL:IS2, LABEL:IS3 together imply the conclusion of Theorem
4.2. Note that if $\Sigma_{\beta}$ is integral then the seed $\Sigma^{\prime}$
is integral, and so LABEL:IS2 automatically implies that
$\phi^{*}\Sigma_{\beta^{\prime}}$ is as well.

######  Definition 4.3.

A move (B1)–(B3) is called _solid_ if all indices involved are solid. For
$i,j\in I$, the (B1) move $(-i)j\leftrightarrow j(-i)$ on indices $c,c+1$ is
called _special_ if $u_{c}s_{i}=s_{j}u_{c}$ and _solid-special_ if it is both
solid and special. A move (B1)–(B5) is called _short_ if it involves at most 3
indices and _long_ otherwise. That is, the long moves are the (B3) moves for
$m_{ij}>3$. Finally, a move (B1)–(B5) is a _mutation move_ if it involves at
least one cluster mutation; otherwise it is a _non-mutation move_.

######  Remark 4.4.

As we will show in Section 4.1, a solid-special (B1) move corresponds to a
single mutation, at the rightmost index involved in the move. The move (B3)
involving $q$ solid indices (for $q\leq m_{ij}$) corresponds to a sequence of
$\binom{q-1}{2}$ mutations on the rightmost $m_{ij}-2$ indices involved in the
move. In particular, a short (B3) move gives rise to one mutation if it is
solid and to zero mutations otherwise.

We will show LABEL:IS0, LABEL:IS2, LABEL:IS3 for short moves directly. This
will complete the proof of Theorem 4.2 in simply-laced types. We then use this
and folding to show LABEL:IS0, LABEL:IS2, LABEL:IS3 for long moves in Sections
5 and 6.

Throughout this section, we fix $\beta,\beta^{\prime}$ related by a short
move. If
$\phi:\accentset{\circ}{R}_{\beta}\xrightarrow{\sim}\accentset{\circ}{R}_{\beta^{\prime}}$
is the corresponding isomorphism, for a rational function or a form $f$ on
$\accentset{\circ}{R}_{\beta^{\prime}}$, we denote its pullback $\phi^{*}f$ by
$f^{\ast}$. For example, $\Delta_{c,k}^{\ast}$ denotes the pullback
$\phi^{*}\Delta_{c,k}$ of a grid minor for
$\accentset{\circ}{R}_{\beta^{\prime}}$.

######  Remark 4.5.

If all indices involved in a move (B1)–(B3) are hollow, then the statements
LABEL:IS0, LABEL:IS2, LABEL:IS3 follow trivially; cf. Corollary 2.27.

###  4.1. Mutation move: (B1), solid-special

Consider the case of a solid-special move (B1) on indices $c,c+1$. Since both
indices are solid, we denote $u:=u_{c-1}=u_{c}=u_{c+1}$ and
$w:=w_{c-1}=w_{c}=w_{c+1}$. The indices $i,j\in\pm I$ are of opposite signs,
and throughout this section, we assume that $i\in-I$ and $j\in I$; the other
case is handled similarly. The solid-special condition yields

(4.1) $u<s_{|i|}u=us_{j}\quad\text{and}\quad s_{|i|^{*}}w=ws_{j}<w.$

######  Proposition 4.6 ([FZ99, Theorem 1.17]).

We have111Our Cartan matrix is the transpose of that of [FZ99]; see [FZ99,
Equation (2.27)].

(4.2)
$\Delta_{c,j}\Delta_{c,j}^{\ast}=\Delta_{c+1,j}\Delta_{c-1,j}+\prod_{k\neq
j}\Delta_{c,k}^{-a_{jk}}.$

###### Proof.

We may choose $t,t^{\prime}\in{\mathbb{C}}$ such that $Z_{c}=Z_{c+1}z_{j}(t)$,
$Z_{c}^{\ast}=\bar{z}_{|i|^{*}}(t^{\prime})^{-1}Z_{c+1}$, and
$Z_{c-1}=Z_{c-1}^{\ast}=\bar{z}_{|i|^{*}}(t^{\prime})^{-1}Z_{c+1}z_{j}(t)=\dot{s}_{|i|^{*}}x_{|i|^{*}}(t^{\prime})Z_{c+1}x_{j}(t)\dot{s}_{j}$.
Let $Z:=x_{|i|^{*}}(t^{\prime})Z_{c+1}x_{j}(t)$. Using properties of
generalized minors from Section 2.5, one can check that each term of (4.2)
equals the corresponding term of

(4.3)
$\Delta_{w\omega_{j},s_{j}\omega_{j}}(Z)\Delta_{ws_{j}\omega_{j},\omega_{j}}(Z)=\Delta_{ws_{j}\omega_{j},s_{j}\omega_{j}}(Z)\Delta_{w\omega_{j},\omega_{j}}(Z)+\prod_{k\neq
j}\Delta_{ws_{j}\omega_{k},\omega_{k}}(Z)^{-a_{jk}}.$

For example, we have

$\Delta_{c,j}=\Delta_{w\omega_{j},\omega_{j}}(Z_{c+1}x_{j}(t)\dot{s}_{j})=\Delta_{\omega_{j},\omega_{j}}(\dot{w}^{-1}Z_{c+1}x_{j}(t)\dot{s}_{j})=\Delta_{\omega_{j},\omega_{j}}(\dot{w}^{-1}Z\dot{s}_{j})=\Delta_{w\omega_{j},s_{j}\omega_{j}}(Z),$

where we have used $\dot{w}^{-1}x_{|i|^{*}}(t^{\prime})\in U_{-}\dot{w}^{-1}$;
cf. (2.10) and (4.1). For $\Delta_{c,k}^{-a_{jk}}$, $k\neq j$, we additionally
used that $s_{j}\omega_{k}=\omega_{k}$. Equation (4.3) holds by [FZ99, Theorem
1.17]. ∎

We shall use the following analog of [GLSBS22, Lemma 8.10].

######  Lemma 4.7.

For $e\in[0,m]$ such that $u_{e}s_{j}=s_{|i|}u_{e}$, we have

(4.4) $\prod_{k\in\pm I}\Delta_{e,k}^{a_{ik}}=\prod_{k\in\pm
I}\Delta_{e,k}^{\epsilon a_{jk}}\quad\text{and}\quad L_{e,i}=\epsilon
L_{e,j},\quad\text{where}\quad\epsilon:=\begin{cases}1,&\text{if
$u_{e}<u_{e}s_{j}$,}\\\ -1,&\text{if $u_{e}>u_{e}s_{j}$.}\\\ \end{cases}$

###### Proof.

We have $\alpha_{j}=\sum_{k\in I}a_{jk}\omega_{k}$ and similarly for
$\alpha_{|i|}$. The first identity in (4.4) therefore becomes
$(h^{-}_{e})^{\alpha_{|i|}}=(h^{+}_{e})^{\epsilon\alpha_{j}}$, which follows
from the assumption $u_{e}\alpha_{j}=\epsilon\alpha_{|i|}$ together with
$h^{-}_{e}=u_{e}\cdot h^{+}_{e}$; cf. (2.7). Taking ${\rm dlog}$ of both
sides, we obtain the second identity. ∎

######  Remark 4.8.

Equations (4.2) and (4.4) are true as stated in the case $i\in I,j\in-I$ as
well.

###### Proof of LABEL:IS0 for (B1), solid-special.

Only the terms $\omega_{\beta,c}$ and $\omega_{\beta,c+1}$ change when doing
the move (B1). Noting that we must have $d_{i}=d_{j}$ and applying (4.4), we
get

$\displaystyle\frac{1}{d_{j}}(\omega_{\beta}-\omega_{\beta^{\prime}}^{\ast})$
$\displaystyle=\frac{1}{d_{j}}\left(\omega_{\beta,c}+\omega_{\beta,c+1}-\omega_{\beta^{\prime},c}^{\ast}-\omega_{\beta^{\prime},c+1}^{\ast}\right)$
$\displaystyle=-L_{c-1,i}\wedge L_{c,i}+L_{c,j}\wedge
L_{c+1,j}-L_{c-1,j}^{\ast}\wedge L_{c,j}^{\ast}+L_{c,i}^{\ast}\wedge
L_{c+1,i}^{\ast}$ $\displaystyle=-L_{c-1,j}\wedge L_{c,j}+L_{c,j}\wedge
L_{c+1,j}-L_{c-1,j}\wedge L_{c,j}^{\ast}+L_{c,j}^{\ast}\wedge L_{c+1,j}$
$\displaystyle=(L_{c,j}+L_{c,j}^{\ast})\wedge(L_{c-1,j}+L_{c+1,j}).$

For $e\in\\{c-1,c,c+1\\}$, let $M_{e}:=\prod_{k\neq j}\Delta_{e,k}^{-a_{jk}}$.
Thus, $M_{c}$ is the third term in (4.2). By (2.8), we have
$h^{+}_{c}=\alpha^{\vee}_{j}(t_{c+1})h^{+}_{c+1}$ and
$h^{-}_{c-1}=\alpha^{\vee}_{|i|}(t_{c})h^{-}_{c}$. This implies that
$h^{+}_{c-1}=\alpha^{\vee}_{j}(t_{c}t_{c+1})h^{+}_{c+1}$ since
$h^{-}_{c}=u\cdot h^{+}_{c}$. Thus, we have $M:=M_{c-1}=M_{c}=M_{c+1}$. Since
$M_{c+1}^{\ast}=M_{c+1}$, we get that
$M=M_{c-1}^{\ast}=M_{c}^{\ast}=M_{c+1}^{\ast}$. Rewrite (4.2) as

$\frac{\Delta_{c,j}\Delta_{c,j}^{\ast}}{M}=\frac{\Delta_{c-1,j}\Delta_{c+1,j}}{M}+1.$

Denote the left-hand side by $A$ and the right-hand side by $B+1$.
Differentiating both sides, we get $dA=dB$, and thus ${\rm
dlog}A=\frac{B}{B+1}{\rm dlog}B$. This implies that ${\rm dlog}A\wedge{\rm
dlog}B=0$. It remains to note that ${\rm dlog}A=L_{c,j}+L_{c,j}^{\ast}$ while
${\rm dlog}B=L_{c-1,j}+L_{c+1,j}$. ∎

###### Proof of LABEL:IS2 for (B1), solid-special.

We do not use the assumption that
$(\accentset{\circ}{R}_{\beta},\Sigma_{\beta})$ is a cluster variety until the
last paragraph of this proof. Let $x:=x_{c+1}$ and $V:=V_{c+1}$. Applying
Propositions 2.21 and 2.22, we see that $x$ is mutable,

(4.5)
${\mathrm{ord}}_{V}\Delta_{c,j}={\mathrm{ord}}_{V}\Delta_{c,i}=1,\quad\text{and}\quad{\mathrm{ord}}_{V}\Delta_{e,k}=0\quad\text{for
$(e,k)\not\in\\{(c,j),(c,i)\\}$.}$

In particular, ${\rm dlog}x$ appears in $\omega_{\beta}$ only in the terms
$L_{c,j}$ and $L_{c,i}$ in $\omega_{\beta,c+1}=d_{j}L_{c,j}\wedge L_{c+1,j}$
and $\omega_{\beta,c}=-d_{i}L_{c-1,i}\wedge L_{c,i}$, respectively. Recall
from (4.4) that we actually have $L_{c,j}=L_{c,i}$. We see from (4.5) that the
coefficient of ${\rm dlog}x$ in $L_{c,j}=L_{c,i}$ is equal to $1$. Collecting
the terms of $\omega_{\beta}$ involving ${\rm dlog}x$ and using $d_{i}=d_{j}$,
we get

