1\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}][\gamma^{*}\otimes
1](\Omega_{0}).\end{split}$ (3.51)

Applying
$[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}]$,
we finally obtain (3.40). ∎

###  3.4 Proof of Proposition 1.4

Let us restate Proposition 1.4 for convenience:

######  Proposition.

If the $r_{0}$ parts of $\pi_{\text{std}}^{\dagger}$ and
$\pi_{\mathbf{\Gamma}}^{\dagger}$ coincide, then the rational map
$\mathcal{Q}:(G,\pi_{\text{std}}^{\dagger})\dashrightarrow(G,\pi_{\mathbf{\Gamma}}^{\dagger})$
is Poisson.

###### Proof.

We apply the pushforward $\mathcal{Q}_{*}$ to each term of (3.11) and combine
the output in the form (3.10). The strategy is to collect the terms (3.20),
(3.23), (3.36) and (3.40), and then apply either (3.6) or (3.7) to cancel out
$\operatorname{Ad}_{\rho(U)^{-1}}$ from the resulting expressions. Let us
notice first that all the terms that end with $[\gamma^{*}\otimes
1](\Omega_{0})$ cancel each other out (this includes a part of the last term
in (3.23)). As for the rest, let us indicate the correspondence between the
terms in the formulas (3.20), (3.23), (3.36) and (3.40), and the terms in
(3.10).

  1. 1)

To obtain the term
$[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}]\left[\frac{\gamma^{*}}{1-\gamma^{*}}\otimes
1\right](\Omega_{>})$, one combines the last two terms of (3.23) with the
$[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}]$
part of the penultimate term in (3.40), and then one applies (3.7);

  2. 2)

To obtain the term $-[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes
1]\left[\frac{1}{1-\gamma^{*}}\otimes 1\right](\Omega_{>})$, one combines the
first term from (3.23), the last term from (3.36) and the
$[\operatorname{Ad}^{\circ}_{\mathcal{Q}(U)}\otimes 1]$ parts from the first
and the penultimate terms of (3.40); then, one applies (3.7);

  3. 3)

To obtain the term
$-[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}]\left[1\otimes\frac{1}{1-\gamma^{*}}\right](\Omega_{<})$,
one combines the
$[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}]$
parts of the first two terms in (3.20), the first two terms in (3.40) and the
fourth term in (3.36); then, one applies (3.6);

  4. 4)

To obtain the term $[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes
1]\left[1\otimes\frac{\gamma^{*}}{1-\gamma^{*}}\right](\Omega_{<})$, one
combines the fifth term from (3.36) together with the
$\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes 1$ parts of the second term
of (3.20) and the second term of (3.40); then, one applies (3.6).

Thus the proposition is proved. ∎

###  3.5 An alternative to $\mathcal{Q}$

In this subsection, we define an alternative version of $\mathcal{Q}$ for the
_$g$ -convention_ (see Section 7). Continuing in the setup of the current
section, let us decompose a generic element $U\in G$ into the product
$U=U_{+}U_{\ominus}$ where $U_{+}\in\mathcal{N}_{+}$ and
$U_{\ominus}\in\mathcal{B}_{-}$. Let
$\mathbf{\Gamma}^{\mathrm{op}}:=(\Gamma_{2},\Gamma_{1},\gamma^{*})$ be the
opposite BD triple. Define the rational map
$\mathcal{Q}^{\mathrm{op}}:(G,\pi_{\text{std}}^{\dagger})\dashrightarrow(G,\pi_{\mathbf{\Gamma}}^{\dagger})$
via

$\mathcal{Q}^{\mathrm{op}}(U):=\rho^{\mathrm{op}}(U)U\rho^{\mathrm{op}}(U)^{-1},\
\ \rho^{\mathrm{op}}(U):=\prod_{i=1}^{\leftarrow}\tilde{\gamma}(U_{+}),\ U\in
G.$ (3.52)

Define a map $T_{*}:\mathfrak{g}\rightarrow\mathfrak{g}$ via setting
$T_{*}|_{\mathfrak{h}}=\operatorname{id}|_{\mathfrak{h}}$,
$T_{*}(e_{-\alpha})=e_{\alpha}$, $T_{*}(e_{\alpha})=e_{-\alpha}$, where
$\langle e_{\alpha},e_{-\alpha}\rangle=1$, as in Section 2.3. Then $T_{*}$
integrates to the _transposition map_ $T:G\rightarrow G$.

######  Lemma 3.14.

Pick any $r_{0}$ element for $\pi_{\mathbf{\Gamma}}^{\dagger}$ and set
$r_{0}^{t}$ to be the $r_{0}$ component for
$\pi_{\mathbf{\Gamma}^{\mathrm{op}}}^{\dagger}$. Then the transposition map
$T:(G,\pi_{\mathbf{\Gamma}}^{\dagger})\rightarrow(G,-\pi_{\mathbf{\Gamma}^{\mathrm{op}}}^{\dagger})$
is a Poisson isomorphism.

###### Proof.

It follows from formula (2.5) that the classical $r$-matrix
$r_{\mathbf{\Gamma}}$ can be expressed as

$r_{\mathbf{\Gamma}}=r_{0}+\left[1\otimes\frac{1}{1-\gamma}\right](\Omega_{>})-\left[\frac{\gamma}{1-\gamma}\otimes
1\right](\Omega_{<}).$ (3.53)

We see that

$[T_{*}\otimes
T_{*}](r_{\mathbf{\Gamma}})=r_{0}+\left[1\otimes\frac{1}{1-\gamma}\right](\Omega_{<})-\left[\frac{\gamma}{1-\gamma}\otimes
1\right](\Omega_{>})=(r_{\mathbf{\Gamma}^{\mathrm{op}}})^{t}.$ (3.54)

Also, $d_{U}^{R}T(\operatorname{Ad}_{U}x)=T_{*}(x)$ and
$d_{U}^{R}T(\operatorname{Ad}_{U}^{\circ}x)=-\operatorname{Ad}_{T(U)}^{\circ}T_{*}(x)$,
$x\in\mathfrak{g}$. Now, applying $[d_{U}^{R}T\otimes d_{U}^{R}T]$ to formula
(3.4), we conclude that $[d_{U}^{R}T\otimes
d_{U}^{R}T]((\pi_{\mathbf{\Gamma}}^{\dagger,R})_{U})=-(\pi_{\mathbf{\Gamma}^{\mathrm{op}}}^{\dagger,R})_{T(U)}$.
∎

######  Proposition 3.15.

If the $r_{0}$ parts of $\pi_{\mathbf{\Gamma}}^{\dagger}$ and
$\pi_{\text{std}}^{\dagger}$ are the same, then the rational map
$\mathcal{Q}^{\mathrm{op}}:(G,\pi_{\text{std}}^{\dagger})\dashrightarrow(G,\pi_{\mathbf{\Gamma}}^{\dagger})$
is Poisson.

###### Proof.

Indeed, let
$\mathcal{Q}:(G,\pi_{\text{std}}^{\dagger})\rightarrow(G,\pi_{\mathbf{\Gamma}^{\mathrm{op}}}^{\dagger})$
be defined relative the opposite BD triple $\mathbf{\Gamma}^{\mathrm{op}}$.
Choose an arbitrary $r_{0}$ for $\pi_{\mathbf{\Gamma}}^{\dagger}$ and set
$r_{0}^{t}$ to be the similar component for
$\pi_{\mathbf{\Gamma}^{\mathrm{op}}}^{\dagger}$. Now notice that the diagram

$\textstyle{(G,\pi_{\text{std}}^{\dagger})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{Q}^{\mathrm{op}}}$$\scriptstyle{T}$$\textstyle{(G,\pi_{\mathbf{\Gamma}}^{\dagger})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T}$$\textstyle{(G,-\pi_{\text{std}}^{\dagger})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{Q}}$$\textstyle{(G,-\pi_{\mathbf{\Gamma}^{\mathrm{op}}}^{\dagger})}$
(3.55)

is commutative. Thus $\mathcal{Q}^{\mathrm{op}}$ is a Poisson isomorphism. ∎

##  4 Birational quasi-isomorphisms

In this section, we describe various birational quasi-isomorphisms between the
generalized cluster structures in the $h$-convention in type $A_{n-1}$ (see
the description in Section 1.2). More specifically, for a BD triple
$\mathbf{\Gamma}:=(\Gamma_{1},\Gamma_{2},\gamma)$ of type $A_{n-1}$ and
$G\in\\{\operatorname{SL}_{n},\operatorname{GL}_{n}\\}$, we describe the
following maps:

  * •

The birational map
$\mathcal{F}:(G,\pi_{\mathbf{\Gamma}}^{\dagger})\dashrightarrow(G,\pi_{\text{std}}^{\dagger})$,
which is a basic ingredient for defining the $h$-variables and birational
quasi-isomorphisms;

  * •

The birational quasi-isomorphism
$\mathcal{Q}:(G,\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma}_{\text{std}}))\dashrightarrow(G,\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma}))$
and its inverse $\mathcal{F}^{c}:=\mathcal{Q}^{-1}$. The results of Section 3
show that
$\mathcal{Q}:(\operatorname{GL}_{n},\pi_{\text{std}}^{\dagger})\dashrightarrow(\operatorname{GL}_{n},\pi_{\mathbf{\Gamma}}^{\dagger})$
is a Poisson isomorphism if the $r_{0}$ parts of $\pi^{\dagger}_{\text{std}}$
and $\pi^{\dagger}_{\mathbf{\Gamma}}$ are the same;

  * •

For any $\tilde{\mathbf{\Gamma}}\prec\mathbf{\Gamma}$, we construct a
birational quasi-isomorphism
$\mathcal{G}:(G,\mathcal{GC}_{h}^{\dagger}(\tilde{\mathbf{\Gamma}}))\dashrightarrow(G,\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma}))$;
as a map
$\mathcal{G}:(G,\pi_{\tilde{\mathbf{\Gamma}}}^{\dagger})\dashrightarrow(G,\pi_{\mathbf{\Gamma}}^{\dagger})$,
it is a Poisson isomorphism if the $r_{0}$ parts of
$\pi_{\tilde{\mathbf{\Gamma}}}$ and $\pi_{\mathbf{\Gamma}}$ are the same. If
$\tilde{\mathbf{\Gamma}}$ is obtained from $\mathbf{\Gamma}$ via removal of a
pair of roots, we provide an even more explicit description of $\mathcal{G}$
in Section 4.5.

If $\tilde{\mathbf{\Gamma}}\prec\mathbf{\Gamma}$ are BD triples of type
$A_{n-1}$, we will decorate the $h$-variables in
$\mathcal{GC}^{\dagger}_{h}(\tilde{\mathbf{\Gamma}})$ with a tilde, to
distinguish them from the $h$-variables in
$\mathcal{GC}^{\dagger}_{h}(\mathbf{\Gamma})$. The marker (see Definition
2.20) will always be chosen in such a way that each $\varphi_{ij}$ and $c_{i}$
are related to themselves, and each $h_{ij}$ is related to $\tilde{h}_{ij}$.
Then the marked variables in $\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma})$ are
given by

$\\{h_{i+1,i+1}\ |\ i\in\Gamma_{2}\setminus\tilde{\Gamma}_{2}\\}.$

###  4.1 Belavin-Drinfeld data for $\operatorname{gl}_{n}(\mathbb{C})$

In this section, we introduce various notions and formulas pertaining to
Belavin-Drinfeld triples.

######  Definition 4.1.

Let $(\Gamma_{1},\Gamma_{2},\gamma)$ be a Belavin-Drinfeld triple. The _degree
of nilpotency_ of $\gamma$ is defined as the number

$\deg\gamma:=\max_{\alpha\in\Gamma_{1}}\\{m\in\mathbb{Z}_{+}\ |\
\gamma^{m}(\alpha)\notin\Gamma_{1}\\}.$ (4.1)

######  Definition 4.2.

Let $\mathbf{\Gamma}:=(\Gamma_{1},\Gamma_{2},\gamma)$ and
$\tilde{\mathbf{\Gamma}}:=(\tilde{\Gamma}_{1},\tilde{\Gamma}_{2},\theta)$ be
two Belavin-Drinfeld triples. We write
$\tilde{\mathbf{\Gamma}}\prec\mathbf{\Gamma}$ if
$\tilde{\Gamma}_{1}\subset\Gamma_{1}$, $\tilde{\Gamma}_{2}\subset\Gamma_{2}$
and $\gamma|_{\tilde{\Gamma}_{1}}=\theta$, and we say that
$\tilde{\mathbf{\Gamma}}$ is obtained from $\mathbf{\Gamma}$ via removal of
the roots $\Gamma_{1}\setminus\tilde{\Gamma}_{1}$ from $\Gamma_{1}$ and
$\Gamma_{2}\setminus\tilde{\Gamma}_{2}$ from $\Gamma_{2}$.

##### Runs and orientability of $\gamma$.

Let us fix a set of simple roots $\Pi$ of type $A_{n-1}$ and identify it with
the interval $[1,n-1]$. Let $\mathbf{\Gamma}:=(\Gamma_{1},\Gamma_{2},\gamma)$
be a BD triple of type $A_{n-1}$. For an arbitrary $i\in\Pi$, set

$i_{-}:=\max\\{j\in[0,n]\setminus\Gamma_{1}\ |\ j<i\\},\ \
i_{+}:=\min\\{j\in[1,n]\setminus\Gamma_{1}\ |\ j\geq i\\}.$ (4.2)

An _$X$ -run_ of $i$ is the interval $\Delta(i):=[i_{-}+1,i_{+}]$. Replacing
$\Gamma_{1}$ with $\Gamma_{2}$ in the above formulas, we obtain the definition
of a _$Y$ -run_ $\bar{\Delta}(i)$ of $i$. The $X$-runs partition the set
$[1,n]$, and likewise the $Y$-runs. A run is called _trivial_ if it consists
of a single element. Evidently, the map $\gamma$ can be viewed as a bijection
between the set of nontrivial $X$-runs and the set of nontrivial $Y$-runs. We
say that a $Y$-run $\bar{\Delta}$ _corresponds_ to an $X$-run $\Delta$ if
$\gamma(i)=j$ for some $i\in\Delta\cap\Gamma_{1}$ and some
$j\in\bar{\Delta}\cap\Gamma_{2}$. The notion of runs was introduced in [23].

######  Definition 4.3.

Let $\mathbf{\Gamma}=(\Gamma_{1},\Gamma_{2},\gamma)$ be a BD triple of type
$A_{n-1}$, and let $\Delta$ be a nontrivial $X$-run. We say that $\gamma$ is
_positively oriented on $\Delta$_ if $\gamma(i+1)=\gamma(i)+1$ for each
$i,i+1\in\Delta\cap\Gamma_{1}$. Likewise, if $\gamma(i+1)=\gamma(i)-1$ for
each $i,i+1\in\Delta\cap\Gamma_{1}$, then $\gamma$ is called _negatively
oriented on $\Delta$_. The former circumstance will be denoted as
$\gamma|_{\Delta\cap\Gamma_{1}}>0$, while the latter as131313If an $X$-run
$\Delta$ consists of $2$ elements, we will view $\gamma$ as being positively
oriented on $\Delta$. However, the distinction is not necessary in this case.
$\gamma|_{\Delta\cap\Gamma_{1}}<0$.

######  Remark 4.4.

In [15, 23], a BD triple was called _oriented_ if
$\gamma|_{\Delta\cap\Gamma_{1}}>0$ for every nontrivial $X$-run $\Delta$; it
was called _nonoriented_ if there exists a nontrivial $X$-run $\Delta$ such
that $\gamma_{\Delta\cap\Gamma_{1}}<0$.

##### The map of indices $\bar{\gamma}$.

Let us set $\bar{\Gamma}_{1}$ to be the union of all the nontrivial
$X$-runs141414Alternatively, $\bar{\Gamma}_{1}$ is the union of $\Gamma_{1}$
and all the right endpoints of all the nontrivial $X$-runs. Similarly for
$\bar{\Gamma}_{2}$.  and $\bar{\Gamma}_{2}$ to be the union of all the
nontrivial $Y$-runs. We define a map
$\bar{\gamma}:\bar{\Gamma}_{1}\rightarrow\bar{\Gamma}_{2}$ in the following
way: for each nontrivial $X$-run $\Delta$ and the corresponding $Y$-run
$\bar{\Delta}$,

  1. a)

If $\gamma|_{\Delta\cap\Gamma_{1}}>0$, then the restriction
$\bar{\gamma}|_{\Delta}:\Delta\rightarrow\bar{\Delta}$ is set to be the unique
increasing bijection;

  2. b)

If $\gamma|_{\Delta\cap\Gamma_{1}}<0$, then the restriction
$\bar{\gamma}|_{\Delta}:\Delta\rightarrow\bar{\Delta}$ is set to be the unique
decreasing bijection151515Note that if $\gamma|_{\Delta\cap\Gamma_{1}}>0$,
then $\bar{\gamma}|_{\Delta\cap\Gamma_{1}}=\gamma|_{\Delta\cap\Gamma_{1}}$;
however, if $\gamma|_{\Delta\cap\Gamma_{1}}<0$, then
$\gamma(i)+1=\bar{\gamma}(i)$, $i\in\Delta\cap\Gamma_{1}$..