(4.6) $d_{j}{\rm dlog}x\wedge(L_{c+1,j}+L_{c-1,i})=d_{j}{\rm
dlog}x\wedge\left({\rm dlog}(\Delta_{c+1,j}\Delta_{c-1,j})-{\rm
dlog}\prod_{k\neq j}\Delta_{c,k}^{-a_{jk}}\right).$

By Proposition 2.21, a cluster variable $x_{e}$ for $e\in J_{\beta}$ may
appear on the right-hand side of (4.6) only for $e\geq c$. Moreover, we have
already observed that $d_{c}=d_{|i|}=d_{j}=d_{c+1}$. Let us denote
$p_{e}:={\mathrm{ord}}_{V_{e}}(\Delta_{c+1,j}\Delta_{c-1,j})$ and
$q_{e}:={\mathrm{ord}}_{V_{e}}\prod_{k\neq j}\Delta_{c,k}^{-a_{jk}}$. Clearly,
$p_{e},q_{e}\geq 0$. We therefore see from (3.1) that for all $e\in
J_{\beta}$, we have ${\tilde{B}}_{e,c+1}=q_{e}-p_{e}$. By Definition 2.14, the
cluster variable $x$ is mutable. Thus, the mutated variable
$x^{\prime}:=x^{\prime}_{c+1}$ satisfies

(4.7) $xx^{\prime}=\prod_{e\in
J_{\beta}:p_{e}>q_{e}}x_{e}^{p_{e}-q_{e}}+\prod_{e\in
J_{\beta}:q_{e}>p_{e}}x_{e}^{q_{e}-p_{e}}.$

We have $V_{e}=V_{e}^{\ast}$ and $x_{e}=x_{e}^{\ast}$ for all $e\in
J_{\beta}\setminus\\{c+1\\}$. Let $V^{\ast}:=V_{c+1}^{\ast}$ and
$x^{\ast}:=x_{c+1}^{\ast}$. A generic point $(X_{\bullet},Y_{\bullet})\in V$
satisfies $X_{c-1}\stackrel{{\scriptstyle\ ws_{j}}}{{\Longleftarrow}}Y_{c+1}$
and $X_{c+1}\stackrel{{\scriptstyle\ w}}{{\Longleftarrow}}Y_{c-1}$, while a
generic point $(X_{\bullet},Y_{\bullet})\in V^{\ast}$ satisfies
$X_{c+1}\stackrel{{\scriptstyle\ ws_{j}}}{{\Longleftarrow}}Y_{c-1}$ and
$X_{c-1}\stackrel{{\scriptstyle\ w}}{{\Longleftarrow}}Y_{c+1}$. Thus, $V\neq
V^{\ast}$.

For $e\in J_{\beta}$, applying ${\mathrm{ord}}_{V_{e}}$ to both sides of
(4.2), we get

(4.8)
${\mathrm{ord}}_{V_{e}}\Delta_{c,j}+{\mathrm{ord}}_{V_{e}}\Delta_{c,j}^{\ast}\geq\min(p_{e},q_{e}).$

For $e=c+1$, we have ${\mathrm{ord}}_{V}\Delta_{c,j}=1$,
${\mathrm{ord}}_{V}\Delta_{c,j}^{\ast}=0$ (since $V\neq V^{\ast}$), and
$p_{c+1}=q_{c+1}=0$. Similarly,
${\mathrm{ord}}_{V^{\ast}}\Delta_{c,j}^{\ast}=1$,
${\mathrm{ord}}_{V^{\ast}}\Delta_{c,j}=0$, and the order of vanishing of
$\Delta_{c+1,j}^{\ast}\Delta_{c-1,j}^{\ast}$ and $\prod_{k\neq
j}(\Delta_{c,k}^{\ast})^{-a_{jk}}$ at $V^{\ast}$ is zero.

Dividing both sides of (4.2) by $\prod_{e\in
J_{\beta}\setminus\\{c+1\\}}x_{e}^{\min(p_{e},q_{e})}$, we get

$xx^{\ast}\prod_{e\in J_{\beta}\setminus\\{c+1\\}}x_{e}^{r_{e}}=\prod_{e\in
J_{\beta}:p_{e}>q_{e}}x_{e}^{p_{e}-q_{e}}+\prod_{e\in
J_{\beta}:q_{e}>p_{e}}x_{e}^{q_{e}-p_{e}},$

where
$r_{e}:={\mathrm{ord}}_{V_{e}}\Delta_{c,j}+{\mathrm{ord}}_{V_{e}}\Delta_{c,j}^{\ast}-\min(p_{e},q_{e})\geq
0$. By (4.7), we get

(4.9) $x^{\prime}=x^{\ast}\prod_{e\in
J_{\beta}\setminus\\{c+1\\}}x_{e}^{r_{e}}.$

Now, assume that $(\accentset{\circ}{R}_{\beta},\Sigma_{\beta})$ is a cluster
variety. We get from Proposition 3.3 that the mutated cluster variable
$x^{\prime}$ is irreducible in ${\mathbb{C}}[\accentset{\circ}{R}_{\beta}]$.
The function $x^{\ast}$ vanishes on
$V^{\ast}\subset\accentset{\circ}{R}_{\beta}$ and therefore is not a unit in
${\mathbb{C}}[\accentset{\circ}{R}_{\beta}]$. It follows that $r_{e}=0$ for
all mutable $e$, and thus the mutated seed
$\Sigma^{\prime}:=\mu_{c+1}\Sigma_{\beta}$ is quasi-equivalent to the pulled
back seed $\Sigma_{\beta^{\prime}}^{\ast}$. ∎

###  4.2. Non-mutation move: (B1), not solid-special

We continue to assume that the move involves indices $c,c+1$, and that
$i\in-I$, $j\in I$.

####  4.2.1. (B1), special, non-solid

Suppose that at least one of the indices is hollow, and that the move is
special. Then it follows that $c+1$ is hollow and $c$ is solid in both $\beta$
and $\beta^{\prime}$. By (2.24), $L_{c,j}=-L_{c+1,j}$ and
$L_{c,i}^{\ast}=-L_{c+1,i}^{\ast}$. Applying (4.4) with $\epsilon=1$ for
$e=c-1,c$ and $\epsilon=-1$ for $e=c+1$ and using $d_{i}=d_{j}$, we obtain

$\frac{\omega_{\beta,c}}{d_{j}}=-L_{c-1,i}\wedge L_{c,i}=L_{c-1,i}\wedge
L_{c+1,i}=L_{c-1,i}^{\ast}\wedge L_{c+1,i}^{\ast}=-L_{c-1,j}^{\ast}\wedge
L_{c+1,j}^{\ast}=L_{c-1,j}^{\ast}\wedge
L_{c,j}^{\ast}=\frac{\omega_{\beta^{\prime},c}^{\ast}}{d_{i}},$

which proves LABEL:IS0. The clusters ${\mathbf{x}}_{\beta}$ and
${\mathbf{x}}_{\beta^{\prime}}^{\ast}$ are identical, which proves LABEL:IS2.

####  4.2.2. (B1), non-special

We start by introducing a formalism for working with the forms $L_{e,k}$. Let
${\lambda}:=\sum_{k\in I}b_{k}\omega_{k}$ with $b_{k}\in\mathbb{Q}$, and let
$h$ be an $H$-valued rational function on $\accentset{\circ}{R}_{\beta}$. We
introduce a rational $1$-form

${\rm dlog}h^{\lambda}:=\sum_{k\in I}b_{k}{\rm dlog}(h^{\omega_{k}}).$

It is clear that

(4.10) ${\rm dlog}h^{{\lambda}_{1}+{\lambda}_{2}}={\rm
dlog}h^{{\lambda}_{1}}+{\rm dlog}h^{{\lambda}_{2}}\quad\text{and}\quad{\rm
dlog}(h_{1}h_{2})^{{\lambda}}={\rm dlog}h_{1}^{\lambda}+{\rm
dlog}h_{2}^{\lambda}.$

For $e\in[0,m]$ and $k\in I$, Lemma 2.10 gives

(4.11) $L_{e,k}={\rm dlog}(h^{+}_{e})^{\alpha_{k}/2}={\rm
dlog}(h^{-}_{e})^{u_{e}\alpha_{k}/2},\quad L_{e,-k}={\rm
dlog}(h^{-}_{e})^{\alpha_{k}/2}={\rm
dlog}(h^{+}_{e})^{u_{e}^{-1}\alpha_{k}/2}.$

Finally, suppose that $h_{1}=h_{2}\alpha^{\vee}_{k}(t)$. Then we have

(4.12) ${\rm dlog}h_{1}^{\lambda}={\rm
dlog}h_{2}^{\lambda}+{\langle}{\lambda},\alpha^{\vee}_{k}{\rangle}{\rm
dlog}t.$

###### Proof of LABEL:IS0 and LABEL:IS2 for (B1), non-special.

Suppose as before that the move involves indices $c,c+1$, and that $i\in-I$,
$j\in I$. Assume first that both $c,c+1$ are solid, and let
$u:=u_{c-1}=u_{c}=u_{c+1}$. Let
$a:={\langle}u^{-1}\alpha_{|i|}/2,\alpha^{\vee}_{j}{\rangle}$ and
$a^{\prime}:={\langle}u\alpha_{j}/2,\alpha^{\vee}_{|i|}{\rangle}$. Using
(4.11)–(4.12) and (2.8), we get

(4.13) $\displaystyle L_{c,i}$ $\displaystyle=L_{c+1,i}+a{\rm dlog}t_{c+1},$
$\displaystyle L_{c-1,i}$ $\displaystyle=L_{c,i}+{\rm dlog}t_{c},$
$\displaystyle L_{c,j}$ $\displaystyle=L_{c+1,j}+{\rm dlog}t_{c+1};$ (4.14)
$\displaystyle L_{c,j}^{\ast}$ $\displaystyle=L_{c+1,j}^{\ast}+a^{\prime}{\rm
dlog}t_{c+1}^{\ast},$ $\displaystyle L_{c-1,j}^{\ast}$
$\displaystyle=L_{c,j}^{\ast}+{\rm dlog}t_{c}^{\ast},$ $\displaystyle
L_{c,i}^{\ast}$ $\displaystyle=L_{c+1,i}^{\ast}+{\rm dlog}t_{c+1}^{\ast}.$

Since the move is non-special, the coroots $\alpha^{\vee}_{j}$ and
$u^{-1}\alpha^{\vee}_{|i|}$ are linearly independent, which implies
$t_{c}^{\ast}=t_{c+1}$ and $t_{c+1}^{\ast}=t_{c}$. Note also that we have
$L_{c+1,i}^{\ast}=L_{c+1,i}$ and $L_{c+1,j}^{\ast}=L_{c+1,j}$. Using
(4.13)–(4.14) to express each $1$-form $L_{e,k}$ in terms of $L_{c+1,i}$,
$L_{c+1,j}$, ${\rm dlog}t_{c}$, and ${\rm dlog}t_{c+1}$, we find

$\omega_{\beta,c}+\omega_{\beta,c+1}-\omega_{\beta^{\prime},c}^{\ast}-\omega_{\beta^{\prime},c+1}^{\ast}=(d_{j}a^{\prime}-d_{i}a){\rm
dlog}t_{c}\wedge{\rm dlog}t_{c+1}.$

Since $d_{j}a^{\prime}=d_{i}a$, we get that
$\omega_{\beta}=\omega_{\beta^{\prime}}^{\ast}$. The clusters
${\mathbf{x}}_{\beta}$ and ${\mathbf{x}}_{\beta^{\prime}}^{\ast}$ differ by a
relabeling $c\leftrightarrow c+1$.