A similar recipe applies to $\bar{\gamma}^{*}$.

##### The map of subsets $\check{\gamma}$.

Let $\Delta$ be any nontrivial $X$-run and $\bar{\Delta}$ be the corresponding
$Y$-run. We define the map $\check{\gamma}:2^{\Delta}\rightarrow 2^{\Delta}$
in the following way. For any $A=\\{a_{i}\\}_{i=1}^{k}\subseteq\Delta$,

  1. a)

If $\gamma|_{\Delta\cap\Gamma_{1}}>0$, then
$\check{\gamma}(A):=\\{\bar{\gamma}(a_{i})\\}_{i=1}^{k}$;

  2. b)

If $\gamma|_{\Delta\cap\Gamma_{1}}<0$, then
$\check{\gamma}(A):=\bar{\Delta}\setminus\\{\bar{\gamma}(a_{i})\\}_{i=1}^{k}$.

##### The extension
$\mathring{\gamma}:\operatorname{gl}_{n}(\mathbb{C})\rightarrow\operatorname{gl}_{n}(\mathbb{C})$.

As was shown in [23], in the case of $\operatorname{GL}_{n}$, it is expedient
to extend the BD map $\gamma:\Gamma_{1}\rightarrow\Gamma_{2}$ in a different
way161616In our previous work [31], we denoted the resulting map by $\gamma$
as well, for the main focus of the paper was the case of
$\operatorname{GL}_{n}$. In this paper, we keep the original notation
introduced in [23].. Let $\Delta_{1},\ldots,\Delta_{k}$ be the list of all
nontrivial $X$-runs, and set $\bar{\Delta}_{1},\ldots,\bar{\Delta}_{k}$ to be
the list of the corresponding $Y$-runs (in other words,
$\gamma(\Delta_{i}\cap\Gamma_{1})=\bar{\Delta}_{i}\cap\Gamma_{2}$,
$i\in[1,k]$). Set $\operatorname{gl}(\Delta_{i})$ to be a subalgebra of
$\operatorname{gl}_{n}(\mathbb{C})$ of the matrices that are zero outside of
the block $\Delta_{i}\times\Delta_{i}$ (and similarly for
$\operatorname{gl}(\bar{\Delta}_{i})$). For each $i\in[1,k]$ define a map
$\mathring{\gamma}_{i}:\operatorname{gl}(\Delta_{i})\rightarrow\operatorname{gl}(\bar{\Delta}_{i})$
in the following way: first, canonically identify
$\operatorname{gl}(\bar{\Delta}_{i})$ and $\operatorname{gl}(\Delta_{i})$ with
$\operatorname{gl}_{|\Delta_{i}|}(\mathbb{C})$; second,

  1. a)

If $\gamma|_{\Delta_{i}\cap\Gamma_{1}}>0$, then
$\mathring{\gamma}_{i}:\operatorname{gl}(\Delta_{i})\rightarrow\operatorname{gl}(\bar{\Delta}_{i})$
acts as the identity map under the identifications
$\operatorname{gl}(\Delta_{i})\cong\operatorname{gl}_{|\Delta_{i}|}(\mathbb{C})$
and
$\operatorname{gl}(\bar{\Delta}_{i})\cong\operatorname{gl}_{|\Delta_{i}|}(\mathbb{C})$;
in other words,
$\mathring{\gamma}_{i}(e_{ks})=e_{\bar{\gamma}(k)\bar{\gamma}(s)}$,
$k,s\in\Delta_{i}$;

  2. b)

If $\gamma|_{\Delta_{i}\cap\Gamma_{1}}<0$, then let us set
$\omega_{0}:=\sum_{j=1}^{|\Delta_{i}|}(-1)^{j-1}e_{|\Delta_{i}|-j+1,j}$; under
the identifications, if $A\in\operatorname{gl}_{|\Delta_{i}|}(\mathbb{C})$,
then
$\mathring{\gamma}_{i}:\operatorname{gl}(\Delta_{i})\rightarrow\operatorname{gl}(\bar{\Delta}_{i})$
acts as

$\mathring{\gamma}_{i}(A)=\omega_{0}(-A)^{T}\omega_{0}^{-1}$ (4.3)

where $A^{T}$ is the transpose of $A$. In other words,
$\mathring{\gamma}_{i}(e_{ks})=(-1)^{s+k-1}e_{\bar{\gamma}(s)\bar{\gamma}(k)}$,
$k,s\in\Delta_{i}$.

The map
$\mathring{\gamma}:\operatorname{gl}_{n}(\mathbb{C})\rightarrow\operatorname{gl}_{n}(\mathbb{C})$
is defined as the direct sum
$\mathring{\gamma}:=\bigoplus_{i=1}^{k}\mathring{\gamma}_{i}$ extended by zero
to $\operatorname{gl}_{n}(\mathbb{C})$. Similarly, one defines the pieces
$\mathring{\gamma}_{i}^{*}:\operatorname{gl}_{n}(\bar{\Delta}_{i})\rightarrow\operatorname{gl}_{n}(\Delta_{i})$,
and then the map
$\mathring{\gamma}^{*}:\operatorname{gl}_{n}(\mathbb{C})\rightarrow\operatorname{gl}_{n}(\mathbb{C})$
is obtained as the direct sum
$\mathring{\gamma}^{*}:=\bigoplus_{i=1}^{k}\mathring{\gamma}_{i}^{*}$ extended
by zero to $\operatorname{gl}_{n}(\mathbb{C})$.

##### The group homomorphism
${\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}$.

In the case of $\operatorname{GL}_{n}$, the restrictions of
$\mathring{\gamma}$ to the Borel subalgebras $\mathfrak{b}_{+}$ and
$\mathfrak{b}_{-}$ of $\operatorname{gl}_{n}(\mathbb{C})$ are Lie algebra
homomorphisms. Therefore, one can integrate $\mathring{\gamma}$ in each case
to a group homomorphism
${\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}:\mathcal{B}_{\pm}\rightarrow\mathcal{B}_{\pm}$,
where $\mathcal{B}_{+}$ and $\mathcal{B}_{-}$ are the subgroups of upper and
lower triangular matrices. Likewise, one integrates
$\mathring{\gamma}^{*}:\mathfrak{b}_{\pm}\rightarrow\mathfrak{b}_{\pm}$ to a
group homomorphism
${\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}^{*}:\mathcal{B}_{\pm}\rightarrow\mathcal{B}_{\pm}$.
Note: since $\gamma$ and $\mathring{\gamma}$ coincide on $\mathfrak{n}_{+}$
and $\mathfrak{n}_{-}$,
$\tilde{\gamma}(N_{\pm})={\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}(N_{\pm})$,
$N_{\pm}\in\mathcal{N}_{\pm}$. We will prefer the notation $\tilde{\gamma}$
over ${\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}$ if the argument
is a unipotent lower or upper triangular matrix. Next, set

$\operatorname{gl}_{n}({\Gamma_{1}}):=\bigoplus_{|\Delta|>1}\operatorname{gl}_{n}(\Delta),\
\
\operatorname{gl}_{n}({\Gamma_{2}}):=\bigoplus_{|\bar{\Delta}|>1}\operatorname{gl}_{n}(\bar{\Delta}),$
(4.4)

where $\operatorname{gl}_{n}(\Delta)$ and
$\operatorname{gl}_{n}(\bar{\Delta})$ are defined above. Let
$\operatorname{GL}_{n}(\Gamma_{i})$ be the connected subgroup of
$\operatorname{GL}_{n}$ that corresponds to
$\operatorname{gl}_{n}(\Gamma_{i})$, $i\in\\{1,2\\}$. The restriction
$\mathring{\gamma}|_{\operatorname{gl}_{n}(\Gamma_{1})}:\operatorname{gl}_{n}(\Gamma_{1})\rightarrow\operatorname{gl}_{n}(\Gamma_{2})$
can be integrated to the group homomorphism
${\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}:\operatorname{GL}_{n}(\Gamma_{1})\rightarrow\operatorname{GL}_{n}(\Gamma_{2})$;
likewise,
$\mathring{\gamma}^{*}:\operatorname{gl}_{n}(\Gamma_{2})\rightarrow\operatorname{gl}_{n}(\Gamma_{1})$
integrates to
${\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}^{*}:\operatorname{GL}_{n}(\Gamma_{2})\rightarrow\operatorname{GL}_{n}(\Gamma_{1})$.

######  Remark 4.5.

Note that $\mathring{\gamma}|_{\operatorname{sl}_{n}(\mathbb{C})}$ coincides
with the map
$\gamma:\operatorname{sl}_{n}(\mathbb{C})\rightarrow\operatorname{sl}_{n}(\mathbb{C})$
introduced in Section 2.1 only on the nilpotent subalgebras $\mathfrak{n}_{-}$
and $\mathfrak{n}_{+}$: the maps are different on the Cartan subalgebra of
$\operatorname{sl}_{n}(\mathbb{C})$. Moreover, the version of $\gamma$ from
Section 2.1 does not integrate to a homomorphism of Borel subgroups.

##### Projections.

We equip $\operatorname{gl}_{n}(\mathbb{C})$ with the trace form and define
the orthogonal projections
$\mathring{\pi}_{\Gamma_{i}}:\operatorname{gl}_{n}(\mathbb{C})\rightarrow\operatorname{gl}_{n}(\Gamma_{i})$
and
$\mathring{\pi}_{\hat{\Gamma}_{i}}:=\operatorname{gl}_{n}(\mathbb{C})\rightarrow\operatorname{gl}_{n}(\Gamma_{i})^{\perp}$,
$i\in\\{1,2\\}$. Given the parabolic subalgebras
$\mathfrak{p}_{\pm}(\Gamma_{i})$ of $\operatorname{gl}_{n}(\mathbb{C})$ and
the corresponding parabolic subgroups $\mathcal{P}_{\pm}(\Gamma_{i})$, we
integrate the restrictions
$(\mathring{\pi}_{\Gamma_{i}})|_{\mathfrak{p}_{\pm}(\Gamma_{i})}$ to the group
projections
$\mathring{\Pi}_{\Gamma_{i}}:\mathcal{P}_{\pm}(\Gamma_{i})\rightarrow\operatorname{GL}_{n}(\Gamma_{i})$,
$i\in\\{1,2\\}$. Let $\mathcal{H}$ be the Cartan subgroup of
$\operatorname{GL}_{n}$, $\mathfrak{h}(\Gamma_{i})$ the Cartan subalgebra of
$\operatorname{gl}_{n}(\Gamma_{i})$, $\mathcal{H}(\Gamma_{i})$ and
$\mathcal{H}(\hat{\Gamma}_{i})$ be the connected subgroups of
$\operatorname{GL}_{n}$ that correspond to $\mathfrak{h}(\Gamma_{i})$ and
$\mathfrak{h}(\Gamma_{i})^{\perp}$, $i\in\\{1,2\\}$. We set
$\mathring{\Pi}_{\hat{\Gamma}_{i}}:\mathcal{H}\rightarrow\mathcal{H}(\hat{\Gamma}_{i})$
to be the group homomorphism that corresponds to
$\mathring{\pi}_{\hat{\Gamma}_{i}}|_{\mathfrak{h}}$.

##### The Gauss decomposition and $\tilde{\gamma}^{*}$.

Let $\mathbf{\Gamma}=(\Gamma_{1},\Gamma_{2},\gamma)$ be a BD triple of type
$A_{n-1}$, and let $U=U_{\oplus}U_{-}$ be the decomposition of a generic
element $U\in\operatorname{GL}_{n}$ into an upper triangular matrix
$U_{\oplus}$ and a unipotent lower triangular matrix $U_{-}$. Let $\Delta$ be
a nontrivial $X$-run and $i,j\in\Delta$, $i>j$. We will need the following
formulas:

  1. a)

If $\gamma|_{\Delta\cap\Gamma_{1}}>0$, then

$[\tilde{\gamma}^{*}(U_{-})]_{ij}=\frac{\det
U^{\\{\bar{\gamma}(j)\\}\cup[\bar{\gamma}(i)+1,n]}_{[\bar{\gamma}(i),n]}}{\det
U^{[\bar{\gamma}(i),n]}_{[\bar{\gamma}(i),n]}};$ (4.5)

  2. b)

If $\gamma|_{\Delta\cap\Gamma_{1}}<0$, then

$[\tilde{\gamma}^{*}(U_{-})]_{ij}=\frac{\det
U^{[\bar{\gamma}(i),n]\setminus\\{\bar{\gamma}(j)\\}}_{[\bar{\gamma}(i)+1,n]}}{\det
U^{[\bar{\gamma}(i)+1,n]}_{[\bar{\gamma}(i)+1,n]}}.$ (4.6)

The above formulas follow from Lemma C.1 and Lemma C.2. Furthermore, if $N$ is
a unipotent lower triangular matrix and $i,j\in\Delta$ such that
$\gamma|_{\Delta\cap\Gamma_{1}}<0$, then

$[\tilde{\gamma}^{*}(N)]_{ij}=(-1)^{i+j}[N^{-1}]_{\bar{\gamma}(j)\bar{\gamma}(i)}=\det
N^{[1,n]\setminus\\{\bar{\gamma}(j)\\}}_{[1,n]\setminus\\{\bar{\gamma}(i)\\}}.$
(4.7)

##### Ringed version of $R_{0}$ for $\operatorname{GL}_{n}$.

As in our previous work [31], it is expedient to choose $R_{0}$ with special
properties in the case of $\operatorname{GL}_{n}$. Any such special solution
will be denoted as $\mathring{R}_{0}$ and called a _ringed_ $R_{0}$ element.
The ringed element $\mathring{R}_{0}$ is a solution of (2.8)-(2.9) that in
addition satisfies any of the following equivalent identities:

$\displaystyle\mathring{R}_{0}(1-\mathring{\gamma})$
$\displaystyle=\mathring{\pi}_{\Gamma_{1}}+\mathring{R}_{0}\mathring{\pi}_{\hat{\Gamma}_{1}}\
\ $ $\displaystyle\ \ \mathring{R}_{0}(1-\mathring{\gamma}^{*})$
$\displaystyle=-\mathring{\gamma}^{*}+\mathring{R}_{0}\mathring{\pi}_{\hat{\Gamma}_{2}}$
(4.8) $\displaystyle\mathring{R}_{0}^{*}(1-\mathring{\gamma})$
$\displaystyle=-\mathring{\gamma}+\mathring{R}_{0}^{*}\mathring{\pi}_{\hat{\Gamma}_{1}}\
\ $ $\displaystyle\ \ \mathring{R}_{0}^{*}(1-\mathring{\gamma}^{*})$
$\displaystyle=\mathring{\pi}_{\Gamma_{2}}+\mathring{R}_{0}^{*}\mathring{\pi}_{\hat{\Gamma}_{2}}$

where all the maps are assumed to be restricted to the diagonal matrices.

###  4.2 The map $\mathcal{F}$

Fix a set of simple roots $\Pi$ of type $A_{n-1}$ and a BD triple
$\mathbf{\Gamma}:=(\Gamma_{1},\Gamma_{2},\gamma)$ of type $A_{n-1}$. Let
$\mathcal{B}_{\pm}$ and $\mathcal{N}_{\pm}$ be the Borel subgroups and their
unipotent radicals, and set $\mathcal{H}$ to be the Cartan subgroup. We
decompose a generic element $U\in\operatorname{GL}_{n}$ as
$U=U_{\oplus}U_{-}$, where $U_{-}\in\mathcal{N}_{-}$ and
$U_{\oplus}\in\mathcal{B}_{+}$. Define a sequence
$\mathcal{F}_{k}:\operatorname{GL}_{n}\dashrightarrow\operatorname{GL}_{n}$
via

$\mathcal{F}_{0}(U):=U,\ \
\mathcal{F}_{k}(U)=\tilde{\gamma}^{*}([\mathcal{F}_{k-1}(U)]_{-})U,\ \
U\in\operatorname{GL}_{n},\ k\geq 1.$ (4.9)

The rational map $\mathcal{F}:G\dashrightarrow G$ is defined as the limit

$\mathcal{F}(U):=\lim_{k\rightarrow\infty}\mathcal{F}_{k}(U),\ \
U\in\operatorname{GL}_{n}.$ (4.10)

######  Lemma 4.6.

Let $N\in\mathcal{N}_{-}$ and $k\in[0,\deg\gamma+1]$. Let
$\\{\hat{\mathcal{F}}_{i}:\operatorname{GL}_{n}\dashrightarrow\operatorname{GL}_{n}\\}_{i\geq
0}$ be a sequence of rational maps defined via

$\hat{\mathcal{F}}_{0}(U):=(\tilde{\gamma}^{*})^{k}(N)U,\ \
\hat{\mathcal{F}}_{i}(U):=\tilde{\gamma}^{*}([\hat{\mathcal{F}}_{i-1}(U)]_{-})U,\
\ U\in G,\ i\geq 1.$ (4.11)

Then $\hat{\mathcal{F}}_{\deg\gamma+1-k}(U)=\mathcal{F}_{\deg\gamma+1-k}(U)$.