Suppose now that one of $c,c+1$ is hollow. For instance, let $c\notin
J_{\beta}$ and $c+1\in J_{\beta}$. By Corollary 2.7, we have
$h^{+}_{c}=h^{+}_{c-1}$, and thus $L_{c,j}=L_{c-1,j}$. Similarly,
$L_{c,j}^{\ast}=L_{c+1,j}^{\ast}$. Recall that $L_{c\pm 1,j}^{\ast}=L_{c\pm
1,j}$. Thus, $\omega_{\beta,c+1}=\omega_{\beta^{\prime},c}^{\ast}$, and so
$\omega_{\beta}=\omega_{\beta^{\prime}}^{\ast}$. The case where $c\in
J_{\beta}$ and $c+1\notin J_{\beta}$ is similar. The clusters
${\mathbf{x}}_{\beta}$ and ${\mathbf{x}}_{\beta^{\prime}}^{\ast}$ differ by a
relabeling $c\leftrightarrow c+1$. For the case $c,c+1\notin J_{\beta}$, see
Remark 4.5. ∎

###  4.3. Non-mutation move: (B2)

Suppose that the move involves indices $c,c+1$. We have
$\omega_{\beta,c}=\omega_{\beta^{\prime},c+1}^{\ast}$ and
$\omega_{\beta,c+1}=\omega_{\beta^{\prime},c}^{\ast}$, so
$\omega_{\beta}=\omega_{\beta^{\prime}}^{\ast}$. The chamber minors satisfy

$\Delta_{c}=\Delta_{c+1}^{\ast}\quad\text{and}\quad\Delta_{c+1}=\Delta_{c}^{\ast}.$

Thus, the clusters ${\mathbf{x}}_{\beta}$ and
${\mathbf{x}}_{\beta^{\prime}}^{\ast}$ differ by a relabeling
$c\leftrightarrow c+1$. This shows LABEL:IS0 and LABEL:IS2.

###  4.4. Mutation move: (B3), solid, short

We proceed analogously to the case of solid-special (B1) in Section 4.1.
Suppose that the move $\beta\to\beta^{\prime}$, $iji\to jij$, involves indices
$c-1,c,c+1$, and that all three indices are solid. Suppose in addition that
$i,j\in I$; the case $i,j\in-I$ is similar.

######  Proposition 4.9 ([FZ99, Theorem 1.16(1)]).

We have

(4.15)
$\Delta_{c,i}\Delta_{c,j}^{\ast}=\Delta_{c+1,i}\Delta_{c-2,j}+\Delta_{c-2,i}\Delta_{c+1,j}.$

###### Proof.

We have $Z_{c-2}=Z_{c+1}z_{i}(t_{1})z_{j}(t_{2})z_{i}(t_{3})$ for some
$t_{1},t_{2},t_{3}\in{\mathbb{C}}$. We have the relation
$z_{i}(t_{1})z_{j}(t_{2})z_{i}(t_{3})=z_{j}(t_{3})z_{i}(t_{2}^{\prime})z_{j}(t_{1})$
for $t_{2}^{\prime}:=t_{1}t_{3}-t_{2}$, which can be checked inside
$\operatorname{SL}_{3}$. Thus, $Z_{c-1}=Z_{c+1}z_{i}(t_{1})z_{j}(t_{2})$,
$Z_{c}=Z_{c+1}z_{i}(t_{1})$, and $Z_{c}^{\ast}=Z_{c+1}z_{j}(t_{3})$. Let
$Z:=Z_{c-2}(\dot{s}_{i}\dot{s}_{j}\dot{s}_{i})^{-1}$. Let
$w:=w_{c-1}=w_{c}=w_{c+1}$. Similarly to the proof of Proposition 4.6, we
claim that each term in (4.15) equals the corresponding term in

(4.16)
$\Delta_{w\omega_{i},s_{i}\omega_{i}}(Z)\Delta_{w\omega_{j},s_{j}\omega_{j}}(Z)=\Delta_{w\omega_{i},\omega_{i}}(Z)\Delta_{w\omega_{j},s_{i}s_{j}\omega_{j}}(Z)+\Delta_{w\omega_{i},s_{j}s_{i}\omega_{i}}(Z)\Delta_{w\omega_{j},\omega_{j}}(Z).$

For example, since
$Z=Z_{c+1}z_{j}(t_{3})z_{i}(t_{2}^{\prime})z_{j}(t_{1})\cdot\dot{s}_{j}^{-1}\dot{s}_{i}^{-1}\dot{s}_{j}^{-1}$
and $z_{k}(t)=x_{k}(t)\dot{s}_{k}$, we have

$\Delta_{w\omega_{j},s_{j}\omega_{j}}(Z)=\Delta_{w\omega_{j},\omega_{j}}(Z_{c+1}x_{j}(t_{3})\dot{s}_{j}x_{i}(t_{2}^{\prime})\dot{s}_{i}x_{j}(t_{1})\dot{s}_{j}\cdot\dot{s}_{j}^{-1}\dot{s}_{i}^{-1})=\Delta_{w\omega_{j},\omega_{j}}(Z_{c+1}x_{j}(t_{3})\dot{s}_{j})=\Delta_{c,j}^{\ast},$

where the second equality holds by (2.10) because
$x_{i}(t_{2}^{\prime})\dot{s}_{i}x_{j}(t_{1})\dot{s}_{i}^{-1}\in U_{+}$.
Equation (4.16) holds by [FZ99, Theorem 1.16(1)]. ∎

###### Proof of LABEL:IS0 for (B3), solid, short.

Let ${\tilde{\beta}}=\beta w_{\circ}$ and
${\tilde{\beta}}^{\prime}=\beta^{\prime}w_{\circ}$. By definition, $\sum_{j\in
J_{{\tilde{\beta}}}}\omega_{j}({\tilde{\beta}})$ and
$\sum_{j\in[m]}\omega_{j}(\beta)$ are identical when expressed in terms of the
symbols $\Delta_{c,i}$. It is known ([SW21, Proposition 3.25] or [BFZ05]) that
Proposition 4.9 implies
$\omega_{{\tilde{\beta}}}=\omega_{{\tilde{\beta}}^{\prime}}$. Since the same
identity for grid minors in Proposition 4.9 holds on
$\accentset{\circ}{R}_{\beta}$, we deduce that
$\omega_{\beta}=\omega_{\beta^{\prime}}$. ∎

###### Proof of LABEL:IS2 for (B3), solid, short.

Let $x:=x_{c+1}$ and $V:=V_{c+1}$. By Propositions 2.21 and 2.22, $x$ is
mutable, ${\mathrm{ord}}_{V}\Delta_{c-1,i}={\mathrm{ord}}_{V}\Delta_{c,i}=1$,
and ${\mathrm{ord}}_{V}\Delta_{e,k}=0$ when $e\notin\\{c-1,c\\}$ or $k\in
I\setminus\\{i\\}$.

Note that $d_{i}=d_{j}$. Collecting the terms of
$\omega_{\beta,c-1}+\omega_{\beta,c}+\omega_{\beta,c+1}$ involving ${\rm
dlog}x$, we get

(4.17) $d_{i}{\rm
dlog}x\wedge\left(L_{c+1,i}-L_{c-2,i}+\frac{1}{2}(L_{c-1,j}-L_{c,j})\right).$

Applying (2.22) and using Corollary 2.11, we get

$\displaystyle L_{c+1,i}-L_{c-2,i}$ $\displaystyle={\rm
dlog}(\Delta_{c+1,i})-{\rm dlog}(\Delta_{c-2,i})+\frac{1}{2}{\rm
dlog}(\Delta_{c-2,j})-\frac{1}{2}{\rm dlog}(\Delta_{c+1,j});$ $\displaystyle
L_{c-1,j}-L_{c,j}$ $\displaystyle={\rm dlog}(\Delta_{c-1,j})-{\rm
dlog}\Delta_{c,j}={\rm dlog}(\Delta_{c-2,j})-{\rm dlog}\Delta_{c+1,j}.$

Thus, (4.17) becomes

$d_{i}{\rm dlog}x\wedge\left({\rm dlog}(\Delta_{c+1,i}\Delta_{c-2,j})-{\rm
dlog}(\Delta_{c-2,i}\Delta_{c+1,j})\right).$

The rest of the proof is entirely analogous to the argument for solid-special
(B1) given at the end of Section 4.1, using (4.15) in place of (4.2). ∎

###  4.5. Non-mutation move: (B3), non-solid, short

Suppose that at least one of the indices $c-1,c,c+1$ is hollow. By Remark 4.5,
we may assume that there are either one or two hollow indices in
$\\{c-1,c,c+1\\}$. Explicitly, underlining the hollow crossings, the possible
moves are $i\,\underline{j}\,i\leftrightarrow j\,i\,\underline{j}$ and
$i\,\underline{j}\,\underline{i}\leftrightarrow\underline{j}\,\underline{i}\,j$
(or the moves obtained from these by swapping the roles of $i$ and $j$).

For $l\in\\{i,j\\}$ and $e\in J_{\beta}$, let us denote

(4.18) $A_{l}:={\rm dlog}\prod_{k\neq i,j}\Delta_{c+1,k}^{a_{lk}},\quad
B_{l}:={\rm dlog}\Delta_{c+1,l},\quad\text{and}\quad T_{e}:={\rm dlog}t_{e}.$

Using (2.15)–(2.16), we can express the ${\rm dlog}$s of grid minors
$\Delta_{e,l}$ for $l\in\\{i,j\\}$ and $e\in\\{c-1,c,c+1\\}$ in the symbols
(4.18). For the case of the move $i\,\underline{j}\,i\to j\,i\,\underline{j}$,
this is done in Table 1. Since the move is short, we have $a_{ij}=a_{ji}=-1$,
and thus

(4.19) $L_{e,i}={\rm dlog}\Delta_{e,i}-\frac{1}{2}({\rm
dlog}\Delta_{e,j}+A_{i})\quad\text{and}\quad L_{e,j}={\rm
dlog}\Delta_{e,j}-\frac{1}{2}({\rm dlog}\Delta_{e,i}+A_{j}).$

Consider the case $i\,\underline{j}\,i\to j\,i\,\underline{j}$. Comparing the
columns labeled $c-2$ in Table 1(a,b), which must coincide by Definition 4.1,
we get $T_{c-1}^{\ast}=T_{c+1}$ and $T_{c}^{\ast}=T_{c+1}+T_{c-1}$. Combining
this with (4.19) allows us to express the forms $\omega_{\beta,e}$,
$\omega_{\beta^{\prime},e}^{\ast}$, $e\in\\{c-1,c,c+1\\}$ in terms of the
symbols (4.18). Using a straightforward computation, we check
$\omega_{\beta}=\omega_{\beta^{\prime}}^{\ast}$. The case of the move
$i\,\underline{j}\,\underline{i}\leftrightarrow\underline{j}\,\underline{i}\,j$
is similar. This finishes the proof of LABEL:IS0.

| $e$ | $c-2$ | $c-1$ | $c$ | $c+1$
---|---|---|---|---
${\rm dlog}\Delta_{e,j}$ | $T_{c+1}+A_{j}+B_{i}-B_{j}$ | $T_{c+1}+A_{j}+B_{i}-B_{j}$ | $B_{j}$ | $B_{j}$
${\rm dlog}\Delta_{e,i}$ | $T_{c+1}+T_{c-1}+B_{i}$ | $T_{c+1}+B_{i}$ | $T_{c+1}+B_{i}$ | $B_{i}$

(a) $i\,\underline{j}\,i$: $c-1\in J_{\beta}$, $c\notin J_{\beta}$, $c+1\in
J_{\beta}$
| $e$ | $c-2$ | $c-1$ | $c$ | $c+1$
---|---|---|---|---
${\rm dlog}\Delta_{e,j}^{\ast}$ | $T_{c-1}^{\ast}+A_{j}+B_{i}-B_{j}$ | $B_{i}+A_{j}-B_{j}$ | $B_{i}+A_{j}-B_{j}$ | $B_{j}$
${\rm dlog}\Delta_{e,i}^{\ast}$ | $T_{c}^{\ast}+B_{i}$ | $T_{c}^{\ast}+B_{i}$ | $B_{i}$ | $B_{i}$

(b) $j\,i\,\underline{j}$: $c-1\in J_{\beta^{\prime}}$, $c\in
J_{\beta^{\prime}}$, $c+1\notin J_{\beta^{\prime}}$
Table 1.  Computing the ${\rm dlog}$s of grid minors via (2.15)–(2.16) for the
move $i\,\underline{j}\,i\to j\,i\,\underline{j}$; cf. Section 4.5.