###### Proof.

For a subset $A\subseteq\Pi$, set $\mathcal{N}_{-}(A)$ to be a subgroup of
$\mathcal{N}_{-}$ generated by $A$. Define
$\mathcal{S}:\mathcal{N}_{-}\rightarrow\mathcal{N}_{-}$ via setting
$\mathcal{S}(N):=\tilde{\gamma}^{*}([NU]_{-})$, and let
$S_{\ell}:=\\{\beta\in\Phi_{+}\ |\ \beta\succ(\gamma^{*})^{\ell}(\alpha)\
\text{for some}\ \alpha\in A\cap\Gamma_{1}\\}$, where we imply the usual order
on the roots. The statement of the lemma is an observation that the entries of
$N$ in the matrix $\mathcal{S}^{\ell}(N)$ are visible only at the root vectors
$e_{-\beta}$ for $\beta\in S_{\ell}$. ∎

######  Proposition 4.7.

The sequence $\mathcal{F}_{k}$ stabilizes at $k=\deg\gamma$ and
$\mathcal{F}(U)=\mathcal{F}_{\deg\gamma}(U)$. Moreover, the rational map
$\mathcal{F}:\operatorname{GL}_{n}\dashrightarrow\operatorname{GL}_{n}$
satisfies the relation

$\mathcal{F}(U)=\tilde{\gamma}^{*}([\mathcal{F}(U)]_{-})U,$ (4.12)

and in particular, $\mathcal{F}$ is birational, with the inverse given by

$\mathcal{F}^{-1}(U)=\tilde{\gamma}^{*}(U_{-})^{-1}U.$ (4.13)

###### Proof.

The proposition follows from Lemma 4.6 via setting
$N:=[\mathcal{F}_{i}(U)]_{-}$ for $i\geq 1$. ∎

######  Proposition 4.8.

Let $N\in\mathcal{N}_{-}$ and $T\in\mathcal{H}$. Set $N_{0}:=N$ and
$N_{k}:=(\tilde{\gamma}^{*})^{k}(N)$, $k\in[1,\deg\gamma]$. For any
$k\in[0,\deg\gamma]$, the following formulas hold:

$\displaystyle\mathcal{F}_{\deg\gamma-k}(N_{k+1}UN_{k}^{-1})=\mathcal{F}_{\deg\gamma-k}(U)N_{k}^{-1};$
(4.14)
$\displaystyle\mathcal{F}_{\deg\gamma-k}(\tilde{\gamma}^{*}(T)UT^{-1})=\tilde{\gamma}^{*}(T)\mathcal{F}_{\deg\gamma-k}(U)T^{-1};$
(4.15)
$\displaystyle\mathcal{F}_{\deg\gamma-k}(T^{-1}U\tilde{\gamma}(T))=T^{-1}\mathcal{F}_{\deg\gamma-k}(U)\tilde{\gamma}(T).$
(4.16)

###### Proof.

Let us set

$\hat{\mathcal{F}}_{i}(U):=\mathcal{F}_{i}(N_{k+1}UN_{k}^{-1})N_{k},\ \ i\geq
0.$

Observe that $\hat{\mathcal{F}}_{0}(U)=N_{k+1}U$ and that
$\tilde{\mathcal{F}}_{i}(U)$ satisfies the recurrence relation
$\hat{\mathcal{F}}_{i}(U)=\tilde{\gamma}^{*}([\hat{\mathcal{F}}_{i-1}(U)]_{-})U$,
$i\geq 0$. Indeed, from the recurrence relation for $\mathcal{F}_{i}(U)$, we
see that

$\hat{\mathcal{F}}_{i+1}(U)=\mathcal{F}_{i}(N_{k+1}UN_{k}^{-1})N_{k}=\tilde{\gamma}^{*}([\mathcal{F}_{i-1}(N_{k+1}UN_{k}^{-1})]_{-})N_{k+1}UN_{k}^{-1}N_{k}=\tilde{\gamma}^{*}([\hat{\mathcal{F}}_{i}(U)]_{-})U.$

By Lemma 4.6, we conclude that
$\hat{\mathcal{F}}_{\deg\gamma-k}(U)=\mathcal{F}_{\deg\gamma-k}(U)$, which
yields the first formula of the proposition.

The second formula is obvious for $k=\deg\gamma$; the other cases follow
easily from induction:

$\begin{split}\mathcal{F}_{\deg\gamma-k+1}({\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}^{*}(T)UT^{-1})=&{\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}^{*}([\mathcal{F}_{\deg\gamma-k}({\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}^{*}(T)UT^{-1})]_{-}){\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}^{*}(T)UT^{-1}=\\\
=&{\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}^{*}([{\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}^{*}(T)\mathcal{F}_{\deg\gamma-k}(U)T^{-1}]_{-}){\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}^{*}(T)UT^{-1}=\\\
=&{\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}^{*}(T[\mathcal{F}_{\deg\gamma-k}(U)]_{-}T^{-1}){\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}^{*}(T)UT^{-1}=\\\
=&{\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}^{*}(T)\mathcal{F}_{\deg\gamma-k+1}(U)T^{-1}.\end{split}$

The last formula is also evident for $k=\deg\gamma$. Assuming it holds for
some $k$, we see that

$\begin{split}\mathcal{F}_{\deg\gamma-k+1}(T^{-1}U{\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}(T))=&{\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}^{*}\left([\mathcal{F}_{\deg\gamma-k}(T^{-1}U{\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}(T))]_{-}\right)T^{-1}U{\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}(T)=\\\
=&{\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}^{*}\left({\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}(T^{-1})[\mathcal{F}_{\deg\gamma-k}(U)]_{-}{\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}(T)\right)T^{-1}U{\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}(T)=\\\
=&\mathring{\Pi}_{\Gamma_{1}}(T^{-1}){\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}^{*}\left([\mathcal{F}_{\deg\gamma-k}(U)]_{-}\right)\mathring{\Pi}_{\hat{\Gamma}_{1}}(T^{-1})U{\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}(T)=\\\
=&\mathring{\Pi}_{\Gamma_{1}}(T^{-1})\mathring{\Pi}_{\hat{\Gamma}_{1}}(T^{-1}){\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}^{*}\left([\mathcal{F}_{\deg\gamma-k}(U)]_{-}\right)U{\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}(T)=\\\
=&T^{-1}\mathcal{F}_{\deg\gamma-k+1}(U){\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}(T).\end{split}$

Thus the identities hold. ∎

######  Corollary 4.9.

The following identities hold :

$\displaystyle\mathcal{F}(\tilde{\gamma}^{*}(N)UN^{-1})=\mathcal{F}(U)N^{-1};$
(4.17)
$\displaystyle\mathcal{F}({{\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}}^{*}(T)UT^{-1}))={{\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}}^{*}(T)\mathcal{F}(U)T^{-1};$
(4.18)
$\displaystyle\mathcal{F}(T^{-1}U{{\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}}(T))=T^{-1}\mathcal{F}(U){{\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}}(T).$
(4.19)

###### Proof.

Set $k=0$ in Proposition 4.8 and invoke Proposition 4.7.∎

######  Corollary 4.10.

Let $h_{ij}$ be any $h$-variable from the initial extended cluster of
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma},\operatorname{GL}_{n})$. Then
there exist characters $\chi$ and $\chi^{\prime}$ such that the following
identities hold:

$\displaystyle h_{ij}(\tilde{\gamma}^{*}(N)UN^{-1})=h_{ij}(U);$ (4.20)
$\displaystyle
h_{ij}({\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}^{*}(T)UT^{-1})=\chi(T)h_{ij}(U);$
(4.21) $\displaystyle
h_{ij}(T^{-1}U{\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}(T))=\chi^{\prime}(T)h_{ij}(U).$
(4.22)

######  Corollary 4.11.

Let $h_{ij}$ be any $h$-variable from the initial extended cluster of
$\mathcal{GC}^{\dagger}_{h}(\mathbf{\Gamma},\operatorname{GL}_{n})$. Then the
following identities hold:

$\displaystyle\nabla_{U}h_{ij}\cdot
U-\mathring{\gamma}\left(U\cdot\nabla_{U}h_{ij}\right)\in\mathfrak{b}_{-};$
(4.23) $\displaystyle\pi_{0}\left(\nabla_{U}\log h_{ij}\cdot
U-\mathring{\gamma}\left(U\cdot\nabla_{U}\log
h_{ij}\right)\right)=\mathrm{const};$ (4.24)
$\displaystyle\pi_{0}\left(U\cdot\nabla_{U}\log
h_{ij}-\mathring{\gamma}^{*}\left(\nabla_{U}\log h_{ij}\cdot
U\right)\right)=\mathrm{const}.$ (4.25)

###### Proof.

The formulas follow from differentiating the identities in Corollary 4.10. ∎

######  Remark 4.12.

The birational map
$\mathcal{F}:(\operatorname{GL}_{n},\pi_{\mathbf{\Gamma}}^{\dagger})\dashrightarrow(\operatorname{GL}_{n},\pi_{\text{std}}^{\dagger})$
is not a Poisson map. Nevertheless, we believe that the pushforward Poisson
bivector $\mathcal{F}_{*}(\pi_{\mathbf{\Gamma}}^{\dagger})$, as well as the
induced compatible generalized cluster structure on $\operatorname{GL}_{n}$
might be of interest. We also believe that there might exist generalized
cluster structures on some interesting subvarieties of $\operatorname{GL}_{n}$
whose variables are constructed as the flag minors of $\mathcal{F}_{m}$ (akin
to the formula (1.13)), and these generalized cluster structures might be
compatible with some other Poisson structures. For a discussion of these
problems, as well as for supporting evidence the reader might refer to the
supplementary file [33].

###  4.3 The maps $\mathcal{Q}$ and $\mathcal{F}^{c}$

Recall from Section 3 that the rational map
$\mathcal{Q}:(\operatorname{GL}_{n},\pi_{\text{std}}^{\dagger})\dashrightarrow(\operatorname{GL}_{n},\pi_{\mathbf{\Gamma}}^{\dagger})$
is given by

$\mathcal{Q}(U):=\rho(U)^{-1}U\rho(U),\ \ \rho(U):=\prod_{i\geq
1}^{\rightarrow}(\tilde{\gamma}^{*})^{i}(U_{-}).$ (4.26)

Let us define the rational map
$\mathcal{F}^{c}:(\operatorname{GL}_{n},\pi_{\mathbf{\Gamma}}^{\dagger})\dashrightarrow(\operatorname{GL}_{n},\pi_{\text{std}}^{\dagger})$
via

$\mathcal{F}^{c}(U):=\mathcal{F}(U)U[\mathcal{F}(U)]^{-1}.$ (4.27)

######  Proposition 4.13.

The rational maps $\mathcal{F}^{c}$ and $\mathcal{Q}$ are inverse to each
other.

###### Proof.

Recall that $\rho(U)$ satisfies the relation

$\tilde{\gamma}^{*}(U_{-})\tilde{\gamma}^{*}(\rho(U))=\rho(U).$

Applying formulas (4.13) and (4.17), we see that

$\mathcal{F}(\mathcal{Q}(U))=\mathcal{F}\left(\tilde{\gamma}^{*}(\rho(U))^{-1}\left[\tilde{\gamma}^{*}(U_{-})^{-1}U\right]\rho(U)\right)=U\cdot\rho(U);$

therefore,

$\mathcal{F}^{c}(\mathcal{Q}(U))=\left[U\cdot\rho(U)\right]\left[\rho(U)^{-1}U\rho(U)\right]\left[\rho(U)^{-1}U^{-1}\right]=U.\qed$

######  Remark 4.14.

The map $\mathcal{F}^{c}$ can also be written as

$\mathcal{F}^{c}(U)=\mathcal{F}(U)\tilde{\gamma}^{*}(\mathcal{F}(U)_{-})^{-1}=\tilde{\gamma}^{*}([\mathcal{F}(U)]_{-})U\tilde{\gamma}^{*}([\mathcal{F}(U)]_{-})^{-1}.$
(4.28)

The identities follow from (4.12).

######  Proposition 4.15.

Let $\psi$ be any variable from the initial extended cluster of
$\mathcal{GC}^{\dagger}_{h}(\mathbf{\Gamma})$. Then the following identities
hold:

  1. 1)

If $\psi$ is a $\varphi$\- or $c$-function, then
$\mathcal{Q}^{*}(\psi(U))=\psi(U)$;

  2. 2)

If $\psi=h_{ij}$ for some $h$-function $h_{ij}(U)$, let
$\\{\alpha_{t}\\}_{t=0}^{m}$ be the $\gamma$-string such that $i-1=\alpha_{k}$
for some $k$; then,

$\mathcal{Q}^{*}(h_{ij}(U))=\tilde{h}_{ij}(U)\prod_{t\geq
k+1}\tilde{h}_{\alpha_{t}+1,\alpha_{t}+1}(U).$ (4.29)

Moreover,
$\mathcal{Q}:(G,\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma}_{\text{std}}))\dashrightarrow(G,\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma}))$
is a birational quasi-isomorphism for
$G\in\\{\operatorname{SL}_{n},\operatorname{GL}_{n}\\}$.

###### Proof.

The statement in Part 1 follows from the invariance properties of $\varphi$\-
and $c$-functions (see (1.6) and (1.7)). For Part 2, notice that for any
$1\leq j\leq n$ and any $I\subseteq[1,n]$ of size $|I|=n-j+1$,

$\mathcal{Q}^{*}\left(\det\mathcal{F}(U)^{[j,n]}_{I}\right)=\det
U^{[j,n]}_{I},$

and then invoke the definition of the $h$-variables (1.11). The last statement
can be verified via Proposition 2.25. The hatted strings $\hat{p}_{ir}$ are
given by the $c$-variables, which in turn are invariant with respect to
$\mathcal{Q}$ by Part 1. To verify that
$\mathcal{Q}^{*}(y_{i})=\tilde{y}_{\kappa(i)}$ for each nonmarked index $i$,
we consider the neighborhoods of the variables as given in Appendix A. For
instance, from Figure 5 we read that $y(h_{ij})$ for $2\leq i\leq j\leq n-1$
is given by

$y(h_{ij})=\frac{h_{i+1,j}h_{i-1,j-1}h_{i,j+1}}{h_{i-1,j}h_{i,j-1},h_{i+1,j+1}}.$
(4.30)

It follows from (4.29) that for any $2\leq j\leq n-1$ and any $2\leq s,k\leq
j$,

$\mathcal{Q}^{*}\left(\frac{h_{kj}}{h_{sj}}\right)=\frac{\tilde{h}_{kj}}{\tilde{h}_{sj}}.$
(4.31)

Therefore, $\mathcal{Q}^{*}(y(h_{ij}))=y(\tilde{h}_{ij})$. A similar reasoning
applies to all the other neighborhoods. By Proposition 2.25, we conclude that
$\mathcal{Q}$ is a quasi-isomorphism. To verify that $\mathcal{Q}$ is a
birational quasi-isomorphism, as in Definition 2.27, we need to verify that
$\mathcal{Q}^{*}$ induces an isomorphism

$\mathcal{Q}^{*}:\mathcal{O}(\operatorname{gl}_{n}(\mathbb{C}))[h_{ii}^{\pm
1}\ |\
i\in[2,n]]\xrightarrow{\sim}\mathcal{O}(\operatorname{gl}_{n}(\mathbb{C}))[\tilde{h}_{ii}^{\pm
1}\ |\ i\in[2,n]].$ (4.32)

From the defining formula (4.26), the formulas from (C.1) for the entries of
$U_{-}$ and the formula (4.29), we see that

$\mathcal{Q}^{*}\left(\mathcal{O}(\operatorname{gl}_{n}(\mathbb{C}))[{h}_{ii}^{\pm
1}\ |\
i\in[2,n]]\right)\subseteq\mathcal{O}(\operatorname{gl}_{n}(\mathbb{C}))[\tilde{h}_{ii}^{\pm
1}\ |\ i\in[2,n]].$ (4.33)

Since $\mathcal{Q}^{*}$ is also an isomorphism of the groups of invertible
elements of the both localizations, we conclude that (4.33) is, in fact, an
equality. Thus $\mathcal{Q}$ is a birational quasi-isomorphism. ∎

##### Equivariance property of $\mathcal{Q}$.

Let $\mathfrak{h}^{s}$ be the Cartan subalgebra of
$\operatorname{sl}_{n}(\mathbb{C})$. Recall that the Lie subalgebra
$\mathfrak{h}_{\mathbf{\Gamma}}\subseteq\mathfrak{h}^{s}$ is given by

$\mathfrak{h}_{\mathbf{\Gamma}}:=\\{h\in\mathfrak{h}^{s}\ |\
\alpha(h)=\beta(h)\ \text{if}\ \gamma^{j}(\alpha)=\beta\ \text{for some}\
j\\}.$ (4.34)

Let $\mathcal{H}_{\mathbf{\Gamma}}$ be the connected subgroup of
$\operatorname{SL}_{n}$ that corresponds to $\mathfrak{h}_{\mathbf{\Gamma}}$.

######  Proposition 4.16.

For any $T\in\mathcal{H}_{\mathbf{\Gamma}}$ and any
$U\in\operatorname{GL}_{n}$, $\mathcal{Q}(TUT^{-1})=T\mathcal{Q}(U)T^{-1}$.

###### Proof.

We see that

$T^{-1}\rho(TUT^{-1})=\prod_{i\geq
1}({\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}^{*})^{i-1}(T^{-1}{\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}^{*}(T))({\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}^{*})^{i}(U_{-}).$
(4.35)

It suffices to show that
$T^{-1}{\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}^{*}(T)$ commutes
with ${\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}^{*}(U_{-})$.
Indeed, we can decompose
$T^{-1}{\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}^{*}(T)$ as

$T^{-1}{\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}^{*}(T)=\mathring{\Pi}_{\Gamma_{1}}(T^{-1}{\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}^{*}(T))\mathring{\Pi}_{\hat{\Gamma}_{1}}(T^{-1}{\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}^{*}(T)).$
(4.36)

Evidently, the second factor commutes with
${\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}^{*}(U_{-})$; as for
the first factor, we see that

$\mathring{\Pi}_{\Gamma_{1}}(T^{-1}{\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}^{*}(T))=\exp\left(\sum_{\Delta}I_{\Delta}t_{\Delta}\right)$
(4.37)

where the sum is over nontrivial $X$-runs $\Delta$,
$I_{\Delta}:=\sum_{i\in\Delta}e_{ii}$ and $t_{\Delta}\in\mathbb{C}$ are some
scalars. Clearly, the last expression also commutes with
${{\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}}^{*}(U_{-})$. Thus
the statement holds. ∎

###  4.4 Birational quasi-isomorphisms $\mathcal{G}$ for
$\tilde{\mathbf{\Gamma}}\prec\mathbf{\Gamma}$

In this subsection, given a pair of BD triples
$\tilde{\mathbf{\Gamma}}:=(\tilde{\Gamma}_{1},\tilde{\Gamma}_{2},\theta)$ and
$\mathbf{\Gamma}:=(\Gamma_{1},\Gamma_{2},\gamma)$ of type $A_{n-1}$, we
construct a birational map
$\mathcal{G}:(\operatorname{GL}_{n},\pi_{\tilde{\mathbf{\Gamma}}}^{\dagger})\dashrightarrow(\operatorname{GL}_{n},\pi_{\mathbf{\Gamma}}^{\dagger})$
from the birational maps
$\mathcal{Q}:(\operatorname{GL}_{n},\pi_{\text{std}}^{\dagger})\dashrightarrow(\operatorname{GL}_{n},\pi_{\mathbf{\Gamma}}^{\dagger})$
and
$\tilde{\mathcal{Q}}:(\operatorname{GL}_{n},\pi_{\text{std}}^{\dagger})\dashrightarrow(\operatorname{GL}_{n},\pi_{\tilde{\mathbf{\Gamma}}}^{\dagger})$
(the map $\tilde{\mathcal{Q}}(U):=\tilde{\rho}(U)^{-1}U\tilde{\rho}(U)$ is
given by (4.26), but relative the BD triple $\tilde{\mathbf{\Gamma}}$, so we
decorate the map itself with a tilde). Proposition 4.17 below is a more
detailed version of Proposition 1.7 for
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma})$. Let us set

$G_{0}(U):=\tilde{\theta}^{*}(\tilde{\mathcal{F}}(U)_{-})^{-1}\tilde{\gamma}^{*}(\tilde{\mathcal{F}}(U)_{-}),\
\ U\in\operatorname{GL}_{n},$ (4.38)

and let $\tilde{\mathcal{F}}$ and $\mathcal{F}$ be the maps given by (4.10)
and constructed relative $\tilde{\mathbf{\Gamma}}$ and $\mathbf{\Gamma}$,
respectively.

######  Proposition 4.17.

Given any pair $\mathbf{\Gamma}$ and $\tilde{\mathbf{\Gamma}}$ of BD triples
of type $A_{n-1}$, set
$\mathcal{G}:=\mathcal{Q}\circ\tilde{\mathcal{Q}}^{-1}$. Then $\mathcal{G}$ is
a birational map given by the following formula:

$\mathcal{G}(U)=G(U)^{-1}\cdot U\cdot G(U),\ \ G(U):=G_{0}(U)\prod_{i\geq
1}^{\rightarrow}(\tilde{\gamma}^{*})^{i}(G_{0}(U)).$ (4.39)

Moreover, for $\tilde{\mathbf{\Gamma}}\prec\mathbf{\Gamma}$ and
$G\in\\{\operatorname{SL}_{n},\operatorname{GL}_{n}\\}$,
$\mathcal{G}:(G,\mathcal{GC}_{h}^{{\dagger}}(\tilde{\mathbf{\Gamma}}))\dashrightarrow(G,\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma}))$
is a birational quasi-isomorphism, and if the $r_{0}$ parts of
$\pi_{\tilde{\mathbf{\Gamma}}}^{\dagger}$ and
$\pi_{\mathbf{\Gamma}}^{\dagger}$ coincide, then
$\mathcal{G}:(G,\pi_{\tilde{\mathbf{\Gamma}}}^{\dagger})\dashrightarrow(G,\pi_{\mathbf{\Gamma}}^{\dagger})$
is a Poisson isomorphism.

In Section 4.5, we will further refine the construction in the case
$\tilde{\mathbf{\Gamma}}$ is obtained from $\mathbf{\Gamma}$ via removal of a
pair of roots. There, we will give a more explicit description of the matrix
$G_{0}(U)$.

###### Proof.

Let us start with the definition
$\mathcal{G}=\mathcal{Q}\circ\tilde{\mathcal{Q}}^{-1}$ and derive formula
(4.39). Recall from (4.28) that

$\tilde{\mathcal{Q}}^{-1}(U)=\tilde{\mathcal{F}}(U)\tilde{\theta}^{*}(\tilde{\mathcal{F}}(U)_{-})^{-1}=\tilde{\theta}^{*}(\tilde{\mathcal{F}}(U)_{-})U\tilde{\theta}^{*}(\tilde{\mathcal{F}}(U)_{-})^{-1}.$

We see that

$\mathcal{G}(U)=\rho\left(\tilde{\mathcal{Q}}^{-1}(U)\right)^{-1}\tilde{\theta}^{*}(\tilde{\mathcal{F}}(U)_{-})\cdot
U\cdot\tilde{\theta}^{*}(\tilde{\mathcal{F}}(U)_{-})^{-1}\rho\left(\tilde{\mathcal{Q}}^{-1}(U)\right).$
(4.40)

Set
$G(U):=\tilde{\theta}^{*}(\tilde{\mathcal{F}}(U)_{-})^{-1}\rho\left(\tilde{\mathcal{Q}}^{-1}(U)\right)$.
Since $\tilde{\gamma}^{*}(\rho(U))=\tilde{\gamma}^{*}(U_{-})^{-1}\rho(U)$, we
see that

$\begin{split}\tilde{\gamma}^{*}(G(U))=&\tilde{\gamma}^{*}\left(\tilde{\theta}^{*}(\tilde{\mathcal{F}}(U)_{-})^{-1}\right)\tilde{\gamma}^{*}\left(\left[\tilde{\mathcal{F}}(U)\tilde{\theta}^{*}(\tilde{\mathcal{F}}(U)_{-})^{-1}\right]_{-}\right)^{-1}\rho\left(\tilde{\mathcal{Q}}^{-1}(U)\right)=\\\
=&\tilde{\gamma}^{*}([\tilde{\mathcal{F}}(U)_{-}])^{-1}\rho(\tilde{\mathcal{Q}}^{-1}(U))=G_{0}(U)^{-1}G(U).\end{split}$
(4.41)

Now, formula (4.39) for $G(U)$ follows from the identity
$G_{0}(U)\tilde{\gamma}^{*}(G(U))=G(U)$.

To show that $\mathcal{G}$ is a birational quasi-isomorphism, let us consider
an increasing chain
$\mathbf{\Gamma}_{1}\prec\mathbf{\Gamma}_{2}\prec\cdots\prec\mathbf{\Gamma}_{m}$
of BD triples $\mathbf{\Gamma}_{i}:=(\Gamma_{1,i},\Gamma_{2,i},\gamma_{i})$
such that $\mathbf{\Gamma}_{1}:=\tilde{\mathbf{\Gamma}}$,
$\mathbf{\Gamma}_{m}:=\mathbf{\Gamma}$ and
$|\Gamma_{1,i+1}\setminus\Gamma_{1,i}|=1$, $i\in[1,m-1]$. Let
$\mathcal{G}_{i}:(\operatorname{GL}_{n},\pi_{\mathbf{\Gamma}_{i}}^{\dagger})\dashrightarrow(\operatorname{GL}_{n},\pi_{\mathbf{\Gamma}_{i+1}}^{\dagger})$
be defined relative $\mathbf{\Gamma}_{i}$ and $\mathbf{\Gamma}_{i+1}$, as
above. In Proposition 4.22 below we show that each
$\mathcal{G}_{i}:(\operatorname{GL}_{n},\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma}_{i}))\dashrightarrow(\operatorname{GL}_{n},\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma}_{i+1}))$
is a birational quasi-isomorphism. In particular, by Proposition 2.25, each
$\mathcal{G}_{i}^{*}$ sends a nonmarked $y$-variable in the initial extended
cluster of $\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma}_{i})$ to a $y$-variable
in the initial extended cluster of
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma}_{i+1})$. Therefore, the same
applies for $\mathcal{G}^{*}$, and by Proposition 2.25, it is a birational
quasi-isomorphism. ∎

######  Remark 4.18.

Note that since we did not assume
$\tilde{\mathbf{\Gamma}}\prec\mathbf{\Gamma}$ in the derivation of formula
(4.39), the formula covers both $\mathcal{G}$ and its inverse
$\mathcal{G}^{-1}$. Explicitly, since
$\mathcal{G}^{-1}=\tilde{\mathcal{Q}}\circ\mathcal{Q}^{-1}$, we set

$\tilde{G}_{0}(U):=\tilde{\theta}^{*}(\mathcal{F}(U)_{-})^{-1}\tilde{\gamma}^{*}(\mathcal{F}(U)_{-}),\
\ U\in\operatorname{GL}_{n};$ (4.42)

then, $\mathcal{G}^{-1}$ is given by

$\mathcal{G}^{-1}(U)=\tilde{G}(U)U\tilde{G}(U)^{-1},\ \
\tilde{G}(U)=\left[\prod_{i\geq
1}^{\leftarrow}(\tilde{\theta}^{*})^{i}(\tilde{G}_{0}(U))\right]\cdot\tilde{G}_{0}(U).$
(4.43)

If $\tilde{\mathbf{\Gamma}}$ is the trivial BD triple, one recovers
$\mathcal{G}^{-1}=\mathcal{F}^{c}$.

###  4.5 Birational quasi-isomorphisms $\mathcal{G}$ for
$|\Gamma_{1}\setminus\tilde{\Gamma}_{1}|=1$

Let $\mathbf{\Gamma}=(\Gamma_{1},\Gamma_{2},\gamma)$ be any Belavin-Drinfeld
triple of type $A_{n-1}$. Fix any pair of roots $p\in\Gamma_{1}$ and
$q:=\gamma(p)\in\Gamma_{2}$, and let $\tilde{\mathbf{\Gamma}}$ be obtained
from $\mathbf{\Gamma}$ via removal of $p$ and $q$ (see Definition 4.2). Let
$\mathcal{F}$ and $\tilde{\mathcal{F}}$ be the birational maps defined by
(4.10) relative the BD triples $\mathbf{\Gamma}$ and
$\tilde{\mathbf{\Gamma}}$, respectively. Let
$\mathcal{G}:=\mathcal{Q}\circ\tilde{\mathcal{Q}}^{-1}$ be the birational map
constructed in Section 4.4. The objective of this subsection is to further
refine formulas (4.38) and (4.39) (the result is the formula (4.47) below).
However, we will not use the obtained formulas anywhere else in the paper. Let
$\Delta$ and $\bar{\Delta}$ be the $X$\- and $Y$-runs that contain $p$ and
$q$, respectively. Set

$\tilde{\Delta}_{1}:=[p_{-}(\Gamma_{1})+1,p],\ \ \
\tilde{\Delta}_{2}:=[p+1,p_{+}(\Gamma_{1})].$ (4.44)

For $1\leq i\leq|\tilde{\Delta}_{2}|$ and $1\leq j\leq|\tilde{\Delta}_{1}|$,
define the following sets of indices:

$J_{ij}:=\begin{cases}\\{\bar{\gamma}(p_{-}(\Gamma_{1})+j)\\}\cup\left([q+1,n]\setminus\\{\bar{\gamma}(p+i)\\}\right)\
&\text{if}\ \gamma|_{\Delta}>0;\\\
\\{\bar{\gamma}(p+i)\\}\cup\left([q+1,n]\setminus\\{\bar{\gamma}(p_{-}(\Gamma_{1})+j)\\}\right)\
&\text{if}\ \gamma|_{\Delta}<0.\\\ \end{cases}$ (4.45)

Formula (4.38) for the matrix $G_{0}(U)$ can be refined as follows.

######  Proposition 4.19.

Under the setup of the current subsection, let us set

$\alpha_{ij}(U):=(-1)^{i+1}\frac{\det[\tilde{\mathcal{F}}(U)]^{J_{ij}}_{[q+1,n]}}{\det[\tilde{\mathcal{F}}(U)]_{[q+1,n]}^{[q+1,n]}}.$
(4.46)

Then

$G_{0}(U)=I+\sum_{i=1}^{|\tilde{\Delta}_{2}|}\sum_{j=1}^{|\tilde{\Delta}_{1}|}\alpha_{ij}(U)e_{p+i,p_{-}(\Gamma_{1})+j}.$
(4.47)

###### Proof.

Due to formula (4.38), the only nontrivial part of $G_{0}(U)$ is given by the
submatrix $[G_{0}(U)]_{\tilde{\Delta}_{2}}^{\tilde{\Delta}_{1}}$. If
$\gamma|_{\Delta\cap\Gamma_{1}}>0$, we see that

$\begin{split}\alpha_{ij}(U)=&(-1)^{i+1}\det\left[\tilde{\mathcal{F}}(U)_{-}\right]_{[q+1,n]}^{\\{q_{-}(\Gamma_{2})+j\\}\cup([q+1,n]\setminus\\{q+i\\})}=\\\
=&\sum_{k\in[1,i]}(-1)^{i+k}\left[\tilde{\mathcal{F}}(U)_{-}\right]_{q+k,q_{-}(\Gamma_{2})+j}\cdot\det\left[\tilde{\mathcal{F}}(U)_{-}\right]_{[q+1,n]\setminus\\{q+k\\}}^{[q+1,n]\setminus\\{q+i\\}}=\\\
=&\sum_{k\in[1,i]}\left[\tilde{\gamma}^{*}\left(\tilde{\mathcal{F}}(U)_{-}\right)\right]_{p+k,p_{-}(\Gamma_{1})+j}\left[\tilde{\theta}^{*}\left(\tilde{\mathcal{F}}(U)_{-}\right)^{-1}\right]_{p+i,p+k}=\\\
=&G_{0}(U)_{p+i,p_{-}(\Gamma_{1})+j}.\end{split}$

In the case of $\gamma|_{\Delta\cap\Gamma_{1}}<0$, let us recall that
$\bar{\gamma}$ is defined in such a way that $\bar{\gamma}(p)=q+1$ (see
Section 4.1). Set $\tau:=q_{+}(\Gamma_{2})-q+j$. We see that

$\begin{split}\alpha_{ij}(U)=&(-1)^{i+1}\det\left[\tilde{\mathcal{F}}(U)_{-}\right]_{[q+1,n]}^{\\{q-i+1\\}\cup\left([q+1,n]\setminus\\{q_{+}(\Gamma_{2})-j+1\\}\right)}=\\\
=&(-1)^{\tau+1}\det\left[\,\left.\tilde{\mathcal{F}}(U)_{-}\right.^{-1}\right]_{[q-i+2,q]\cup\\{q_{+}(\Gamma_{2})-j+1\\}}^{[q-i+1,q]}=\\\
=&(-1)^{\tau+1}\sum_{k=1}^{j}(-1)^{k-1}\det\left[\,\left.\tilde{\mathcal{F}}(U)_{-}\right.^{-1}\right]_{[q-i+1,q]\setminus\\{q-i+1\\}}^{[q-i+1,q]\setminus\\{q-k+1\\}}\left[\,\left.\tilde{\mathcal{F}}(U)_{-}\right.^{-1}\right]_{q_{+}(\Gamma_{2})-j+1,q-k+1}=\\\
=&\sum_{k=1}^{i}\left[\tilde{\theta}^{*}\left(\,\left.\tilde{\mathcal{F}}(U)_{-}\right.^{-1}\right)\right]_{p+i,p+k}\left[\tilde{\gamma}^{*}\left(\tilde{\mathcal{F}}(U)_{-}\right)\right]_{p+k,p_{-}(\Gamma_{1})+j}=G_{0}(U)_{p+i,p_{-}(\Gamma_{1})+j}.\end{split}$

where we also used formula (4.7). Thus the result follows. ∎

######  Remark 4.20.

The reader might notice similarities between the above formulas and the
formulas for birational quasi-isomorphisms $\mathcal{U}$ in the case of
$D(\operatorname{GL}_{n})$ from [31, Section 5.1].