To show LABEL:IS2, consider again the case $i\,\underline{j}\,i\to
j\,i\,\underline{j}$. By Corollary 2.11, $\Delta_{c}^{\ast}=\Delta_{c-1}$, and
we see from Table 1 that the ratio $\Delta_{c-1}^{\ast}/\Delta_{c+1}$ is a
monomial in the cluster variables $x_{e}=x_{e}^{\ast}$ for $e>c+1$. It follows
that the clusters ${\mathbf{x}}_{\beta}$ and
${\mathbf{x}}_{\beta^{\prime}}^{\ast}$ differ by a relabeling
$x_{c}^{\ast}:=x_{c-1}$, $x_{c-1}^{\ast}=x_{c+1}$. The case of the move
$i\,\underline{j}\,\underline{i}\leftrightarrow\underline{j}\,\underline{i}\,j$
is similar. We are done with LABEL:IS2.

###  4.6. Non-mutation move: (B4)

Suppose that $i\in I$. The isomorphism
$\phi:\accentset{\circ}{R}_{\beta}\xrightarrow{\sim}\accentset{\circ}{R}_{\beta^{\prime}}$
sending
$(X_{\bullet},Y_{\bullet})\mapsto(X_{\bullet}^{\prime},Y_{\bullet}^{\prime})$
is given by $X^{\prime}_{m-1}=X^{\prime}_{m}=Y^{\prime}_{m}:=X_{m-1}$,
$Y^{\prime}_{m-1}:=Y_{m-1}$, and
$(X^{\prime}_{c},Y^{\prime}_{c}):=(X_{c},Y_{c})$ for all $0\leq c<m-1$. The
last crossing in $\beta_{0}i$ and $\beta_{0}(-i^{\ast})$ is always hollow, and
thus the statements LABEL:IS0 and LABEL:IS2 follow trivially.

###  4.7. Non-mutation move: (B5)

Suppose that $i\in I$. The isomorphism
$\phi:\accentset{\circ}{R}_{\beta}\xrightarrow{\sim}\accentset{\circ}{R}_{\beta^{\prime}}$
sending
$(X_{\bullet},Y_{\bullet})\mapsto(X_{\bullet}^{\prime},Y_{\bullet}^{\prime})$
is defined as follows. For $c\in[m]$, we set
$(X^{\prime}_{c},Y^{\prime}_{c}):=(X_{c},Y_{c})$. Suppose that $i\in I$; thus,
$Y_{0}=Y_{1}=Y_{1}^{\prime}$ and $X^{\prime}_{0}=X^{\prime}_{1}=X_{1}$. Recall
that $X_{0}\stackrel{{\scriptstyle w_{\circ}\ }}{{\Longrightarrow}}Y_{0}$. We
let $Y_{0}^{\prime}$ be the unique weighted flag satisfying
$X_{0}\stackrel{{\scriptstyle w_{\circ}s_{i^{\ast}}\
}}{{\Longrightarrow}}Y_{0}^{\prime}\stackrel{{\scriptstyle s_{i^{\ast}}\
}}{{\longrightarrow}}Y_{0}$. It follows that
$X^{\prime}_{0}\stackrel{{\scriptstyle\
w_{\circ}}}{{\Longleftarrow}}Y^{\prime}_{0}$ and
$Y^{\prime}_{0}\stackrel{{\scriptstyle s_{i^{\ast}}\
}}{{\longrightarrow}}Y^{\prime}_{1}$, so
$(X_{\bullet}^{\prime},Y_{\bullet}^{\prime})\in\accentset{\circ}{R}_{\beta^{\prime}}$.
The inverse map is defined similarly: $X_{0}$ may be recovered as the unique
weighted flag satisfying $X^{\prime}_{0}\stackrel{{\scriptstyle s_{i}\
}}{{\longrightarrow}}X_{0}\stackrel{{\scriptstyle s_{i}w_{\circ}\
}}{{\Longrightarrow}}Y^{\prime}_{0}$.

The statement LABEL:IS0 is trivial if the first crossing of $\beta$ is hollow.
If the first crossing of $\beta$ is solid, we have

$X_{0}^{\prime}=X_{1}^{\prime}=X_{1}\stackrel{{\scriptstyle s_{i}\
}}{{\longrightarrow}}X_{0}\stackrel{{\scriptstyle s_{i}w_{\circ}\
}}{{\Longrightarrow}}Y_{0}^{\prime}\stackrel{{\scriptstyle s_{i^{\ast}}\
}}{{\longrightarrow}}Y_{0}=Y_{1}=Y_{1}^{\prime}.$

It follows that after acting on all these flags by some $g\in G$, we can find
$t,t^{\prime}\in{\mathbb{C}}$ and $h\in H$ such that

$X_{0}^{\prime}=X_{1}^{\prime}=X_{1}=\dot{w}_{\circ}\dot{s}_{i}h\bar{z}_{i}(t)U_{+},\quad
X_{0}=\dot{w}_{\circ}\dot{s}_{i}hU_{+},\quad Y_{0}^{\prime}=U_{+},\quad
Y_{0}=Y_{1}=Y_{1}^{\prime}=z_{i^{\ast}}(t^{\prime})U_{+}.$

Here, we have $X_{0}\stackrel{{\scriptstyle s_{i}w_{\circ}\
}}{{\Longrightarrow}}Y_{0}^{\prime}$ and thus $X_{0}\stackrel{{\scriptstyle\
w_{\circ}s_{i}}}{{\Longleftarrow}}Y_{0}^{\prime}$, and we have used a
representative $\dot{w}_{\circ}\dot{s}_{i}$ of $w_{\circ}s_{i}$ in $N_{G}(H)$.
Let us denote $h_{0}:=h^{+}_{0}=h^{-}_{0}$ and
$h_{0}^{\ast}:=(h^{+}_{0})^{\ast}=(h^{-}_{0})^{\ast}$. We have
$Z_{0}=Y_{0}^{-1}X_{0}=z_{i^{\ast}}(t^{\prime})^{-1}\dot{w}_{\circ}\dot{s}_{i}h$,
and thus, proceeding as in the proof of Lemma 2.6, we get $h_{0}=h$.
Similarly,
$Z_{0}^{\prime}=(Y_{0}^{\prime})^{-1}X_{0}^{\prime}=\dot{w}_{\circ}\dot{s}_{i}h\bar{z}_{i}(t)$,
so $h_{0}^{\ast}=s_{i}\cdot h$.

We conclude that $h_{0}^{\ast}=s_{i}\cdot h_{0}$. Applying (4.11), we find

$L_{0,-i}^{\ast}={\rm dlog}(s_{i}\cdot h_{0})^{\alpha_{i}/2}={\rm
dlog}(h_{0})^{-\alpha_{i}/2}=-L_{0,i}.$

Applying (4.4) for $e=0,1$, we obtain $L_{0,i}=L_{0,-i}$ and
$L_{1,i}=L_{1,-i}$. Recall that $L_{1,i}=L_{1,i}^{\ast}$. Thus, we get
$\omega_{\beta,1}=\omega_{\beta^{\prime},1}^{\ast}$, and therefore
$\omega_{\beta}=\omega_{\beta^{\prime}}^{\ast}$, finishing the proof of
LABEL:IS0.

We now prove LABEL:IS2. Let $\beta=i\beta_{0}$ and
$\beta^{\prime}=(-i)\beta_{0}$. If the first crossing is hollow, the claim is
trivial. Suppose that the first crossing is solid. We have
$\Delta_{c,k}=\Delta_{c,k}^{\ast}$ for all $c\geq 1$ and $k\in\pm I$. Thus,
$x_{c}=x_{c}^{\ast}$ for all $c\in J_{\beta}$ such that $c>1$. Since
$h_{0}^{\ast}=s_{i}\cdot h_{0}$, Lemma 2.10 implies that
$x_{1}^{\ast}=x_{1}^{-1}M$, where $M$ is a Laurent monomial in the grid minors
$\Delta_{0,k}$ for $k\neq i$ of the same sign as $i$. It follows from
Propositions 2.21 and 2.22 that $M$ is a Laurent monomial in the frozen
variables other than $x_{1}$. This shows LABEL:IS2.

###  4.8. Proof of LABEL:IS3 for short braid moves (B1)–(B4)

Suppose that $\beta$ and $\beta^{\prime}$ are related by a short move
(B1)–(B4), and suppose that $(\accentset{\circ}{R}_{\beta},\Sigma_{\beta})$ is
a cluster variety. By LABEL:IS2, there is a seed $\Sigma^{\prime}$ which
differs from $\Sigma_{\beta}$ by mutation and possibly a relabeling, such that
the seeds $\Sigma^{\prime}$ and $\Sigma_{\beta^{\prime}}^{\ast}$ are quasi-
equivalent. We claim that these seeds are actually identical. To see that,
choose a long enough double braid word $\beta_{0}$ such that all cluster
variables of $\beta,\beta^{\prime}$ become mutable in
${\tilde{\beta}}:=\beta_{0}\beta$,
${\tilde{\beta}}^{\prime}:=\beta_{0}\beta^{\prime}$; cf. Lemma 2.26. Let
$\tilde{\Sigma}^{\prime}$ be obtained from $\Sigma_{{\tilde{\beta}}}$ using
the same mutations and relabeling as how $\Sigma^{\prime}$ was obtained from
$\Sigma_{\beta}$. By LABEL:IS2, $\tilde{\Sigma}^{\prime}$ is quasi-equivalent
to $\Sigma_{{\tilde{\beta}}^{\prime}}^{\ast}$. Since all frozen variables of
$\Sigma^{\prime}$ are mutable in $\tilde{\Sigma}^{\prime}$, it follows that
the seeds $\Sigma^{\prime}$ and $\Sigma_{\beta^{\prime}}^{\ast}$ are
identical. This finishes the proof of LABEL:IS3 and of Theorem 4.2. ∎

###  4.9. Deletion-contraction for double braid varieties

We apply the cluster algebraic results from Section 3.3 to the seeds
$\Sigma_{\beta}$.

######  Theorem 4.10.

Let $i\in I$ and consider a double braid word $\beta=ii\beta^{\prime}$ on
positive letters. If
$(\accentset{\circ}{R}_{i\beta^{\prime}},\Sigma_{i\beta^{\prime}}),(\accentset{\circ}{R}_{\beta^{\prime}},\Sigma_{\beta^{\prime}})$
are sink-recurrent cluster varieties, then
$(\accentset{\circ}{R}_{\beta},\Sigma_{\beta})$ is a sink-recurrent cluster
variety.