We will use the following lemma in order to prove that $\mathcal{G}$ is a
quasi-isomorphism induced by toric actions (in the sense of Proposition 2.26),
as well as to prove Part 3 of Theorem 1.5 in Section 5.2.

######  Lemma 4.21.

The action of $\mathbb{C}^{*}$ upon $\operatorname{GL}_{n}$ given by
$(a,U)\mapsto a\cdot U$, $a\in\mathbb{C}^{*}$, $U\in\operatorname{GL}_{n}$,
induces a global toric action of rank $1$ upon
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma},\operatorname{GL}_{n})$.

###### Proof.

To prove that the local toric action is global, we appeal to Proposition 2.19.
Notice that all the variables in the initial extended cluster $\Psi_{0}$ of
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma})$ are homogeneous polynomials. The
weight vector $\omega$ of the induced local toric action upon $\Psi_{0}$ is
given by the polynomial degrees of the variables; however, we assign the value
$0$ to the components of $\omega$ that correspond to the $c$-variables (this
ensures that the coefficients $\hat{p}_{ir}$ from the proposition are
invariant with respect to the toric action). Showing that $B\omega=0$ for the
initial extended exchange matrix $B$ amounts to showing that the $y$-variables
in $\Psi_{0}$ have polynomial degree $0$. From (1.3), we see that

$\deg\varphi_{kl}=l-1+\frac{1}{2}(n-k-l+1)(n-k-l+2).$ (4.48)

In Section 6.2, we will also show that for any $\gamma$-string
$\\{\alpha_{i}\\}_{i=0}^{m}$ and each $j\in[\alpha_{i}+1,n]$, the polynomial
degree of $h_{\alpha_{i}+1,j}$ is given by

$\deg h_{\alpha_{i}+1,j}=n-j+1+\sum_{\ell=1}^{m-i}(n-\alpha_{i+\ell}).$ (4.49)

Now, to show that the $y$-variables in $\Psi_{0}$ have polynomial degree $0$,
we analyze each of the neighborhoods from Section A.1 and Section A.3. For
instance, consider Figure 9d, which is the neighborhood of $h_{jj}$ for $2\leq
j\leq n-1$, $j-1,j\in\Gamma_{2}$, $i:=\gamma^{*}(j-1)+1$ and
$\gamma|_{\Delta(i)\cap\Gamma_{1}}<0$. We see that

$\begin{split}\deg y(h_{jj})=&(\deg h_{j,j+1}-\deg h_{in})-(\deg
h_{j-1,j}-\deg h_{j-1,j-1})-\\\ &-(\deg h_{j+1,j+1}-\deg
h_{i-1,n})=2-1-1=0.\end{split}$ (4.50)

Thus the action $(a,U)\mapsto a\cdot U$, $a\in\mathbb{C}^{*}$,
$U\in\operatorname{GL}_{n}$ induces a global toric action of rank $1$ upon
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma},\operatorname{GL}_{n})$. ∎

######  Proposition 4.22.

Let $\psi$ be any variable from the initial extended cluster of
$\mathcal{GC}^{\dagger}_{h}(\mathbf{\Gamma})$. Then the following identities
hold:

  1. 1)

If $\psi$ is a $\varphi$\- or $c$-function, then
$\mathcal{G}^{*}(\psi(U))=\psi(U)$;

  2. 2)

If $\psi=h_{ij}$ for some $h$-function $h_{ij}(U)$, then

$\mathcal{G}^{*}(h_{ij}(U))=\begin{cases}\tilde{h}_{ij}(U)\tilde{h}_{q+1,q+1}(U)\
&\text{if}\ p=\gamma^{\nu}(i-1)\ \text{for some}\ \nu>0;\\\ \tilde{h}_{ij}(U)\
&\text{otherwise.}\end{cases}$ (4.51)

Moreover, $\mathcal{G}$ is a birational quasi-isomorphism.

###### Proof.

Due to the invariance properties of $\varphi$\- and $c$-functions (see (1.6)
and (1.7)) and the fact that $G(U)$ is a unipotent lower triangular matrix, we
see that Part 1 holds. For the second part, we see that for any $j\in[1,n]$
and $I\subseteq[1,n]$, $|I|=n-j+1$,

$\det[\mathcal{F}(\mathcal{G}(U))]_{I}^{[j,n]}=\det[\tilde{\mathcal{F}}(U)]_{I}^{[j,n]}.$

We also notice that if $S^{\gamma}(\alpha_{0})$ is a $\gamma$-string such that
$q\notin S^{\gamma}(\alpha_{0})$, then
$S^{\gamma}(\alpha_{0})=S^{\theta}(\alpha_{0})$; otherwise, if $q\in
S^{\gamma}(\alpha_{0})$, then
$S^{\gamma}(\alpha_{0})=S^{\theta}(\alpha_{0})\cup S^{\theta}(q)$. Now Part 2
follows from an application of $\mathcal{G}^{*}$ to the definition of the
$h$-functions given in (1.11).

The last statement can be proved either via Proposition 2.25 or Proposition
2.26. Let us use the latter proposition, for it also shows that, as a quasi-
isomorphism, $\mathcal{G}^{*}$ is generated by toric actions when
$|\Gamma_{1}\setminus\tilde{\Gamma}_{1}|=1$. First of all, by Lemma 5.10,
there are global toric action of rank $1$ acting upon
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma},\operatorname{GL}_{n})$ and
$\mathcal{GC}_{h}^{\dagger}(\tilde{\mathbf{\Gamma}},\operatorname{GL}_{n})$
induced by the polynomial degrees of the variables. From formula (4.49), we
also see that

$\deg h_{ij}-\deg\tilde{h}_{ij}=\begin{cases}\deg\tilde{h}_{q+1,q+1}\
&\text{if}\ p=\gamma^{\nu}(i-1)\ \text{for some}\ \nu>0;\\\ 0\
&\text{otherwise.}\end{cases}$ (4.52)

It follows from Proposition 2.26 that $\mathcal{G}^{*}$ is a quasi-
isomorphism. Since $\mathcal{G}^{*}$ induces an isomorphism between
$\mathcal{O}(\operatorname{GL}_{n})[h_{q+1,q+1}^{\pm 1}]$ and
$\mathcal{O}(\operatorname{GL}_{n})[\tilde{h}_{q+1,q+1}^{\pm 1}]$, we conclude
that it is a birational quasi-isomorphism. ∎

##  5 Proofs of Theorems 1.5 and 1.6 for
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma})$

In this section, we prove Theorem 1.5 and Theorem 1.6 for
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma})$. Each subsection corresponds to
one part of a theorem (note that Part 1 follows from Part 4 due to Part 2 of
Proposition 2.16). The main idea is to use an inductive argument on the size
$|\Gamma_{1}|$ along with the birational quasi-isomorphisms constructed in
Section 4. The statements for $|\Gamma_{1}|=0$ follow from the earlier work
[20], the statements for $|\Gamma_{1}|=1$ require a manual verification only
for the marked variables, and the statements for $|\Gamma_{1}|>1$ follow from
the existence of complementary birational quasi-isomorphisms.

##### Conventions.

Throughout the section, when the indices of the $h$-variables are out of
range, the following conventions are used:

$\displaystyle h_{1j}(U):=\varphi_{j-1,n-j+1}(U)=(-1)^{\varepsilon_{1j}}\det
U^{[j,n]}_{[1,n-j+1]},\ $ $\displaystyle j\in[2,n];$ (5.1) $\displaystyle
h_{i,n+1}(U):=\begin{cases}h_{jj}(U)\ &\text{if}\ i-1\in\Gamma_{2}\\\ 1\
&\text{otherwise}\end{cases}\ $ $\displaystyle i\in[1,n]$ (5.2)

where $j:=\gamma^{*}(i-1)+1$ if $i-1\in\Gamma_{2}$.

###  5.1 Completeness

In this subsection, we prove that
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma},\operatorname{GL}_{n})$ and
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma},\operatorname{SL}_{n})$ are
complete. Along the way, we prove that all the variables in the initial
extended cluster are irreducible (as elements of
$\mathcal{O}(\operatorname{GL}_{n})$) and (for the initial cluster variables)
coprime with their mutations. The final result of the subsection is
Proposition 5.9, which corresponds to Part 2 of Theorem 1.5 and Part 2 of
Theorem 1.6.

####  5.1.1 The case of $|\Gamma_{1}|=|\Gamma_{2}|=1$ (shifted cluster).

Let $\mathbf{\Gamma}=(\Gamma_{1},\Gamma_{2},\gamma)$ be a BD triple of type
$A_{n-1}$ such that $\Gamma_{1}:=\\{p\\}$ and $\Gamma_{2}:=\\{q\\}$ for some
simple roots $p$ and $q$. We decorate the $h$-variables in
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma}_{\text{std}})$ with a tilde, to
distinguish them from the $h$-variables in
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma})$. Recall that we choose the
marker (see Definition 2.20) between
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma}_{\text{std}})$ and
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma})$ in such a way that each
$\varphi$\- and $c$\- variable are related to themselves, and each $h_{ij}$ is
related to $\tilde{h}_{ij}$. The only marked variable is given by
$\psi_{\square}:=h_{q+1,q+1}$. The map $\mathcal{F}^{c}$ assumes the form

$\mathcal{F}^{c}(U)=\tilde{\gamma}^{*}(U_{-})U\tilde{\gamma}^{*}(U_{-})^{-1},\
\ \tilde{\gamma}^{*}(U_{-})=I+\alpha(U)e_{p+1,p},\ \
\alpha(U)=\frac{1}{\psi_{\square}(U)}\det U^{\\{q\\}\cup[q+2,n]}_{[q+1,n]}.$
(5.3)

##### Shifting the initial cluster.

We will prove
$\bar{\mathcal{A}}_{\mathbb{C}}(\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma},\operatorname{GL}_{n}))=\mathcal{O}(\operatorname{GL}_{n})$
starting not in the initial extended cluster, but in a more convenient one.
Let us set $\Psi$ to be the extended cluster in
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma})$ obtained via mutating the
initial extended cluster $\Psi_{0}$ along the following mutation
sequence:171717If $p=n-1$, the mutation sequence is empty, so we set $\Psi$ to
be the initial extended cluster.

${h}_{p+1,n}\rightarrow{h}_{p+1,n-1}\rightarrow\cdots\rightarrow{h}_{p+1,p+2}.$
(5.4)

We set $\tilde{\Psi}$ to be the extended cluster in
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma}_{\text{std}})$ related to $\Psi$
(see Definition 2.23) and $\Psi^{\prime}$ to be the extended cluster in
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma})$ obtained by mutating $\Psi$ in
the direction of $\psi_{\square}$.

######  Lemma 5.1.

The extended cluster $\Psi$ contains the variables

${h}^{\prime}_{p+1,n-s}(U)=-(-1)^{s(n-(p+1))}\det
U_{\\{p\\}\cup[p+2,s+p+2]}^{[n-s-1,n]},\ \ s\in[0,n-p-2].$ (5.5)

Moreover, for any $\psi\in\Psi\setminus\\{h_{p+1,p+1}\\}$ and the related
variable $\tilde{\psi}\in\tilde{\Psi}\setminus\\{\tilde{h}_{p+1,p+1}\\}$,
$\tilde{\psi}(U)=\psi(U)$.

###### Proof.

It is well-known (see, for instance, [31, Lemma 5.3], with adjustments of
signs) that the related extended cluster $\tilde{\Psi}$ contains the variables

${h}^{\prime}_{p+1,n-s}(U)=-(-1)^{s(n-(p+1))}\det
U_{\\{p\\}\cup[p+2,s+p+2]}^{[n-s-1,n]},\ \ s\in[0,n-p-2].$ (5.6)

Applying $(\mathcal{F}^{c})^{*}$ to (5.6), we see that

$(\mathcal{F}^{c})^{*}[\tilde{h}^{\prime}_{p+1,n-s}(U)]=-(-1)^{s(n-(p+1))}\det
U_{\\{p\\}\cup[p+2,s+p+2]}^{[n-s-1,n]}.$ (5.7)

Since $\mathcal{Q}$ is a quasi-isomorphism, we conclude that $\Psi$ contains
the variables (5.5) as well. In particular, $\Psi$ and $\tilde{\Psi}$ differ
only in the variable $h_{p+1,p+1}$. ∎

For the next lemma, let us mention that the frozen variable $h_{p+1,p+1}$ in
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma})$ is given by

$h_{p+1,p+1}(U)=\det U^{[p+1,n]}_{[p+1,n]}\det U^{[q+1,n]}_{[q+1,n]}+\det
U^{[p+1,n]}_{\\{p\\}\cup[p+2,n]}\det U^{\\{q\\}\cup[q+2,n]}_{[q+1,n]}.$ (5.8)

######  Lemma 5.2.

The mutation of $\Psi$ at the marked variable $\psi_{\square}$ produces the
variable

$\psi_{\square}^{\prime}(U)=(-1)^{\varepsilon_{\square}}\left[\det
U^{[p+1,n]}_{[p+1,n]}\det U_{I_{\square}}^{[q+1,n]}+\det
U^{[p+1,n]}_{\\{p\\}\cup[p+2,n]}\det
U^{\\{q\\}\cup[q+2,n]}_{I_{\square}}\right]$ (5.9)

where

$\varepsilon_{\square}:=\begin{cases}n-q\ &\text{if}\ q-p\neq 1,\\\ n-q+1\
&\text{if}\ q-p=1;\end{cases}\ \ \ \ I_{\square}:=\begin{cases}[q,n-1]\
&\text{if}\ q-p\neq 1,\\\ \\{q-1\\}\cup[q+1,n-1]\ &\text{if}\
q-p=1.\end{cases}$ (5.10)

###### Proof.

_Case $q-p=1$, $q\leq n-2$._ The mutation relation for $\psi_{\square}$ reads

$\psi_{\square}\psi_{\square}^{\prime}=(h_{p+1,p+2}^{\prime})^{2}h_{q+1,q+2}+h_{p+1,p+3}^{\prime}h_{p+1,p+1}.$
(5.11)

Expanding the right-hand side, we see that

$\begin{split}(-1)^{n-p}&\psi_{\square}(U)\psi_{\square}^{\prime}(U)=\left[\det
U^{[p+1,n]}_{\\{p\\}\cup[p+2,n]}\right]^{2}\det U^{[p+3,n]}_{[p+2,n-1]}+\\\
+&\det U^{[p+2,n]}_{\\{p\\}\cup[p+2,n-1]}\left[\det U^{[p+1,n]}_{[p+1,n]}\det
U^{[q+1,n]}_{[q+1,n]}+\det U^{\\{q\\}\cup[q+2,n]}_{[q+1,n]}\det
U^{[p+1,n]}_{\\{p\\}\cup[p+2,n]}\right].\end{split}$ (5.12)

Applying the Desnanot-Jacobi identity (see Proposition C.3) to the matrix
$A:=U^{[p+1,n]}_{\\{p\\}\cup[p+2,n]}$, we see that

$\begin{split}\det U^{[p+1,n]}_{\\{p\\}\cup[p+2,n]}\det
U^{[p+3,n]}_{[p+2,n-1]}+\det&U^{[p+2,n]}_{\\{p\\}\cup[p+2,n-1]}\det
U^{\\{p+1\\}\cup[p+3,n]}_{[p+2,n]}=\\\ =&\det U^{[p+2,n]}_{[p+2,n]}\det
U^{\\{p+1\\}\cup[p+3,n]}_{\\{p\\}\cup[p+2,n-1]}.\end{split}$ (5.13)

Now, combining the first and the third terms in (5.12) via (5.13), and then
dividing both sides by $(-1)^{n-p}\det U^{[q+1,n]}_{[q+1,n]}$, formula (5.9)
follows.

_Case $q-p=1$, $q=n-1$._ In this case, the mutation at $\psi_{\square}$ reads

$\psi_{\square}\psi_{\square}^{\prime}=h_{n-2,n}h_{n-1,n-1}+(h_{n-1,n}^{\prime})^{2}.$
(5.14)

Expanding the latter relation, we see that

$\begin{split}\psi_{\square}(U)\psi_{\square}^{\prime}(U)=&u_{n-2,n}\left[\det
U^{[n-1,n]}_{[n-1,n]}u_{nn}+\det
U^{[n-1,n]}_{\\{n-2\\}\cup\\{n\\}}u_{n-1,n}\right]+\left[\det
U^{[n-1,n]}_{\\{n-2\\}\cup\\{n\\}}\right]^{2}=\\\
=&u_{nn}\left[u_{n-2,n-1}\det U^{[n-1,n]}_{\\{n-2\\}\cup\\{n\\}}+u_{n-2,n}\det
U^{[n-1,n]}_{[n-1,n]}\right].\end{split}$ (5.15)

Dividing both sides by $\psi_{\square}(U)=u_{nn}$, we obtain (5.9).

_Case $q-p\neq 1$._ In this case, the mutation at $\psi_{\square}$ reads

$\psi_{\square}\psi_{\square}^{\prime}=h_{p+1,p+2}^{\prime}h_{qq}h_{q+1,q+2}+h_{q,q+1}h_{p+1,p+1}.$
(5.16)

Expanding the right-hand side, we see that

$\begin{split}(-1)^{n-q}\psi_{\square}(U)&\psi_{\square}^{\prime}(U)=\det
U^{[p+1,n]}_{\\{p\\}\cup[p+2,n]}\det U^{[q,n]}_{[q,n]}\det
U^{[q+2,n]}_{[q+1,n-1]}+\\\ +&\det U^{[q+1,n]}_{[q,n-1]}\left[\det
U^{[p+1,n]}_{[p+1,n]}\det U^{[q+1,n]}_{[q+1,n]}+\det
U^{[p+1,n]}_{\\{p\\}\cup[p+2,n]}\det
U^{\\{q\\}\cup[q+2,n]}_{[q+1,n]}\right].\end{split}$ (5.17)

Applying the Desnanot-Jacobi identity (see Proposition C.3) to the matrix
$A:=U^{[q,n]}_{[q,n]}$, we see that

$\det U^{[q,n]}_{[q,n]}\det U^{[q+2,n]}_{[q+1,n-1]}+\det
U^{[q+1,n]}_{[q,n-1]}\det U^{\\{q\\}\cup[q+2,n]}_{[q+1,n]}=\det
U^{[q+1,n]}_{[q+1,n]}\det U^{\\{q\\}\cup[q+2,n]}_{[q,n-1]}.$ (5.18)

Combining the first and the third terms in (5.17) with (5.18) and dividing
both sides by $(-1)^{n-q}\psi_{\square}$, we obtain formula (5.9). ∎

######  Lemma 5.3.

The regular functions $\psi_{\square}$ and $\psi^{\prime}_{\square}$ are
coprime.

###### Proof.

Evidently, $\psi_{\square}$ is irreducible, so it suffices to show that
$\psi^{\prime}_{\square}(U)$ is not divisible by $\psi_{\square}(U)$. If
$p+1<q+1$, then $\psi_{\square}^{\prime}$ is generically nonzero on the
variety given by $u_{n,q+1}=u_{n,q+2}=\cdots=u_{nn}=0$, whereas
$\psi_{\square}$ vanishes. If $p+1>q+1$, then let $V$ be the variety given by
the equations

$u_{i,q+1}-u_{i,q+2}=0,\ \ i\in[q+1,n].$ (5.19)

We see that $\psi_{\square}(U)|_{V}=0$; on the other hand,

$\psi^{\prime}_{\square}(U)|_{V}=(-1)^{\varepsilon_{\square}}\det
U^{[p+1,n]}_{\\{p\\}\cup[p+2,n]}\det U^{\\{q\\}\cup[q+2,n]}_{[q,n-1]},$ (5.20)

which is generically nonzero on $V$. Thus $\psi_{\square}$ and
$\psi_{\square}^{\prime}$ are coprime. ∎

######  Lemma 5.4.

Let $\mathcal{F}$ be a field of characteristic zero, and let $\alpha$ and
$\beta$ be distinct transcendental elements over $\mathcal{F}$ such that
$\mathcal{F}(\alpha)=\mathcal{F}(\beta)$. If there is a relation

$\sum_{k=1}^{m}(\alpha^{k}-\beta^{k})p_{k}=0$ (5.21)

for $p_{k}\in\mathcal{F}$, then all $p_{k}=0$.

###### Proof.

See [31, Lemma 5.15]. ∎

######  Lemma 5.5.

For any $f\in\mathcal{O}(\operatorname{GL}_{n})$, if $\mathcal{Q}^{*}(f)=f$,
then $f\in\mathcal{L}_{\mathbb{C}}(\Psi^{\prime})$.

###### Proof.

Since
$\mathcal{O}(\operatorname{GL}_{n})\subseteq\mathcal{L}_{\mathbb{C}}(\tilde{\Psi})$,
one can find a sequence of elements
$f_{k}\in\mathcal{L}_{\mathbb{C}}(\tilde{\Psi})$ that do not involve the
variable $h_{p+1,p+1}$, such that

$f(U)=\sum_{k\geq 0}f_{k}(U)[\tilde{h}_{p+1,p+1}(U)]^{k}.$ (5.22)

Note that $\tilde{h}_{p+1,p+1}$ is frozen in
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma}_{\text{std}})$, so the expression
is polynomial in $\tilde{h}_{p+1,p+1}$. Applying $(\mathcal{F}^{c})^{*}$ to
(5.22) and using Lemma 5.1 together with the assumption
$(\mathcal{F}^{c})^{*}(f)=f$ yields

$f(U)=\sum_{k\geq
0}f_{k}(U)\left[\frac{h_{p+1,p+1}(U)}{\psi_{\square}(U)}\right]^{k}.$ (5.23)

Subtracting (5.22) from (5.23) and invoking Lemma 5.4, we conclude that
$f(U)=f_{0}(U)$. Since $\tilde{\psi}_{\square}(U)$ is frozen in
$\mathcal{GC}_{h}^{\dagger}(\tilde{\mathbf{\Gamma}})$, it enters $f_{0}(U)$
only with nonnegative exponents, so one can rewrite it in terms of a fraction
with $\psi_{\square}^{\prime}$ in the denominator. Thus
$f\in\mathcal{L}_{\mathbb{C}}(\Psi^{\prime})$. ∎

######  Proposition 5.6.

Under the setup of the current subsection, the following statements hold:

  1. 1)

All the variables in $\Psi$ are regular and irreducible;

  2. 2)

For any cluster variable $\psi\in\Psi$, its mutation $\psi^{\prime}$ is
regular and coprime with $\psi$;

  3. 3)

The generalized cluster structures
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma},\operatorname{GL}_{n})$ and
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma},\operatorname{SL}_{n})$ are
complete.

###### Proof.

By Lemma 5.3, the variable $\psi_{\square}^{\prime}$ is regular and coprime
with $\psi_{\square}$ in the ring $\mathcal{O}(\operatorname{GL}_{n})$. By
Proposition 2.28, it remains to show that
$\mathcal{O}(\operatorname{GL}_{n})\subseteq\mathcal{L}_{\mathbb{C}}(\Psi^{\prime})$.
Since
$\mathcal{O}(\operatorname{GL}_{n})\subseteq\mathcal{L}_{\mathbb{C}}(\tilde{\Psi})$,
it suffices to show that
$\mathcal{L}_{\mathbb{C}}(\tilde{\Psi})\subseteq\mathcal{L}_{\mathbb{C}}(\Psi^{\prime})$;
in turn, due to Lemma 5.1, proving the latter inclusion reduces to showing
that $\det U^{[p+1,n]}_{[p+1,n]}\in\mathcal{L}_{\mathbb{C}}(\Psi^{\prime})$.
First, let us notice that

$(\mathcal{F}^{c})^{*}\left(\det
U^{\\{q\\}\cup[q+2,n]}_{I_{\square}}\right)=\det
U^{\\{q\\}\cup[q+2,n]}_{I_{\square}},$ (5.24)

where $I_{\square}$ is given by (5.10). By Lemma 5.5, $\det
U^{\\{q\\}\cup[q+2,n]}_{I_{\square}}\in\mathcal{L}_{\mathbb{C}}(\Psi^{\prime})$.
Second, notice that $\det U^{[q+1,n]}_{I_{\square}}$ is a cluster variable in
$\Psi^{\prime}$ (hence an invertible element of
$\mathcal{L}_{\mathbb{C}}(\Psi^{\prime})$), for

$\det U^{[q+1,n]}_{I_{\square}}=\begin{cases}(-1)^{n-q}h_{q,q+1}(U)\
&\text{if}\ q-p\neq 1;\\\ (-1)^{n-p}h^{\prime}_{p+1,p+3}(U)\ &\text{if}\
q-p=1,\ p\leq n-3;\\\ (-1)^{n-p}h_{pn}(U)\ &\text{if}\ q-p=1,\
p=n-2.\end{cases}$ (5.25)

It follows that one can solve the equation (5.9) for $\det
U^{[p+1,n]}_{[p+1,n]}$ in the ring $\mathcal{L}_{\mathbb{C}}(\Psi^{\prime})$.
Due to the discussion in the beginning of the proof, we finally conclude that
the statement holds. ∎

####  5.1.2 The case of $|\Gamma_{1}|=|\Gamma_{2}|=1$ (initial cluster).

Although in the previous subsection we proved
$\bar{\mathcal{A}}(\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma},\operatorname{GL}_{n}))=\mathcal{O}(\operatorname{GL}_{n})$
for $\mathbf{\Gamma}=(\Gamma_{1},\Gamma_{2},\gamma)$ with $|\Gamma_{1}|=1$, we
proved regularity, irreducibility and coprimality with mutations not in the
initial extended cluster $\Psi_{0}$, but in a certain shifted extended cluster
$\Psi$. According to Proposition 2.28, in order to claim the same properties
for $\Psi_{0}$, we need to verify them for the marked variable in $\Psi_{0}$.
As above, let us assume that $\Gamma_{1}=\\{p\\}$ and $\Gamma_{2}=\\{q\\}$,
and set $\psi_{\square}:=h_{q+1,q+1}$.

######  Lemma 5.7.

The mutation of $\Psi_{0}$ at the marked variable $\psi_{\square}$ produces
the variable

$\psi_{\square}^{\prime}(U)=\begin{cases}(-1)^{p+q+1}\left[u_{p+1,n}\det
U^{[q+1,n]}_{[q,n-1]}+u_{pn}\det U^{\\{q\\}\cup[q+2,n]}_{[q,n-1]}\right]\
&q\neq p+1;\\\\[10.0pt] \det U^{[q+1,n]}_{[q+1,n]}\left[u_{p+1,n}\det
U^{[q+1,n]}_{[q,n-1]}+u_{pn}\det
U^{\\{q\\}\cup[q+2,n]}_{[q,n-1]}\right]+\\\\[7.0pt] +\det
U^{\\{q\\}\cup[q+2,n]}_{[q+1,n]}\left[u_{p+1,n}\det
U^{[q+1,n]}_{\\{q-1\\}\cup[q+1,n-1]}+u_{pn}\det
U^{\\{q\\}\cup[q+2,n]}_{\\{q-1\\}\cup[q+1,n-1]}\right]\ &q=p+1.\end{cases}$

###### Proof.

The exchange relation is given by

$\psi_{\square}\psi_{\square}^{\prime}=h_{q,q+1}h_{p+1,n}+h_{qq}h_{pn}h_{q+1,q+2}.$
(5.26)

Let us study two cases:

_Case $q\neq p+1$._ Writing out all the functions, the relation becomes

$\begin{split}(-1)^{p+q+1}\psi_{\square}(U)\psi_{\square}^{\prime}(U)=\det&U^{[q+1,n]}_{[q,n-1]}u_{p+1,n}\det
U^{[q+1,n]}_{[q+1,n]}+\\\ +&u_{pn}\left(\det U^{[q+1,n]}_{[q,n-1]}\det
U^{\\{q\\}\cup[q+2,n]}_{[q+1,n]}+\det U^{[q,n]}_{[q,n]}\det
U^{[q+2,n]}_{[q+1,n-1]}\right).\end{split}$ (5.27)

Applying the Desnanot-Jacobi relation (see Proposition C.3) to the last term,
we see that

$\det U^{[q+1,n]}_{[q,n-1]}\det U^{\\{q\\}\cup[q+2,n]}_{[q+1,n]}+\det
U^{[q,n]}_{[q,n]}\det U^{[q+2,n]}_{[q+1,n-1]}=\det U^{[q+1,n]}_{[q+1,n]}\det
U^{\\{q\\}\cup[q+2,n]}_{[q,n-1]};$

hence, after dividing both sides by $\det U^{[q+1,n]}_{[q+1,n]}$ in (5.27),
the result follows.

_Case $q=p+1$._ Writing out all functions and rearranging the terms, the
relation becomes

$\begin{split}\psi_{\square}^{\prime}(U)=u_{p+1,n}&\left[\det
U^{[q+1,n]}_{[q+1,n]}\det U^{[q+1,n]}_{[q,n-1]}+\det
U^{[q+1,n]}_{\\{q-1\\}\cup[q+1,n-1]}\det
U^{\\{q\\}\cup[q+2,n]}_{[q+1,n]}\right]+\\\ +u_{pn}&\left[\det
U^{[q+1,n]}_{[q,n-1]}\det U^{\\{q\\}\cup[q+2,n]}_{[q+1,n]}+\det
U^{[q,n]}_{[q,n]}\det U^{[q+2,n]}_{[q+1,n-1]}\right]+\\\ +&\frac{u_{pn}\det
U^{\\{q\\}\cup[q+2,n]}_{[q+1,n]}}{\det U^{[q+1,n]}_{[q+1,n]}}\cdot
A\end{split}$ (5.28)

where $A$ is given by

$A=\det U^{[q+1,n]}_{\\{q-1\\}\cup[q+1,n-1]}\det
U^{\\{q\\}\cup[q+2,n]}_{[q+1,n]}+\det U^{[q,n]}_{\\{q-1\\}\cup[q+1,n]}\det
U^{[q+2,n]}_{[q+1,n-1]}.$ (5.29)

The second term in (5.28) can be simplified via the Desnanot-Jacobi identity
as

$\det U^{[q+1,n]}_{[q,n-1]}\det U^{\\{q\\}\cup[q+2,n]}_{[q+1,n]}+\det
U^{[q,n]}_{[q,n]}\det U^{[q+2,n]}_{[q+1,n-1]}=\det U^{[q+1,n]}_{[q+1,n]}\det
U^{\\{q\\}\cup[q+2,n]}_{[q,n-1]};$ (5.30)

likewise, formula (5.29) can be simplified as

$\begin{split}\det U^{[q+1,n]}_{\\{q-1\\}\cup[q+1,n-1]}\det
U^{\\{q\\}\cup[q+2,n]}_{[q+1,n]}+\det&U^{[q,n]}_{\\{q-1\\}\cup[q+1,n]}\det
U^{[q+2,n]}_{[q+1,n-1]}=\\\ =&\det U^{[q+1,n]}_{[q+1,n]}\det
U^{\\{q\\}\cup[q+2,n]}_{\\{q-1\\}\cup[q+1,n-1]}.\end{split}$ (5.31)

Combining the above terms, the formula for $\psi_{\square}^{\prime}$ follows.
∎

######  Proposition 5.8.

Under the setup of the current subsection, the following statements hold:

  1. 1)

All the variables in $\Psi_{0}$ are regular and irreducible;

  2. 2)

For any cluster variable $\psi\in\Psi_{0}$, its mutation $\psi^{\prime}$ is
regular and coprime with $\psi$.

###### Proof.

According to Proposition 2.28, it is enough to verify the above statements
only for $\psi_{\square}$. Evidently, $\psi_{\square}$ is irreducible and
$\psi_{\square}^{\prime}$ is regular (see Lemma 5.7). If $q\neq p+1$, we see
that $\psi_{\square}$ vanishes on the variety given by
$u_{n,q+1}=u_{n,q+2}=\cdots=u_{nn}=0$, whereas $\psi_{\square}^{\prime}(U)$ is
generically nonzero. If $q=p+1$, then it suffices to check that

$u_{p+1,n}\det U^{[q+1,n]}_{\\{q-1\\}\cup[q+1,n-1]}+u_{pn}\det
U^{\\{q\\}\cup[q+2,n]}_{\\{q-1\\}\cup[q+1,n-1]}$ (5.32)

is not divisible by $\psi_{\square}(U)$, which immediately follows from the
same argument. ∎

####  5.1.3 Completeness for any BD triple

In this subsection, we finalize the proofs of Part 2 of Theorem 1.5 and Part 2
of Theorem 1.6.

######  Proposition 5.9.

Let $\mathbf{\Gamma}:=(\Gamma_{1},\Gamma_{2},\gamma)$ be a BD triple of type
$A_{n-1}$ and $G\in\\{\operatorname{SL}_{n},\operatorname{GL}_{n}\\}$. Then
the following statements hold:

  1. 1)

The generalized cluster structure
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma},G)$ is regular;

  2. 2)

All the variables in the initial extended cluster of
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma},G)$ are irreducible (as elements
of $\mathcal{O}(G)$) and (for cluster variables) coprime with their mutations;

  3. 3)

The generalized cluster structure
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma},G)$ is complete.