###### Proof.

Suppose first that at least one of the first two crossings in $\beta$ is
hollow, in which case $1$ must be solid and $2$ must be hollow. Consider an
arbitrary point $(X_{\bullet},Y_{\bullet})\in\accentset{\circ}{R}_{\beta}$.
Since the letters in $\beta$ are positive, we have
$Y_{0}=Y_{1}=\dots=Y_{m}=X_{m}$. Since $w_{2}\leq w_{\circ}s_{i}$ and
$w_{0}=w_{\circ}$, we must have $X_{1}\stackrel{{\scriptstyle\
w_{\circ}}}{{\Longleftarrow}}X_{m}$ and $X_{2}\stackrel{{\scriptstyle\
w_{\circ}s_{i}}}{{\Longleftarrow}}X_{m}$. It follows that $h^{\pm}_{1}$ and
$h^{\pm}_{2}$ are regular functions on $\accentset{\circ}{R}_{\beta}$. Choose
a representative $Z_{2}=\dot{w}_{\circ}h^{-}_{2}\dot{s}_{i}^{-1}$ as in (2.7),
and let $t,t^{\prime}\in{\mathbb{C}}$ be such that $Z_{1}=Z_{2}z_{i}(t)$ and
$Z_{0}=Z_{2}z_{i}(t)z_{i}(t^{\prime})$. Thus, $t,t^{\prime}$ are regular
functions on $\accentset{\circ}{R}_{\beta}$. Proceeding as in the proof of
Lemma 2.6, we find $h^{+}_{1}=h^{-}_{2}$ and
$h^{+}_{0}=h^{-}_{2}\alpha^{\vee}_{i}(t^{\prime})$, where
$h^{+}_{0},h^{+}_{1}$ are regular on $\accentset{\circ}{R}_{\beta}$. It
follows that $\Delta_{0,i}=t^{\prime}\Delta_{1,i}$. For any $e\in J_{\beta}$
such that $e>1$, the function $x_{e}$ depends on $Z_{2},Z_{3},\dots,Z_{m}$ but
does not depend on $t,t^{\prime}$. By Proposition 2.21, we have
$\Delta_{0,i}=x_{1}M$ for some monomial $M$ in $\\{x_{e}\\}_{e>1}$. The
Deodhar hypersurface $V_{1}$ is clearly given by the equation $t^{\prime}=0$.
We conclude that $x_{1}=t^{\prime}$. We have an isomorphism

(4.20)
$r:\accentset{\circ}{R}_{\beta}\xrightarrow{\sim}\accentset{\circ}{R}_{i\beta^{\prime}}\times{\mathbb{C}}^{\times},\quad(X_{\bullet},Y_{\bullet})\mapsto\left((X_{1},\dots,X_{m},Y_{1},\dots,Y_{m}),x_{1}\right).$

For the inverse map, we put $Z_{2}$ in the form
$Z_{2}=\dot{w}_{\circ}h^{-}_{2}\dot{s}_{i}^{-1}$, then recover $t$ from
$Z_{1}=Z_{2}z_{i}(t)$, and then set $Z_{0}:=Z_{2}z_{i}(t)z_{i}(t^{\prime})$,
where $t^{\prime}:=x_{1}$. To summarize, we have an isomorphism
$r:\accentset{\circ}{R}_{\beta}\xrightarrow{\sim}\accentset{\circ}{R}_{i\beta^{\prime}}\times{\mathbb{C}}^{\times}$
where the second coordinate is given by the frozen variable $x_{1}$. Moreover,
since $\Delta_{1,i}=M$ involves only frozen variables, we see that $1$ is
connected to only frozen indices in $\tilde{\Gamma}(\Sigma_{\beta})$. It
follows that the principal parts of $\Sigma_{\beta}$ and
$\Sigma_{i\beta^{\prime}}$ agree, and therefore
$(\accentset{\circ}{R}_{\beta},\Sigma_{\beta})$ is a sink-recurrent cluster
variety. Moreover, if $\Sigma_{i\beta^{\prime}}$ is really full rank then so
is $\Sigma_{\beta}$.

Suppose now that the first two crossings are both solid. Our goal is to apply
Theorem 3.13. First, we show that $\Sigma_{\beta}$ is sink-recurrent. Let
$\Gamma:=\Gamma(\Sigma_{\beta})$. The seed $\Sigma_{\beta}^{\backslash 2}$ is
obtained from $\Sigma_{i\beta^{\prime}}$ by adding an isolated frozen variable
$x_{1}$, and
$\Sigma_{\beta}^{\backslash\widehat{N}^{\operatorname{in}}_{s}(\Gamma)}$ is
obtained from $\Sigma_{\beta^{\prime}}$ by adding isolated frozen variables
$x_{1}$ and $x_{2}$, so both of these seeds are sink-recurrent. Further, in
$\Sigma_{\beta}$, the variable $x_{1}$ is frozen, ${\tilde{B}}_{12}=-1$, and
${\tilde{B}}_{1c}=0$ for mutable $c>2$. Next, by Corollary 2.11, the sum of
terms of $\omega_{\beta}$ involving $x_{2}$ is clearly of the form $d_{i}{\rm
dlog}x_{2}\wedge{\rm dlog}M$ for a Laurent monomial $M$ in ${\mathbf{x}}$, and
thus $\Sigma_{\beta}$ is integral at $2$. We have shown that $\Sigma_{\beta}$
satisfies 3.11.

Next, we show that the mutated cluster variable $x_{2}^{\prime}$ is regular on
$\accentset{\circ}{R}_{\beta}$. We apply the moves
$ii\beta^{\prime}\xrightarrow{\ref{bm5}}(-i)i\beta^{\prime}\xrightarrow{\ref{bm1}}i(-i)\beta^{\prime}$.
Denote $\dot{\mathbf{x}}:={\mathbf{x}}_{(-i)i\beta^{\prime}}$ and
$\ddot{\mathbf{x}}:={\mathbf{x}}_{i(-i)\beta^{\prime}}$. It follows from the
argument in Section 4.1 that $\dot{x}_{2}$ is mutable in
$\Sigma_{(-i)i\beta^{\prime}}$, and by (4.9), its mutation
$\dot{x}_{2}^{\prime}$ is regular on
$\accentset{\circ}{R}_{(-i)i\beta^{\prime}}$, as it equals the pullback
$\ddot{x}_{2}^{\ast}$ times a monomial in the other cluster variables in
$\dot{\mathbf{x}}$ with nonnegative exponents. As explained in Section 4.7,
the seeds $\Sigma_{\beta}$ and $\Sigma_{(-i)i\beta^{\prime}}$ are quasi-
equivalent. By Lemma 3.9, we find that the mutation $x_{2}^{\prime}$ differs
from $\dot{x}_{2}^{\prime}$ by a unit (cf. part (2) of Corollary 2.25), and
thus $x_{2}^{\prime}$ is regular on $\accentset{\circ}{R}_{\beta}$.

Let $W:=\\{x_{2}\neq 0\\}$ $V:=V_{2}=\\{x_{2}=0\\}$ be the open-closed
covering of $\accentset{\circ}{R}_{\beta}$ coming from $x_{2}$. Our final goal
is to construct isomorphisms

$W\cong\accentset{\circ}{R}_{i\beta^{\prime}}\times{\mathbb{C}}^{\times}\cong\mathcal{X}(\Sigma_{\beta}^{\backslash
2})\quad\text{and}\quad
V\cong\operatorname{Spec}({\mathbb{C}}[x_{2}^{\prime}])\times\accentset{\circ}{R}_{\beta^{\prime}}\cong{\mathbb{C}}\times\mathcal{X}(\Sigma_{\beta}^{/2})$

satisfying the conditions of Theorem 3.13. By Proposition 2.22, for $e\in
J_{\beta}$, $e>2$, we have ${\mathrm{ord}}_{V_{e}}\Delta_{1,i}=0$ if and only
if ${\mathrm{ord}}_{V_{e}}\Delta_{0,i}=0$. Moreover, the same proposition
implies ${\mathrm{ord}}_{V_{2}}\Delta_{0,i}=0$. It follows by Proposition 2.21
that $\Delta_{1,i}$ is equal to $x_{2}$ times a monomial in the frozen
variables, and that $\Delta_{0,i}$ is equal to $x_{1}$ times a monomial in the
same set of frozen variables. Since $W$ is the complement of $V_{2}$, we see
that $(X_{\bullet},Y_{\bullet})\in W$ if and only if
$X_{1}\stackrel{{\scriptstyle\ w_{\circ}}}{{\Longleftarrow}}Y_{1}=X_{m}$.
Thus, $h^{+}_{1}$ is a regular function on $W$. We choose a representative
$Z_{1}=\dot{w}_{\circ}h^{+}_{1}$ and let $t\in{\mathbb{C}}$ be such that
$Z_{0}=Z_{1}z_{i}(t^{\prime})$. Then we get
$t^{\prime}=\Delta_{0,i}/\Delta_{1,i}=Mx_{1}/x_{2}$, where $M$ is a Laurent
monomial in the frozen variables other than $x_{1}$. Similarly to (4.20), we
let $r:W\to\accentset{\circ}{R}_{i\beta^{\prime}}\times{\mathbb{C}}^{\times}$
be the map sending $(X_{\bullet},Y_{\bullet})$ to
$\left((X_{1},\dots,X_{m},Y_{1},\dots,Y_{m}),Mx_{1}/x_{2}\right)$. By
assumption, we have
$\accentset{\circ}{R}_{i\beta^{\prime}}\cong\mathcal{X}(\Sigma_{i\beta^{\prime}})$.
The frozen index $1$ is only connected to other frozen indices in
$\tilde{\Gamma}(\Sigma_{\beta})$. Thus, the seed $\Sigma_{\beta}^{\backslash
2}$ is obtained from $\Sigma_{i\beta^{\prime}}$ by adding an isolated frozen
vertex, and therefore $\mathcal{X}(\Sigma_{\beta}^{\backslash
2})\cong\mathcal{X}(\Sigma_{i\beta^{\prime}})\times{\mathbb{C}}$. Adjusting
the isolated frozen variable by a Laurent monomial in the other frozen
variables, we see that the pullbacks of $x_{1},\dots,x_{{n+m}}$ under the
inclusion $\mathcal{X}(\Sigma_{\beta}^{\backslash 2})\cong W\hookrightarrow X$
are indeed the same-named cluster variables in $\Sigma_{\beta}^{\backslash
2}$. This verifies assumption (1) of Theorem 3.13.

Now suppose that $(X_{\bullet},Y_{\bullet})\in V$. We have
$X_{0}\stackrel{{\scriptstyle\ w_{\circ}}}{{\Longleftarrow}}X_{m}$ but not
$X_{1}\stackrel{{\scriptstyle\ w_{\circ}}}{{\Longleftarrow}}X_{m}$, so we must
have $X_{1}\stackrel{{\scriptstyle\ w_{\circ}s_{i}}}{{\Longleftarrow}}X_{m}$,
and therefore $X_{2}\stackrel{{\scriptstyle\
w_{\circ}}}{{\Longleftarrow}}X_{m}$. Consider a map
$r:V\to\accentset{\circ}{R}_{\beta^{\prime}}\times{\mathbb{C}}$ sending
$(X_{\bullet},Y_{\bullet})$ to
$\left((X_{2},\dots,X_{m},Y_{2},\dots,Y_{m}),x_{2}^{\prime}\right)$. We claim
that this map is an isomorphism. To construct an inverse, we need to show how
to recover $X_{0},X_{1},Y_{0},Y_{1}$ from the image of $r$. We have
$Y_{0}=Y_{1}=X_{m}$. Also, $X_{1}$ is uniquely determined by $Y_{1}=Y_{2}$ and
$X_{2}$, since $Y_{1}\stackrel{{\scriptstyle\
s_{i}w_{\circ}}}{{\Longleftarrow}}X_{1}\stackrel{{\scriptstyle\
s_{i}}}{{\longleftarrow}}X_{2}$. It remains to recover $X_{0}$. Note also that
we can recover the cluster variables $x_{e}$, $e>2$, as well as the mutated
cluster variable $x^{\prime}_{2}$, from the image of $r$. Since
$(X_{\bullet},Y_{\bullet})\in V$, the frozen variable $x_{1}$ is also
recovered from the exchange relation $0=x_{2}x_{2}^{\prime}=M_{1}+x_{1}M_{2}$
for $x_{2}$.

In order to recover $X_{1}$, we apply moves
$ii\beta^{\prime}\xrightarrow{\ref{bm5}}(-i)i\beta^{\prime}\xrightarrow{\ref{bm1}}i(-i)\beta^{\prime}$
as we did above. Let $(\dot{X}_{\bullet},\dot{Y}_{\bullet})$ and
$(\ddot{X}_{\bullet},\ddot{Y}_{\bullet})$ denote the images of
$(X_{\bullet},Y_{\bullet})$ in $\accentset{\circ}{R}_{(-i)i\beta^{\prime}}$
and $\accentset{\circ}{R}_{i(-i)\beta^{\prime}}$, respectively, under the
isomorphism $\phi$. As in Section 4.7, let $Y_{0}^{\prime}$ be the unique
weighted flag satisfying $X_{0}\stackrel{{\scriptstyle w_{\circ}s_{i^{\ast}}\
}}{{\Longrightarrow}}Y_{0}^{\prime}\stackrel{{\scriptstyle s_{i^{\ast}}\
}}{{\longrightarrow}}Y_{0}$. We have $\dot{X}_{2}=\ddot{X}_{2}=X_{2}$ and
$\dot{Y}_{2}=\ddot{Y}_{2}=Y_{2}$. Further, we have
$(\dot{X}_{1},\dot{X}_{0},\dot{Y}_{0},\dot{Y}_{1})=(X_{1},X_{1},Y_{0}^{\prime},Y_{0})$
and
$(\ddot{X}_{1},\ddot{X}_{0},\ddot{Y}_{0},\ddot{Y}_{1})=(X_{2},X_{1},Y_{0}^{\prime},Y_{0}^{\prime})$.
Let $\dot{\mathbf{x}}:={\mathbf{x}}_{(-i)i\beta^{\prime}}$ and
$\ddot{\mathbf{x}}:={\mathbf{x}}_{i(-i)\beta^{\prime}}$. We claim that
$Y^{\prime}_{0}$ is uniquely determined by $X_{2}$, $Y_{2}$, and
$\ddot{\mathbf{x}}$. Indeed, if $\ddot{x}_{2}=0$ then $Y_{0}^{\prime}$ is
uniquely determined by $Y_{2}\stackrel{{\scriptstyle\
s_{i^{\ast}}}}{{\longleftarrow}}Y_{0}^{\prime}\stackrel{{\scriptstyle\
w_{\circ}s_{i^{\ast}}}}{{\Longleftarrow}}X_{2}$. Otherwise, we have
$X_{2}\stackrel{{\scriptstyle\ w_{\circ}}}{{\Longleftarrow}}Y_{0}^{\prime}$,
and the $U_{+}\times U_{+}$-double coset
$\ddot{Z}_{1}:=(Y_{0}^{\prime})^{-1}X_{2}$ is uniquely determined by the
values of the characters in $\ddot{\mathbf{x}}$. From $\ddot{Z}_{1}$ and
$X_{2}$, we recover $Y_{0}^{\prime}$. The weighted flag $X_{0}$ is then
uniquely determined by $X_{1}\stackrel{{\scriptstyle s_{i}\
}}{{\longrightarrow}}X_{0}\stackrel{{\scriptstyle s_{i}w_{\circ}\
}}{{\Longrightarrow}}Y_{0}^{\prime}$. It suffices to show that
$\ddot{\mathbf{x}}$ is uniquely determined by the image of $r$. For $e\in
J_{\beta}$, $e>2$, we have $\ddot{x}_{e}=x_{e}$. Moreover,
$\ddot{x}_{1}=M/x_{1}$ for some monomial $M$ in the frozen variables $x_{e}$
other than $x_{1}$ (all of which must satisfy $e>2$ since $x_{2}$ is mutable).
Finally, by (4.9), $\ddot{x}_{2}$ differs from $x_{2}^{\prime}$ by a monomial
in the cluster variables other than $x_{2}$. We are done with verifying
assumption (2) of Theorem 3.13.

We have verified all conditions in 3.11 and Theorem 3.13. Thus,
$(\accentset{\circ}{R}_{\beta},\Sigma_{\beta})$ is a cluster variety. We have
already shown that it is sink-recurrent. ∎

###  4.10. Proof of Theorem 1.1 for $G$ simply-laced

We proceed by induction on the number $m$ of indices in $\beta$. Recall that
we always assume $\pi(\beta)=w_{\circ}$. The base case is $m=\ell(w_{\circ})$,
where all indices are hollow. Thus, the cluster algebra is
${\mathcal{A}}_{\beta}={\mathbb{C}}$. We claim that the double braid variety
$\accentset{\circ}{R}_{\beta}$ is a single point. Indeed, since the letters of
$\beta$ form a reduced word for $w_{\circ}$, a point
$(X_{\bullet},Y_{\bullet})\in\accentset{\circ}{R}_{\beta}$ must satisfy
$X_{0}\stackrel{{\scriptstyle\ w_{\circ}}}{{\longleftarrow}}Y_{0}$ (as opposed
to $X_{0}\stackrel{{\scriptstyle\ w_{\circ}}}{{\Longleftarrow}}Y_{0}$), and
moreover, all other flags $X_{c},Y_{c}$, $c\in[m]$, are uniquely determined by
$X_{0},Y_{0}$. Acting by $G$, we can set $Y_{0}=U_{+}$ and
$X_{0}=\dot{w}_{\circ}hU_{+}$ for $h\in H$, and in order to have
$X_{0}\stackrel{{\scriptstyle\ w_{\circ}}}{{\longleftarrow}}Y_{0}$, we must
have $h=1$. This finishes the proof for the base case.

Suppose now that $m>\ell(w_{\circ})$. Applying (B1) and (B4), we can assume
that all letters of $\beta$ belong to $I$. Since $G$ is simply-laced, all
braid moves are automatically short. Applying (B2)–(B3), we may therefore
transform $\beta$ into a braid word of the form $\beta_{1}ii\beta_{2}$ for
some braid words $\beta_{1},\beta_{2}$ and $i\in I$. We can also apply
_conjugation moves_ to $\beta$: if $\beta=j\beta_{0}$, the conjugation move
consists of the moves

(4.21)
$\beta=j\beta_{0}\xrightarrow{\text{\ref{bm5}}}(-j)\beta_{0}\xrightarrow{\text{\ref{bm1}}}\cdots\xrightarrow{\text{\ref{bm1}}}\beta_{0}(-j)\xrightarrow{\text{\ref{bm4}}}\beta_{0}j^{\ast}.$

Applying conjugation moves, we may further transform $\beta$ into the form
$\beta^{\prime}:=ii\beta_{2}\beta_{1}^{\ast}$, where $\beta_{1}^{\ast}$ is
obtained from $\beta_{1}$ by applying the map $j\mapsto j^{\ast}$ to each
letter. Applying Theorem 4.10 to $\beta^{\prime}$, we find that
$(\accentset{\circ}{R}_{\beta^{\prime}},\Sigma_{\beta^{\prime}})$ is a cluster
variety. It follows from Theorem 4.2 (for short moves) and Corollary 3.10 that
$(\accentset{\circ}{R}_{\beta},\Sigma_{\beta})$ is therefore also a cluster
variety. ∎

######  Remark 4.11.

It follows that the seed $\Sigma_{\beta}$ is really full rank when $G$ is
simply-laced. Indeed, this property is clearly preserved under moves
(B1)–(B5), and the proof of Theorem 4.10 together with Remark 3.15 implies
that if $\beta=ii\beta^{\prime}$ and $\Sigma_{i\beta^{\prime}}$ is really full
rank then so is $\Sigma_{\beta}$. Proceeding by induction as in the above
proof of Theorem 1.1, we see that all seeds $\Sigma_{\beta}$ are really full
rank when $G$ is simply-laced.

##  5\. Folding

We review some background on folding before completing the proof in Section 6.

###  5.1. Pinnings

Let $G$ be a simply connected, complex, semisimple algebraic group with
pinning $(H,B_{+},B_{-},x_{i},y_{i};i\in I)$. Then there exists an algebraic
group $\dot{G}$ of simply-laced type with pinning
$(\dot{H},\dot{B}_{+},\dot{B}_{-},\dot{x}_{i^{\prime}},\dot{y}_{i^{\prime}};{i^{\prime}}\in\dot{I})$;
see [Lus94, §1.6]. We have a bijection $\sigma:\dot{I}\to\dot{I}$ which
extends to an automorphism $\sigma:\dot{G}\to\dot{G}$, and a map
$\iota:G\to\dot{G}$ which yields algebraic group isomorphisms

$\iota:G\xrightarrow{\sim}\dot{G}^{\sigma},\quad
H\xrightarrow{\sim}\dot{H}^{\sigma},\quad
B_{\pm}\xrightarrow{\sim}(\dot{B}_{\pm})^{\sigma},\quad
U_{\pm}\xrightarrow{\sim}(\dot{U}_{\pm})^{\sigma}.$

The maps $gB_{+}\mapsto\iota(g)\dot{B}_{+}$ and
$gU_{+}\mapsto\iota(g)\dot{U}_{+}$ induce isomorphisms of varieties:

(5.1)
$\iota:G/B_{+}\xrightarrow{\sim}(\dot{G}/\dot{B}_{+})^{\sigma}\quad\text{and}\quad
G/U_{+}\xrightarrow{\sim}(\dot{G}/\dot{U}_{+})^{\sigma}.$

For the first isomorphism, see [Lus94, §8.8]. The surjectivity and injectivity
of the second map follow from that of the first by a straightforward
computation.

For an element $i\in I$, we denote by $\bm{i}\subset\dot{I}$ the associated
$\sigma$-orbit, i.e., the orbit under the cyclic group generated by $\sigma$.
We also let
$\bm{-i}:=\\{-{i^{\prime}}\mid{i^{\prime}}\in\bm{i}\\}\subset-\dot{I}$. We let
$\\{\dot{\alpha}_{{i^{\prime}}}\mid{i^{\prime}}\in\dot{I}\\}$,
$\\{\dot{\alpha}^{\vee}_{{i^{\prime}}}\mid{i^{\prime}}\in\dot{I}\\}$, and
$\\{\dot{\omega}_{i^{\prime}}\mid{i^{\prime}}\in\dot{I}\\}$ be the simple
roots, simple coroots, and fundamental weights of the root system of
$\dot{G}$. Letting
$\dot{a}_{{i^{\prime}}{j^{\prime}}}:={\langle}\dot{\alpha}_{i^{\prime}},\dot{\alpha}^{\vee}_{j^{\prime}}{\rangle}$
be the entries of the associated Cartan matrix (and setting
$\dot{a}_{(-{i^{\prime}})(-{j^{\prime}})}:=\dot{a}_{{i^{\prime}}{j^{\prime}}}$
and $\dot{a}_{(\pm{i^{\prime}})(\mp{j^{\prime}})}:=0$ as before), we have

(5.2) $d_{i}=|\bm{i}|\quad\text{and}\quad
a_{ij}=\sum_{{j^{\prime}}\in\bm{j}}\dot{a}_{{i^{\prime}}{j^{\prime}}}\quad\text{for
all $i,j\in\pm I$ and ${i^{\prime}}\in\bm{i}$.}$

The Coxeter generators of the Weyl group $\dot{W}$ of $\dot{G}$ are denoted by
$\\{{\tilde{s}}_{i^{\prime}}\mid{i^{\prime}}\in\dot{I}\\}$. Restricting
$\iota$ to the normalizer of $H$, we get a group isomorphism

(5.3) $\iota:W\xrightarrow{\sim}\dot{W}^{\sigma},\quad
s_{i}\mapsto\prod_{{i^{\prime}}\in\bm{i}}{\tilde{s}}_{i^{\prime}}.$

Here the order inside $\bm{i}$ is immaterial since the corresponding elements
${\tilde{s}}_{i^{\prime}}$ commute. It follows that the longest element
$w_{\circ}\in W$ gets mapped under (5.3) to the longest element
${\tilde{w}}_{\circ}$ of $\dot{W}$, because $\sigma:\dot{W}\to\dot{W}$
preserves Coxeter length and therefore
${\tilde{w}}_{\circ}\in\dot{W}^{\sigma}$. The following result is immediate.