###### Proof.

The regularity of the initial extended cluster follows from Proposition 6.4.
As in the previous papers [23, 31], we run an induction on the size
$|\Gamma_{1}|$ for a fixed $n$. For $|\Gamma_{1}|=0$, the statements were
proved in [20, Theorem 3.8, Theorem 3.10]; for $|\Gamma_{1}|=1$, the
statements are contained in Proposition 5.6 and Proposition 5.8. For
$|\Gamma_{1}|>1$, the statements follow from the existence of complementary
birational quasi-isomorphisms $\mathcal{G}$ and Proposition 2.30. ∎

###  5.2 Toric action

In this subsection, we prove Part 3 of Theorems 1.5 and 1.6. Given a BD triple
$\mathbf{\Gamma}:=(\Gamma_{1},\Gamma_{2},\gamma)$ of type $A_{n-1}$, let us
recall that

$\mathfrak{h}_{\mathbf{\Gamma}}=\\{h\in\mathfrak{h}^{s}\ |\
\alpha(h)=\beta(h)\ \text{if}\ \gamma^{j}(\alpha)=\beta\ \text{for some}\
j\\}.$ (5.33)

where $\mathfrak{h}^{s}$ is the Cartan subalgebra of
$\operatorname{sl}_{n}(\mathbb{C})$ (relative the a priori fixed root system).
The dimension of $\mathfrak{h}_{\mathbf{\Gamma}}$ is equal to
$k_{\mathbf{\Gamma}}:=|\Pi\setminus\Gamma_{1}|$. Let
$\mathcal{H}_{\mathbf{\Gamma}}$ be the connected subgroup of
$\operatorname{SL}_{n}$ with Lie algebra $\mathfrak{h}_{\mathbf{\Gamma}}$. We
let the group $\mathcal{H}_{\mathbf{\Gamma}}$ act upon $\operatorname{SL}_{n}$
by conjugation.

######  Lemma 5.10.

Let $\Gamma_{1}=\\{p\\}$ and $\Gamma_{2}=\\{q\\}$ for some simple roots $p$
and $q$. Let $\psi_{\square}:=h_{q+1,q+1}$ be the marked variable. Then the
$y$-variable $y(\psi_{\square})$ is invariant with respect to the action by
$\mathcal{H}_{\mathbf{\Gamma}}$.

###### Proof.

Let us recall from Proposition 4.16 that $\mathcal{Q}$ is equivariant with
respect to the action by $\mathcal{H}_{\mathbf{\Gamma}}$. In the setup of the
lemma, $\mathcal{H}_{\mathbf{\Gamma}}$ is given by

$\mathcal{H}_{\mathbf{\Gamma}}=\\{T=\operatorname{diag}(t_{1},t_{2},\ldots,t_{n})\
|\ \det T=1,\ t_{p}t_{p+1}^{-1}t_{q+1}t_{q}^{-1}=1\\}.$ (5.34)

Applying $\mathcal{Q}^{*}$ to $y(\psi_{\square})$, we see that

$\mathcal{Q}^{*}(y(\psi_{\square}))=\frac{\tilde{h}_{qq}}{\tilde{h}_{q,q+1}}\frac{\tilde{h}_{pn}}{\tilde{h}_{p+1,n}}\frac{\tilde{h}_{q+1,q+2}}{\tilde{h}_{q+1,q+1}}.$
(5.35)

Now, it is a simple observation that $\tilde{h}_{q+1,q+1}(U)$ and
$\tilde{h}_{qq}(U)$ are invariant with respect to
$\mathcal{H}_{\mathbf{\Gamma}}$, as well as

$\displaystyle\tilde{h}_{pn}(TUT^{-1})=t_{p}t_{n}^{-1}\tilde{h}_{pn}(U),\ $
$\displaystyle\tilde{h}_{q+1,q+2}(TUT^{-1})=t_{q+1}t_{n}^{-1}\tilde{h}_{q+1,q+2}(U),$
(5.36) $\displaystyle\tilde{h}_{q,q+1}(TUT^{-1})=t_{q}t_{n}^{-1}\tilde{h}(U),\
$
$\displaystyle\tilde{h}_{p+1,n}(TUT^{-1})=t_{p+1}t_{n}^{-1}\tilde{h}_{p+1,n}(U).$

Therefore,

$\mathcal{Q}^{*}\left(y(\psi_{\square})(TUT^{-1})\right)=\mathcal{Q}^{*}\left(y(\psi_{\square})(U)\right)\frac{t_{p}t_{n}^{-1}t_{q+1}t_{n}^{-1}}{t_{q}t_{n}^{-1}t_{p+1}t_{n}^{-1}}=\mathcal{Q}^{*}\left(y(\psi_{\square})(U)\right).$
(5.37)

It follows that $y(\psi_{\square})$ is invariant with respect to the action by
$\mathcal{H}_{\mathbf{\Gamma}}$. ∎

######  Proposition 5.11.

Let $\mathbf{\Gamma}$ be a BD triple of type $A_{n-1}$. The following
statements hold:

  1. 1)

The action of $\mathcal{H}_{\mathbf{\Gamma}}$ upon $\operatorname{SL}_{n}$
induces a global toric action upon
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma},\operatorname{SL}_{n})$ of rank
$k_{\mathbf{\Gamma}}$;

  2. 2)

The action of $\mathcal{H}_{\mathbf{\Gamma}}\times\mathbb{C}^{*}$ upon
$\operatorname{GL}_{n}$ induces a global toric action upon
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma},\operatorname{GL}_{n})$ of rank
$k_{\mathbf{\Gamma}}+1$.

###### Proof.

The statement for $\mathbb{C}^{*}$ was already proved in Lemma 4.21, so let us
focus on $\mathcal{H}_{\mathbf{\Gamma}}$; the proof is based on Proposition
2.19. Let us notice that due to Proposition 4.16, all birational quasi-
isomorphisms $\mathcal{G}$ that were introduced in Sections 4.4 and 4.5 are
equivariant with respect to $\mathcal{H}_{\mathbf{\Gamma}}$. This implies that
$\mathcal{H}_{\mathbf{\Gamma}}$ induces a local toric action upon the initial
extended cluster of $\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma})$. The fact
that the action is of maximal rank follows from a standard argument181818Let
us briefly recall the argument. Assume that the toric action is of rank less
than maximal. This means that there exists a $1$-dimensional subgroup of
$\mathcal{H}_{\mathbf{\Gamma}}$ that leaves all cluster and frozen variables
from the initial extended cluster $\Psi_{0}$ invariant. However, since
$\mathcal{O}(\operatorname{GL}_{n})\subseteq\mathcal{L}_{\mathbb{C}}(\Psi_{0})$,
we see that the action leaves _all_ regular functions invariant. In other
words, it is trivial. that was utilized in [23, 31]. The invariance of the
Casimirs $\hat{p}_{ir}$ (which are the $c$-variables in our case) was proved
before in [20, Theorem 6.1]; therefore, it remains to show that all the
$y$-variables in the initial extended cluster $\Psi_{0}$ of
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma})$ are invariant with respect to
the action by $\mathcal{H}_{\mathbf{\Gamma}}$.

We proceed by induction on the size $|\Gamma_{1}|$. If $|\Gamma_{1}|=0$, then
the statement was already proved in [20, Theorem 6.1]. Let $\mathbf{\Gamma}$
be a BD triple of type $A_{n-1}$ and $\tilde{\mathbf{\Gamma}}$ be obtained
from $\mathbf{\Gamma}$ via removal of a pair of roots. Let $\psi_{\square}$ be
the corresponding marked variable. Notice that
$\mathfrak{h}_{\tilde{\mathbf{\Gamma}}}\supset\mathfrak{h}_{\mathbf{\Gamma}}$.
If $|\Gamma_{1}|=1$ and $y(\psi)$ is a $y$-variable of
$\psi\neq\psi_{\square}$, we see that

$\mathcal{G}^{*}\left(y(\psi)(TUT^{-1})\right)=y(\psi)(T\mathcal{G}(U)T^{-1})=y(\psi)(\mathcal{G}(TUT^{-1}))=y(\tilde{\psi})(U).$
(5.38)

If $\psi=\psi_{\square}$, then the invariance follows from Lemma 5.10. The
case of $|\Gamma_{1}|>1$ follows from the existence of complementary
birational quasi-isomorphisms $\mathcal{G}$ (see Section 4.4). ∎

###  5.3 Compatibility

In this subsection, we use the Poisson properties of the birational quasi-
isomorphism $\mathcal{Q}$ in order to prove Part 4 of Theorem 1.5 and Theorem
1.6; that is, the generalized cluster structure
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma})$ is compatible with
$\pi^{\dagger}_{\mathbf{\Gamma}}$ for any BD triple $\mathbf{\Gamma}$ of type
$A_{n-1}$.

######  Lemma 5.12.

Let $\mathbf{\Gamma}:=(\Gamma_{1},\Gamma_{2},\gamma)$ be a BD triple such that
$\Gamma_{1}=\\{p\\}$, $\Gamma_{2}=\\{q\\}$ for some simple roots $p$ and $q$.
Then for any cluster or frozen variable $\psi$ from the initial extended
cluster of $\mathcal{GC}^{\dagger}(\mathbf{\Gamma})$,

$\\{\log y(\psi_{\square}),\log\psi\\}=\begin{cases}1\ &\text{if}\
\psi=h_{q+1,q+1},\\\ 0\ &\text{otherwise}\end{cases}$ (5.39)

where $\psi_{\square}:=h_{q+1,q+1}$ is the marked variable.

###### Proof.

Let us apply the birational quasi-isomorphism $\mathcal{Q}$ and prove an
equivalent statement for the trivial BD triple. We see that

$\mathcal{Q}^{*}(y(\psi_{\square}))=\frac{\tilde{h}_{qq}}{\tilde{h}_{q,q+1}}\frac{\tilde{h}_{pn}}{\tilde{h}_{p+1,n}}\frac{\tilde{h}_{q+1,q+2}}{\tilde{h}_{q+1,q+1}}.$
(5.40)

Let us first consider the case when $\psi$ is an $h$-variable. If $h_{1}$ and
$h_{2}$ are two functions on $\operatorname{GL}_{n}$ such that
$h_{i}(UN_{-})=h_{i}(U)$, where $N_{-}$ is a unipotent lower triangular matrix
and $i\in\\{1,2\\}$, then the Poisson bracket
$\\{\cdot,\cdot\\}_{\text{std}}^{\dagger}$ between them is given by

$\\{h_{1},h_{2}\\}_{\text{std}}^{\dagger}=\langle\pi_{>}\nabla_{U}^{R}h_{1},\nabla_{U}^{R}h_{2}\rangle+\langle
R_{0}[\nabla_{U}h_{1},U],[\nabla_{U}h_{2},U]\rangle-\langle\pi_{0}[\nabla_{U}h_{1},U],\nabla_{U}h_{2}U\rangle.$
(5.41)

It follows from Lemma C.4 that

$\langle\pi_{>}\left(\nabla_{U}^{R}\log\frac{\tilde{h}_{pn}}{\tilde{h}_{p+1,n}}\right),\nabla_{U}^{R}\log\tilde{h}_{ks}\rangle=-\chi_{n-s=p-k}$
(5.42)
$\langle\pi_{>}\left(\nabla_{U}^{R}\log\frac{\tilde{h}_{qq}}{\tilde{h}_{q,q+1}}\frac{\tilde{h}_{q+1,q+2}}{\tilde{h}_{q+1,q+1}}\right),\nabla_{U}^{R}\log\tilde{h}_{ks}\rangle=-\chi_{k\leq
q}\cdot\chi_{s=q+1}$ (5.43)

where $\chi_{(\cdot)}\in\\{0,1\\}$ is an indicator that returns true or false
depending on whether the corresponding condition is satisfied. It follows from
Lemma C.5 that

$\pi_{0}\left(\nabla_{U}^{L}\log\mathcal{Q}^{*}y(\psi_{\square})\right)=e_{qq}-e_{q+1,q+1},\
\
\pi_{0}\left(\nabla_{U}^{R}\mathcal{Q}^{*}y(\psi_{\square})\right)=e_{pp}-e_{p+1,p+1};$
(5.44)

hence,

$\begin{split}R_{0}[\nabla_{U}\log\mathcal{Q}^{*}&y(\psi_{\square}),U]=R_{0}(e_{qq}-e_{q+1,q+1}-(e_{pp}-e_{p+1,p+1}))=\\\
=&-R_{0}(1-\gamma)(e_{pp}-e_{p+1,p+1})=-e_{pp}+e_{p+1,p+1}.\end{split}$ (5.45)

Let us set $\Delta(i,j):=\sum_{k=i}^{j}e_{kk}$. We see that

$\begin{split}\\{\log\mathcal{Q}^{*}y(\psi_{\square}),\log\hat{h}_{ks}\\}_{\text{std}}^{\dagger}=&-\chi_{n-s=p-k}-\chi_{k\leq
q}\cdot\chi_{s=q+1}+\\\ +&\langle
e_{pp}-e_{p+1,p+1},\Delta(k,n-s+k)\rangle-\langle
e_{qq}-e_{q+1,q+1},\Delta(s,n)\rangle=\\\ =&-\chi_{k=p+1}+(1-\chi_{k\leq
q})\chi_{s=q+1}=-\chi_{k=p+1}+\chi_{k=q+1}\chi_{s=q+1}.\end{split}$ (5.46)

Now let us recall that $\mathcal{Q}^{*}({h_{ks}})=\tilde{h}_{ks}$ if $k\neq
p+1$ and $\mathcal{Q}^{*}(h_{p+1,s})=\tilde{h}_{p+1,s}\tilde{h}_{q+1,q+1}$.
Together with formula (5.46), the result follows in the case $\psi$ is an
$h$-function. If $\psi=\varphi_{kl}$ for some $\varphi$-function
$\varphi_{kl}$, then for any regular function $h$ such that $h(UN_{-})=h(U)$,

$\\{\log h,\log\varphi_{kl}\\}_{\text{std}}^{\dagger}=\langle
R_{0}\pi_{0}[\nabla_{U}\log
h,U],[\nabla_{U}\log\varphi_{kl},U]\rangle-\langle\pi_{0}\nabla_{U}^{R}\log
h,[\nabla_{U}\log\varphi_{kl},U]\rangle.$ (5.47)

Setting $h:=\log\mathcal{Q}^{*}y(\psi_{\square})$ and applying
$\pi_{0}[\nabla_{U}\log\varphi_{kl},U]=\mathrm{const}$ together with (5.44)
and (5.45), we obtain that

$\\{\log\mathcal{Q}^{*}y(\psi_{\square}),\log\varphi_{kl}\\}_{\text{std}}^{\dagger}=0.$
(5.48)

Thus the result follows. ∎

######  Proposition 5.13.

Let $\mathbf{\Gamma}$ be a BD triple of type $A_{n-1}$. For any choice of
$r_{0}$ relative the triple $\mathbf{\Gamma}$, the generalized cluster
structure $\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma})$ is compatible with
$\pi_{\mathbf{\Gamma}}^{\dagger}$.

###### Proof.

Let us fix $n$ and run an induction on the size $|\Gamma_{1}|$. For
$|\Gamma_{1}|=0$, the result was proved in [20, Theorem 3.8]. Let
$\mathbf{\Gamma}$ be any BD triple of type $A_{n-1}$ and
$\tilde{\mathbf{\Gamma}}$ be the BD triple obtained from $\mathbf{\Gamma}$ via
removal of $p\in\Gamma_{1}$ and $q\in\Gamma_{2}$. Let
$\mathcal{G}:(G,\pi_{\tilde{\mathbf{\Gamma}}}^{\dagger})\dashrightarrow(G,\pi_{\mathbf{\Gamma}}^{\dagger})$
be the birational quasi-isomorphism constructed in Section 4.5, where
$G\in\\{\operatorname{SL}_{n},\operatorname{GL}_{n}\\}$. Let
$\tilde{\Psi}_{0}$ and $\Psi_{0}$ be the initial extended clusters of
$\mathcal{GC}_{h}^{\dagger}(\tilde{\mathbf{\Gamma}},G)$ and
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma},G)$, and let
$\psi_{\square}\in\Psi_{0}$ be the marked variable. Since
$\tilde{\mathbf{\Gamma}}\prec\mathbf{\Gamma}$, any $r_{0}$ component for
$\pi_{\mathbf{\Gamma}}^{\dagger}$ is also an $r_{0}$ component for
$\pi_{\tilde{\mathbf{\Gamma}}}^{\dagger}$; hence, we can choose the same
$r_{0}$ to make $\mathcal{G}$ a Poisson isomorphism (the result of Proposition
4.17). Now, since $\tilde{\Psi}_{0}$ is assumed to be log-canonical and
$\mathcal{G}$ is a Poisson isomorphism, $\Psi_{0}$ is log-canonical as well.
Let $\psi_{1}\in\Psi_{0}\setminus\\{\psi_{\square}\\}$ be a cluster variable
and $\psi_{2}\in\Psi_{0}$ be any cluster or frozen variable. Since

$\\{\log
y(\tilde{\psi}_{1}),\log\tilde{\psi}_{2}\\}=\chi_{\tilde{\psi}_{1}=\tilde{\psi}_{2}},$
(5.49)

an application of $(\mathcal{G}^{*})^{-1}$ yields

$\\{\log y(\psi_{1}),\log\psi_{2}\\}=\chi_{\psi_{1}=\psi_{2}}.$ (5.50)

If $|\Gamma_{1}|=1$, it follows from Lemma 5.12 (which treats the case of
$\psi_{1}=\psi_{\square}$) and Proposition 2.16 that
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma},G)$ is compatible with
$\pi_{\mathbf{\Gamma}}^{\dagger}$. For $|\Gamma_{1}|>1$, to verify (5.50) for
the marked variables, one can use another birational quasi-isomorphism
$\mathcal{G}$ whose marked variables are disjoint from the former one. Thus
the statement of the proposition holds. ∎

##  6 Further properties of $h$-variables

In this section, we provide several additional formulas for the $h$-variables,
including their explicit expansion in terms of minors of an element
$U\in\operatorname{GL}_{n}$. Also, we study the Poisson properties of the
frozen $h$-variables and prove Proposition 1.8.