######  Lemma 5.1.

Let $B_{1},B_{2}\in G/U_{+}$. If $B_{1}\xrightarrow{w}B_{2}$ then
$\iota(B_{1})\xrightarrow{\iota(w)}\iota(B_{2})$. If
$B_{1}\stackrel{{\scriptstyle w\ }}{{\Longrightarrow}}B_{2}$ then
$\iota(B_{1})\stackrel{{\scriptstyle\iota(w)\
}}{{\Longrightarrow}}\iota(B_{2})$. In particular, if
$B_{1}\stackrel{{\scriptstyle w_{\circ}\ }}{{\Longrightarrow}}B_{2}$ then
$\iota(B_{1})\stackrel{{\scriptstyle{\tilde{w}}_{\circ}\
}}{{\Longrightarrow}}\iota(B_{2})$.

###  5.2. Braid varieties

Let $\beta=i_{1}i_{2}\dots i_{m}\in(\pm I)^{m}$ be a double braid word. Let
${\tilde{\beta}}={i^{\prime}_{1}}{i^{\prime}_{2}}\dots
i^{\prime}_{\tilde{m}}\in(\pm\dot{I})^{\tilde{m}}$ be obtained by
concatenating the letters in $\bm{i_{1}},\bm{i_{2}},\dots,\bm{i_{m}}$
(choosing the order inside each $\bm{i_{c}}$ arbitrarily), where
$\tilde{m}:=|\bm{i_{1}}|+|\bm{i_{2}}|+\dots+|\bm{i_{m}}|$. We let
${\lambda}_{\beta}:[\tilde{m}]\to[m]$ denote the unique order-preserving map
satisfying $|{\lambda}_{\beta}^{-1}(c)|=|\bm{i_{c}}|$ for all $c\in[m]$. It is
clear that an index $c\in[m]$ is solid (resp., hollow) if and only if all
indices in ${\lambda}_{\beta}^{-1}(c)$ are solid (resp., hollow). In other
words, the set $\dot{J}_{{\tilde{\beta}}}$ of solid crossings for
${\tilde{\beta}}$ is given by

(5.4) $\dot{J}_{{\tilde{\beta}}}={\lambda}_{\beta}^{-1}(J_{\beta}).$

Let $\accentset{\circ}{\mathcal{Y}}^{\prime}_{\beta}$ be the variety of tuples
$(\dot{X}_{\bullet},\dot{Y}_{\bullet})$ of weighted flags in
$\dot{G}/\dot{U}_{+}$ satisfying

${\dot{X}_{0}}$${\dot{X}_{1}}$${\cdots}$${\dot{X}_{m}}$${\dot{Y}_{0}}$${\dot{Y}_{1}}$${\cdots}$${\dot{Y}_{m}.}$$\scriptstyle{\iota(s_{i_{1}}^{+})}$$\scriptstyle{\iota(s_{i_{2}}^{+})}$$\scriptstyle{\iota(s_{i_{m}}^{+})}$$\scriptstyle{\iota(s_{i_{1}^{\ast}}^{-})}$$\scriptstyle{{\tilde{w}}_{\circ}}$$\scriptstyle{\iota(s_{i_{2}^{\ast}}^{-})}$$\scriptstyle{\iota(s_{i_{m}^{\ast}}^{-})}$

Let $\mathcal{Y}^{\prime}_{\beta}$ be obtained by omitting the condition
$\dot{X}_{0}\stackrel{{\scriptstyle\
{\tilde{w}}_{\circ}}}{{\Longleftarrow}}\dot{Y}_{0}$. Lemma 2.1 yields
isomorphisms
$\accentset{\circ}{\mathcal{Y}}^{\prime}_{\beta}\cong\accentset{\circ}{\mathcal{Y}}_{\tilde{\beta}}$
and $\mathcal{Y}^{\prime}_{\beta}\cong\mathcal{Y}_{\tilde{\beta}}$. Let
$\accentset{\circ}{R}^{\prime}_{\beta}$ be the quotient of
$\accentset{\circ}{\mathcal{Y}}^{\prime}_{\beta}$ by the free
$\dot{G}$-action. Then
$\accentset{\circ}{R}^{\prime}_{\beta}\cong\accentset{\circ}{R}_{\tilde{\beta}}$.

The map $\sigma$ acts on the varieties
$\accentset{\circ}{\mathcal{Y}}^{\prime}_{\beta}$,
$\mathcal{Y}^{\prime}_{\beta}$, and $\accentset{\circ}{R}^{\prime}_{\beta}$
termwise by acting on each $\dot{X}_{c}$ and $\dot{Y}_{c}$. Let
$T^{\prime}_{\beta}\subset\accentset{\circ}{R}^{\prime}_{\beta}$ be the image
of the Deodhar torus
$T_{{\tilde{\beta}}}\subset\accentset{\circ}{R}_{\tilde{\beta}}$ under the
isomorphism
$\accentset{\circ}{R}_{\tilde{\beta}}\cong\accentset{\circ}{R}^{\prime}_{\beta}$.
We have the following straightforward result.

######  Proposition 5.2.

Applying $\iota$ termwise yields isomorphisms

(5.5)
$\accentset{\circ}{\mathcal{Y}}_{\beta}\xrightarrow{\sim}(\accentset{\circ}{\mathcal{Y}}^{\prime}_{\beta})^{\sigma},\quad\mathcal{Y}_{\beta}\xrightarrow{\sim}(\mathcal{Y}^{\prime}_{\beta})^{\sigma},\quad\accentset{\circ}{R}_{\beta}\xrightarrow{\sim}(\accentset{\circ}{R}^{\prime}_{\beta})^{\sigma},\quad\text{and}\quad
T_{\beta}\xrightarrow{\sim}(T^{\prime}_{\beta})^{\sigma}.$

###  5.3. Grid minors

Recall that we have the character and cocharacter lattices
$X^{\ast}(H):=\operatorname{Hom}(H,{\mathbb{C}}^{\times})$,
$X_{\ast}(H):=\operatorname{Hom}({\mathbb{C}}^{\times},H)$. The map
$\iota:H\to\dot{H}$ induces a map $\iota_{*}:X_{\ast}(H)\to X_{\ast}(\dot{H})$
sending
$\alpha^{\vee}_{i}\mapsto\sum_{{i^{\prime}}\in\bm{i}}\dot{\alpha}^{\vee}_{{i^{\prime}}}$
for $i\in I$, so that
$\iota(\alpha^{\vee}_{i}(t))=\prod_{{i^{\prime}}\in\bm{i}}\dot{\alpha}^{\vee}_{{i^{\prime}}}(t)$
for $t\in{\mathbb{C}}^{\times}$. It also induces a map
$\iota^{\ast}:X^{\ast}(\dot{H})\to X^{\ast}(H)$ sending
$\dot{\omega}_{i^{\prime}}\mapsto\omega_{i}$ for all $i\in I$ and
${i^{\prime}}\in\bm{i}$, so that
$\iota(h)^{\dot{\omega}_{i^{\prime}}}=h^{\omega_{i}}$ for $h\in H$. It follows
that for all $g\in G$, $v,w\in W$, $i\in I$, and ${i^{\prime}}\in\bm{i}$, we
have

(5.6)
$\Delta_{v\omega_{i},w\omega_{i}}(g)=\Delta_{\iota(v)\dot{\omega}_{i^{\prime}},\iota(w)\dot{\omega}_{i^{\prime}}}(\iota(g)).$

Let $(X_{\bullet},Y_{\bullet})\in\accentset{\circ}{R}^{\prime}_{\beta}$. As
usual, for $c=0,1,\dots,m$, we denote $Z_{c}:=Y_{c}^{-1}X_{c}$. Let
$\tilde{u}_{c}:=\iota(u_{c})$ and $\tilde{w}_{c}:=\iota(w_{c})$. For
${i^{\prime}}\in\dot{I}$, consider analogs of grid minors for
$\accentset{\circ}{R}^{\prime}_{\beta}$:

(5.7)
$\Delta^{\prime}_{c,{i^{\prime}}}(X_{\bullet},Y_{\bullet})=\Delta_{\tilde{w}_{c}\dot{\omega}_{i^{\prime}},\dot{\omega}_{i^{\prime}}}(Z_{c}),\qquad\Delta^{\prime}_{c,-{i^{\prime}}}(X_{\bullet},Y_{\bullet})=\Delta_{{\tilde{w}}_{\circ}\dot{\omega}_{i^{\prime}},\tilde{u}_{c}^{-1}\dot{\omega}_{i^{\prime}}}(Z_{c}).$

Comparing (5.6)–(5.7) to Definition 2.9, we find that the grid minors on
$\accentset{\circ}{R}_{\beta}$ are pullbacks of the minors defined in (5.7):
for $c=0,1,\dots,m$, $i\in\pm I$, and ${i^{\prime}}\in\bm{i}$, we have

(5.8) $\iota^{\ast}\Delta^{\prime}_{c,{i^{\prime}}}=\Delta_{c,i}.$

Using Corollary 2.11, we obtain the following description of chamber minors on
$\accentset{\circ}{R}_{\tilde{\beta}}$, which we denote by
$\tilde{\Delta}_{{c^{\prime}}}$, ${c^{\prime}}\in\dot{J}_{{\tilde{\beta}}}$;
cf. (5.4).

######  Lemma 5.3.

Let ${c^{\prime}}\in\dot{J}_{{\tilde{\beta}}}$ be a solid crossing for
${\tilde{\beta}}$. Set ${i^{\prime}}:={i^{\prime}_{{c^{\prime}}}}$ and
$c:={\lambda}_{\beta}({c^{\prime}})\in J$. Then the isomorphism
$\accentset{\circ}{R}_{\tilde{\beta}}\cong\accentset{\circ}{R}^{\prime}_{\beta}$
sends the chamber minor $\tilde{\Delta}_{{c^{\prime}}}$ to
$\Delta^{\prime}_{c-1,{i^{\prime}}}$, and we have
$\iota^{\ast}\Delta^{\prime}_{c-1,{i^{\prime}}}=\Delta_{c}$.

###  5.4. $2$-form

Our next goal is to show the following result.

######  Lemma 5.4.

Let $\omega^{\prime}_{\beta}$ be the pullback of the $2$-form
$\omega_{{\tilde{\beta}}}$ on $\accentset{\circ}{R}_{{\tilde{\beta}}}$ under
the isomorphism
$\accentset{\circ}{R}^{\prime}_{\beta}\cong\accentset{\circ}{R}_{\tilde{\beta}}$.
We have

$\iota^{\ast}\omega^{\prime}_{\beta}=\omega_{\beta}.$

###### Proof.

Recall from (2.22)–(2.23) that we have $1$-forms
$L_{c,i}=\frac{1}{2}\sum_{k\in\pm I}a_{ik}{\rm dlog}\Delta_{c,k}$ on
$\accentset{\circ}{R}_{\beta}$ for $(c,i)\in[m]\times(\pm I)$, and that for
$c\in[m]$ and $i:=i_{c}$, we set $\omega_{c}(\beta):={\rm
sign}(i)d_{i}L_{c-1,i}\wedge L_{c,i}$. For ${i^{\prime}}\in\pm\dot{I}$,
introduce a $1$-form
$L^{\prime}_{c,{i^{\prime}}}:=\frac{1}{2}\sum_{{j^{\prime}}\in\pm\dot{I}}\dot{a}_{{i^{\prime}}{j^{\prime}}}{\rm
dlog}\Delta^{\prime}_{c,{j^{\prime}}}$. By Corollary 2.11, we have

(5.9) $\omega^{\prime}_{\beta}=\sum_{c\in J_{\beta}}{\rm
sign}(i_{c})\sum_{{i^{\prime}}\in\bm{i_{c}}}L^{\prime}_{c-1,{i^{\prime}}}\wedge
L^{\prime}_{c,{i^{\prime}}}.$

Next, applying (5.8) and (5.2), we see that for all $c\in[m]$, $i\in\pm I$,
and ${i^{\prime}}\in\bm{i}$, we have

(5.10)
$\iota^{\ast}L^{\prime}_{c,{i^{\prime}}}=\iota^{\ast}\left(\frac{1}{2}\sum_{{j^{\prime}}\in\pm\dot{I}}\dot{a}_{{i^{\prime}}{j^{\prime}}}{\rm
dlog}\Delta^{\prime}_{c,{j^{\prime}}}\right)=\frac{1}{2}\sum_{j\in\pm
I}\left(\sum_{{j^{\prime}}\in\bm{j}}\dot{a}_{{i^{\prime}}{j^{\prime}}}\right){\rm
dlog}\Delta_{c,j}=L_{c,i}.$

The result follows by combining (5.9)–(5.10) with (5.2). ∎

###  5.5. Folding seeds

We briefly review the notion of folding seeds, following [FWZ16, Section 4.4],
though translating into our conventions.

######  Definition 5.5.

Let $\dot{\Sigma}=(\dot{T},\dot{x},\dot{\mathbf{d}},\dot{\omega})$ be a seed
with $\dot{\mathbf{d}}=(1,\dots,1)$, with mutable indices $\dot{J}^{{\rm
mut}}$ and frozen indices $\dot{J}^{{\rm fro}}$. Let $\sigma$ be a bijection
acting on $\dot{J}:=\dot{J}^{{\rm mut}}\sqcup\dot{J}^{{\rm fro}}$. Let $J$ be
the set of $\sigma$-orbits, and for $j\in J$, we denote the corresponding
orbit by $\bm{j}$. An orbit is _mutable_ (resp., _frozen_) if it consists
entirely of mutable (resp., frozen) indices. The bijection $\sigma$ also acts
on the set of cluster variables by
$\sigma(\dot{x}_{{j^{\prime}}})=\dot{x}_{\sigma({j^{\prime}})}$. We call
$\dot{\Sigma}$ _weakly $\sigma$-admissible_222Our notion of weak
$\sigma$-admissibility differs from the notion of admissibility in [FWZ16,
Definition 4.4.1] in that we do not require that for any
$a^{\prime},a^{\prime\prime}$ in the same orbit and any $k^{\prime}$ mutable,
$\dot{\tilde{B}}_{a^{\prime}k^{\prime}}{\tilde{B}}_{a^{\prime\prime}k^{\prime}}\geq
0$. if:

  1. (1)

Every orbit is either mutable or frozen.

  2. (2)

The 2-form $\omega$ is invariant under the $\sigma$-action.

  3. (3)

For all $a^{\prime},a^{\prime\prime}\in\dot{J}^{{\rm mut}}$ in the same
$\sigma$-orbit, $\dot{\tilde{B}}_{a^{\prime}a^{\prime\prime}}=0$, where
$\dot{\tilde{B}}$ is the exchange matrix of $\dot{\Sigma}$.

Part (1) implies a natural decomposition $J=J^{{\rm mut}}\sqcup J^{{\rm
fro}}$. The map $\sigma$ also acts on the torus $\dot{T}$ by permuting
coordinates. Notice that $T^{\sigma}$ is isomorphic to
$({\mathbb{C}}^{*})^{|J|}$. We denote by $\iota:T^{\sigma}\hookrightarrow T$
the inclusion map.

######  Definition 5.6.

Suppose $\dot{\Sigma}$ is weakly $\sigma$-admissible, with notation as in
Definition 5.5. The _folded seed_ is a seed with index set $J=J^{{\rm
mut}}\sqcup J^{{\rm fro}}$, defined as
$\iota^{\ast}\dot{\Sigma}=\Sigma:=(T,{\mathbf{x}},\mathbf{d},\omega)$ where

  * •

$T=T^{\sigma}$;

  * •

${\mathbf{x}}=(x_{j})_{j\in J}$, where for $j\in J$,
$x_{j}:=\iota^{*}\dot{x}_{j^{\prime}}$ for any ${j^{\prime}}\in\bm{j}$;

  * •

$d_{j}=|\bm{j}|$ for $j\in J$;

  * •

$\omega=\iota^{*}\dot{\omega}.$

Note that $x_{j}$ is well-defined, since
$\iota^{*}\dot{x}_{j^{\prime}}=\iota^{*}\dot{x}_{\sigma({j^{\prime}})}$ for
all ${j^{\prime}}\in\bm{j}$. The exchange matrix ${\tilde{B}}$ of $\Sigma$ is
therefore written in terms of the exchange matrix $\dot{\tilde{B}}$ of
$\dot{\Sigma}$ as

(5.11)
${\tilde{B}}_{ab}=\sum_{a^{\prime}\in\bm{a}}\dot{\tilde{B}}_{a^{\prime}b^{\prime}},\quad\text{where
$b^{\prime}\in\bm{b}$ is arbitrary}.$

In particular, if $\dot{\Sigma}$ is integral then so is $\Sigma$. For the rest
of this subsection, we assume that $\dot{\Sigma}$ and $\Sigma$ are integral.

For weakly $\sigma$-admissible $\dot{\Sigma}$ and $j\in J^{{\rm mut}}$, we
denote by $\mu_{\bm{j}}\dot{\Sigma}=\mu_{\bm{j}}(\dot{\Sigma})$ the result of
mutating $\dot{\Sigma}$ once at each index in $\bm{j}$, and call
$\mu_{\bm{j}}$ an _orbit-mutation_. Note that $\mu_{\bm{j}}\dot{\Sigma}$ does
not depend on the order of mutation. The seed $\mu_{\bm{j}}\dot{\Sigma}$ may
not be weakly $\sigma$-admissible; we introduce the following notion to avoid
such mutations.

######  Definition 5.7.

Let $\dot{\Sigma}$ be a weakly $\sigma$-admissible seed and $j\in J^{{\rm
mut}}$. We call $\mu_{\bm{j}}$ _quasi-admissible_ if for all $k\in J^{{\rm
mut}}$, we have
${\tilde{B}}_{k^{\prime}{j^{\prime}}}{\tilde{B}}_{k^{\prime\prime}{j^{\prime}}}\geq
0$ for all $k^{\prime},k^{\prime\prime}\in\bm{k}$ and ${j^{\prime}}\in\bm{j}$.

The name “quasi-admissible” is justified by the following proposition.

######  Proposition 5.8.

Let $\dot{\Sigma}$ be a weakly $\sigma$-admissible seed and $j\in J^{{\rm
mut}}$. If $\mu_{\bm{j}}$ is quasi-admissible, then
$\mu_{\bm{j}}(\dot{\Sigma})$ is weakly $\sigma$-admissible and

$\mu_{j}(\iota^{\ast}\dot{\Sigma})\sim\iota_{j}^{\ast}(\mu_{\bm{j}}\dot{\Sigma}),$

where $\iota_{j}$ is an inclusion of the associated tori.

###### Proof.

We use the notation of Definitions 5.5 and 5.6. In particular, let
$\Sigma:=\iota^{\ast}\dot{\Sigma}$ be the folded seed on index set $J$. Since
the exchange matrix $\dot{\tilde{B}}$ of $\dot{\Sigma}$ is skew-symmetric, it
is equivalent to a quiver $\dot{Q}$; we will use the two interchangeably.

It is clear that $\mu_{\bm{j}}\dot{\Sigma}$ satisfies condition (1) of
Definition 5.5. Because there are no arrows between vertices in $\bm{j}$,
mutating at all vertices of $\bm{j}$ shows that the exchange matrix of
$\mu_{\bm{j}}\dot{\Sigma}$ satisfies
$(\mu_{\bm{j}}\dot{\tilde{B}})_{a^{\prime}b^{\prime}}=(\mu_{\bm{j}}\dot{\tilde{B}})_{\sigma(a^{\prime})\sigma(b^{\prime})}$
for $a^{\prime},b^{\prime}\in\dot{J}$. The assumption that $\mu_{\bm{j}}$ is
quasi-admissible implies that for $k\in J^{{\rm mut}}$,
$(\mu_{\bm{j}}\dot{\tilde{B}})_{k^{\prime}k^{\prime\prime}}=0$ for all
$k^{\prime},k^{\prime\prime}\in\bm{k}$. Thus, $\mu_{\bm{j}}\dot{\Sigma}$ is
weakly $\sigma$-admissible.

Let $\Sigma_{1}:=\mu_{j}(\Sigma)$ and
$\Sigma_{2}:=\iota^{\ast}_{j}(\mu_{\bm{j}}\dot{\Sigma})$. Let ${\mathbf{y}}$
and $\mathbf{z}$ be the clusters in $\Sigma_{1}$ and $\Sigma_{2}$,
respectively. For $k\neq j$, $y_{k}=z_{k}$ because both are equal to the
cluster variable $x_{k}$ of $\Sigma$.

To analyze the relationship between $y_{j}$ and $z_{j}$, we need the following
notions. Let $a,k\in J$ and choose $a^{\prime}\in\bm{a}$,
$k^{\prime}\in\bm{k}$. We call a path $a^{\prime}\to k^{\prime}\to
a^{\prime\prime}$ in $\dot{Q}$ a _bad path_ if $a^{\prime},a^{\prime\prime}$
are in the same orbit; condition (2) of Definition 5.7 implies that no bad
path in $\dot{Q}$ begins or ends in a mutable orbit. Let $P_{k^{\prime}}$ be a
maximal (by inclusion) collection of arrow-disjoint bad paths with middle
vertex $k^{\prime}$.

In $\dot{\Sigma}$, for ${j^{\prime}}\in\bm{j}$, the mutation
$\dot{x}_{j^{\prime}}^{\prime}$ of $\dot{x}_{j^{\prime}}$ is defined by the
exchange relation

$\dot{x}_{j^{\prime}}\dot{x}_{j^{\prime}}^{\prime}=M^{\prime}N^{\prime}+M^{\prime\prime}N^{\prime\prime},\quad\text{where}$
$N^{\prime}:=\prod_{(a^{\prime}\to{j^{\prime}}\to a^{\prime\prime})\in
P_{j^{\prime}}}\dot{x}_{a^{\prime}}\qquad\text{and}\qquad
N^{\prime\prime}:=\prod_{(a^{\prime}\to{j^{\prime}}\to a^{\prime\prime})\in