###  6.1 Preliminaries

In this subsection, we derive preliminary formulas that we will need in order
to compute the $h$-variables explicitly in terms of minors of $U$.

##### Notation.

Let $A\subseteq[1,n]$, $i\in[1,n]$. Set $A^{\leq}(i):=A\cap[1,i]$,
$A^{\geq}(i):=A\cap[i,n]$, and for a number $t\in[1,|A|]$ denote by ${A\choose
t}$ the set of all subsets of $A$ of size $t$.

######  Lemma 6.1.

Let $\mathbf{\Gamma}:=(\Gamma_{1},\Gamma_{2},\gamma)$ be a BD triple of type
$A_{n-1}$. Let $i\in[1,n]$, $r\in[i,n]$, $k\geq 0$ and $J\subseteq[1,n]$,
$|J|=r-i+1$. Let $\Delta$ be the $X$-run that contains $i$. For
$i-1\in\Gamma_{1}$, set

$L_{i,r}:=\begin{cases}[\bar{\gamma}(i)_{+}(\Gamma_{2})+1,n]\ &\text{if}\
r\notin\Delta\ \text{or}\ \gamma|_{\Delta\cap\Gamma_{2}}<0;\\\
[\bar{\gamma}(r)+1,n]\ &\text{if}\ r\in\Delta\ \text{and}\
\gamma|_{\Delta\cap\Gamma_{1}}>0;\end{cases}$ (6.1)
$D_{i,I}(U):=\begin{cases}\dfrac{\det U^{\check{\gamma}(I)\cup
L_{i,r}}_{[\bar{\gamma}(i),n]}}{\det
U^{[\bar{\gamma}(i),n]}_{[\bar{\gamma}(i),n]}}\ &\text{if}\
\gamma|_{\Delta\cap\Gamma_{1}}>0;\\\\[20.0pt] \dfrac{\det
U^{\check{\gamma}(I)\cup L_{i,r}}_{[\bar{\gamma}(i)+1,n]}}{\det
U^{[\bar{\gamma}(i)+1,n]}_{[\bar{\gamma}(i)+1,n]}}\ &\text{if}\
\gamma|_{\Delta\cap\Gamma_{1}}<0.\end{cases}$ (6.2)

Then for any $k\geq 0$,

$\det[\mathcal{F}_{k}(U)]_{[i,r]}^{J}=\begin{cases}\sum_{I\in{\Delta^{\leq}(r)\choose|\Delta\cap[i,r]|}}\mathcal{D}_{i,I}(\mathcal{F}_{k-1}(U))\cdot\det
U^{J}_{I\cup[i_{+}(\Gamma_{1})+1,r]}\ &\text{if}\
i-1\in\Gamma_{1},\\\\[12.0pt] \det U_{[i,r]}^{J}\
&\text{otherwise.}\end{cases}$ (6.3)

###### Proof.

By formula (4.9) (the definition of $\mathcal{F}_{k}(U)$) and the Cauchy-Binet
formula,

$\det[\mathcal{F}_{k}(U)]^{K}_{[i,r]}=\sum_{K,\
|K|=|i,r|}\det[\tilde{\gamma}^{*}(\mathcal{F}_{k-1}(U)_{-})]_{[i,r]}^{K}\det
U^{J}_{K}.$ (6.4)

Let us notice that the shape of the matrix
$[\tilde{\gamma}^{*}(U_{-})]_{[i,n]}^{[1,n]}$ is given by

$[\tilde{\gamma}^{*}(U_{-})]_{[i,n]}^{[1,n]}=\begin{bmatrix}0&[\tilde{\gamma}^{*}(U_{-})]_{[i,i_{+}(\Gamma_{1})]}^{\Delta}&0\\\
0&0&[\tilde{\gamma}^{*}(U_{-})]_{[i_{+}(\Gamma_{1})+1,n]}^{[i_{+}(\Gamma_{1})+1,n]}\end{bmatrix};$
(6.5)

therefore, $\det[\tilde{\gamma}^{*}(U_{-})]_{[i,r]}^{K}$ is nontrivial for a
generic $U\in\operatorname{GL}_{n}$ if and only if
$K=I\cup[i_{+}(\Gamma_{1})+1,r]$ where
$I\in{\Delta^{\leq}(r)\choose|\Delta\cap[i,r]|}$. Hence
$\det[\tilde{\gamma}^{*}(U_{-})]^{K}_{[i,r]}=\det[\tilde{\gamma}^{*}(U_{-})]^{I}_{\Delta\cap[i,r]}$.
Now, formula (6.2) follows from formulas (C.2)–(C.3) and the definitions of
$\bar{\gamma}$ and $\check{\gamma}$ given in Section 4.1. ∎

######  Lemma 6.2.

Let $\mathbf{\Gamma}=(\Gamma_{1},\Gamma_{2},\gamma)$ be a BD triple of type
$A_{n-1}$ and $S^{\gamma}(\alpha_{0}):=\\{\alpha_{i}\\}_{i\geq 0}^{m}$ be a
$\gamma$-string. Then for any $i\in[0,m]$, $r\geq\alpha_{m-i}+1$,
$J\subseteq[1,n]$, $|J|=|[\alpha_{m-i}+1,r]|$, and any $\ell\geq 0$,

$\det[\mathcal{F}(U)]_{[\alpha_{m-i}+1,r]}^{J}=\det[\mathcal{F}_{i+\ell}(U)]_{[\alpha_{m-i}+1,r]}^{J}.$
(6.6)

###### Proof.

Let us prove the statement by induction on $i$. For $i=0$, the statement is
contained in the second line of formula (6.3), so let us assume that the
statement holds for $i\geq 0$ and prove it for $i+1$. For each $t\in[0,m]$,
denote by $\Delta_{t}$ the $X$-run that contains $\alpha_{t}$. Notice that

$\displaystyle\bar{\gamma}(\alpha_{t}+1)$ $\displaystyle=\alpha_{t+1}+1\ $
$\displaystyle\text{if}\ \gamma|_{\Delta_{i}\cap\Gamma_{1}}>0,$ (6.7)
$\displaystyle\bar{\gamma}(\alpha_{t}+1)+1$ $\displaystyle=\alpha_{t+1}+1\ $
$\displaystyle\text{if}\ \gamma|_{\Delta_{i}\cap\Gamma_{1}}<0.$

For $I\in{\Delta_{m-i-1}^{\leq}(r)\choose|\Delta_{i}\cap[\alpha_{m-i-1},r]}$,
the latter formulas together with formula (6.2) imply that

$D_{\alpha_{m-i-1}+1,I}(U)=\frac{\det U^{\check{\gamma}(I)\cup
L_{\alpha_{m-i-1}+1,r}}_{[\alpha_{m-i}+1,n]}}{\det
U^{[\alpha_{m-i}+1,n]}_{[\alpha_{m-i}+1,n]}}$ (6.8)

regardless of the orientation of $\gamma$ on $\Delta_{m-i-1}$. Now, it follows
from formula (6.3), the assumption of the induction and formula (6.8) that

$\begin{split}\det[\mathcal{F}_{i+1+\ell}(U)]^{J}_{[\alpha_{m-i-1}+1,r]}=&\sum_{I\in{\Delta_{m-i-1}^{\leq}(r)\choose|\Delta_{m-i-1}\cap[\alpha_{m-i-1}+1,r]}}D_{\alpha_{m-i-1}+1,I}(\mathcal{F}_{i+\ell}(U))\det
U^{J}_{I\cup[(\alpha_{m-i-1})_{+}(\Gamma_{1})+1,r]}=\\\
=&\sum_{I\in{\Delta_{m-i-1}^{\leq}(r)\choose|\Delta_{m-i-1}\cap[\alpha_{m-i-1}+1,r]}}D_{\alpha_{m-i-1}+1,I}(\mathcal{F}(U))\det
U^{J}_{I\cup[(\alpha_{m-i-1})_{+}(\Gamma_{1})+1,r]}=\\\
=&\det[\mathcal{F}(U)]_{[\alpha_{m-i-1}+1,r]}^{J}.\end{split}$

Thus the statement holds. ∎

As a consequence of Lemma 6.2, the defining formula for the $h$-variables
(1.11) can be recast in the following way:

######  Proposition 6.3.

Let $\mathbf{\Gamma}:=(\Gamma_{1},\Gamma_{2},\gamma)$ be a BD triple of type
$A_{n-1}$ and $S^{\gamma}(\alpha_{0}):=\\{\alpha_{i}\\}_{i=0}^{m}$ be a
$\gamma$-string. For any $i\in[0,m]$ and $j\in[\alpha_{i}+1,n]$, the
$h$-variable $h_{\alpha_{i}+1,j}(U)$ can be written as

$h_{\alpha_{i}+1,j}(U)=(-1)^{\varepsilon_{\alpha_{i}+1,j}}\det[\mathcal{F}_{m-i}(U)]^{[j,n]}_{[\alpha_{i}+1,n-j+\alpha_{i}+1]}\prod_{t\geq
i+1}^{m}\det[\mathcal{F}_{m-t}(U)]^{[\alpha_{t}+1,n]}_{[\alpha_{t}+1,n]}$
(6.9)

where $\varepsilon_{\alpha_{i}+1,j}\in\\{0,1\\}$ is given by formula (1.12).

###### Proof.

The statement is a straightforward consequence of Lemma 6.2. ∎

###  6.2 Explicit expansion of the $h$-variables

In this subsection, we derive an explicit formula for the $h$-variables in
terms of minors of $U\in\operatorname{GL}_{n}$. Let us fix a BD triple
$\mathbf{\Gamma}:=(\Gamma_{1},\Gamma_{2},\gamma)$ and a $\gamma$-string
$S^{\gamma}(\alpha_{0}):=\\{\alpha_{i}\\}_{i=0}^{m}$. Define the following
data:

$\displaystyle\Delta_{i}$ $\displaystyle:=\Delta(\alpha_{i})=\text{the
}X\text{-run that contains }\alpha_{i},$ $\displaystyle 0\leq i\leq m;$ (6.10)
$\displaystyle s_{ij}$ $\displaystyle:=n-j+\alpha_{i}+1,$ $\displaystyle
i\in[0,m],\ j\in[\alpha_{i}+1,n];$ $\displaystyle K_{i}$
$\displaystyle:=[(\alpha_{i})_{+}(\Gamma_{1})+1,n],$ $\displaystyle
i\in[0,m-1];$ $\displaystyle L_{i}$
$\displaystyle:=[(\alpha_{i})_{+}(\Gamma_{2})+1,n],$ $\displaystyle
i\in[2,m];$ $\displaystyle L^{\prime}_{i}(j)$
$\displaystyle:=\begin{cases}[\max\check{\gamma}(\Delta_{i-1}^{\leq}(s_{i-1,j}))+1,n]\
&\text{if }\gamma|_{\Delta_{i}\cap\Gamma_{1}}>0;\\\
[(\alpha_{i})_{+}(\Gamma_{2})+1,n]\ &\text{if
}\gamma|_{\Delta_{i}\cap\Gamma_{1}}<0.\end{cases}$ $\displaystyle i\in[1,m],\
j\in[\alpha_{i}+1,n];$ $\displaystyle\mathcal{N}_{i}(j)$
$\displaystyle:={\Delta_{i}^{\leq}(s_{ij})\choose|\Delta_{i}\cap[\alpha_{i}+1,s_{ij}]|},\
\ \ \mathcal{N}_{i}:={\Delta_{i}\choose|\Delta_{i}^{\geq}(\alpha_{i}+1)|},$
$\displaystyle i\in[0,m-1],\ j\in[\alpha_{i}+1,n].$

######  Proposition 6.4.

For each $i\in[0,m]$ and each $j\in[\alpha_{i}+1,n]$, the $h$-variable
$h_{\alpha_{i}+1,j}$ is given by:

  1. 1.

If $i=m$, then

$h_{\alpha_{m}+1,j}(U)=(-1)^{\varepsilon_{\alpha_{m}+1,j}}\det
U^{[j,n]}_{[\alpha_{m}+1,s_{mj}]};$ (6.11)

  2. 2.

If $i=m-1$, then

$h_{\alpha_{m-1}+1,j}(U)=(-1)^{\varepsilon_{\alpha_{m-1}+1,j}}\sum_{I_{m-1}:\,I_{m-1}\in\mathcal{N}_{m-1}(j)}\det
U^{[j,n]}_{I_{m-1}\cup K_{m-1}^{\leq}(s_{m-1,j})}\det
U^{\check{\gamma}(I_{m-1})\cup L_{m}^{\prime}(j)}_{[\alpha_{m}+1,n]};$ (6.12)

  3. 3.

If $i\in[0,m-2]$, then

$\displaystyle h_{\alpha_{i}+1,j}$ $\displaystyle(U)=$ (6.13)
$\displaystyle=(-1)^{\varepsilon_{\alpha_{i}+1,j}}\sum_{\begin{subarray}{c}(I_{i},I_{i+1},\ldots,I_{m-1}):\\\
I_{i}\in\mathcal{N}_{i}(j),\ I_{t}\in\mathcal{N}_{t},\\\ \
t\in[i+1,m-1]\end{subarray}}$ $\displaystyle\det U^{[j,n]}_{I_{i}\cup
K_{i}^{\leq}(s_{ij})}\det U^{\check{\gamma}(I_{i})\cup
L_{i+1}^{\prime}(j)}_{I_{i+1}\cup K_{i+1}}\det U^{\check{\gamma}(I_{i+1})\cup
L_{i+2}}_{I_{i+2}\cup K_{i+2}}\cdots$ $\displaystyle\cdots\det
U^{\check{\gamma}(I_{m-2})\cup L_{m-1}}_{I_{m-1}\cup K_{m-1}}\det
U^{\check{\gamma}(I_{m-1})\cup L_{m}}_{[\alpha_{m}+1,n]}.$

###### Proof.

The case $i=m$ is already contained in Lemma 6.1. Comparing (6.1) with
$L_{i}^{\prime}(j)$ from (6.10), we see that
$L_{\alpha_{i}+1,s_{ij}}=L_{i+1}^{\prime}(j)$. Rewriting (6.3) in terms of
(6.10), we see that

$\begin{split}\det[\mathcal{F}_{m-i}(U)]_{[\alpha_{i}+1,s_{ij}]}^{[j,n]}\cdot&\det[\mathcal{F}_{m-i-1}(U)]_{[\alpha_{i+1}+1,n]}^{[\alpha_{i+1}+1,n]}=\\\
=&\sum_{I_{i}\in\mathcal{N}_{i}(j)}\det U^{[j,n]}_{I_{i}\cup
K_{i}^{\leq}(s_{ij})}\det[\mathcal{F}_{m-i-1}(U)]_{[\alpha_{i+1}+1,n]}^{\check{\gamma}(I_{i})\cup
L^{\prime}_{i+1}(j)}.\end{split}$ (6.14)

Setting $i=m-1$ in formula (6.14), we obtain (up to a sign) formula (6.12). If
$m-i-1\geq 1$, let us expand
$\det[\mathcal{F}_{m-i-1}(U)]_{[\alpha_{i+1}+1,n]}^{\check{\gamma}(I_{i})\cup
L_{i+1}^{\prime}(j)}$ via formula (6.3); in the notation of (6.10), we obtain

$\begin{split}\det[\mathcal{F}_{m-i}(U)&]_{[\alpha_{i}+1,s_{ij}]}^{[j,n]}\det[\mathcal{F}_{m-i-1}(U)]_{[\alpha_{i+1}+1,n]}^{[\alpha_{i+1}+1,n]}\det[\mathcal{F}_{m-i-2}(U)]_{[\alpha_{i+2}+1,n]}^{[\alpha_{i+2}+1,n]}=\\\
=&\sum_{\begin{subarray}{c}I_{i}\in\mathcal{N}_{i}(j),\\\
I_{i+1}\in\mathcal{N}_{i+1}\end{subarray}}\det U^{[j,n]}_{I_{i}\cup
K_{i}^{\leq}(s_{ij})}\det U_{[\alpha_{i+1}+1,n]}^{\check{\gamma}(I_{i})\cup
L^{\prime}_{i+1}(j)}\det[\mathcal{F}_{m-i-2}(U)]_{[\alpha_{i+2}+1,n]}^{\check{\gamma}(I_{i+1})\cup
L_{i+2}}.\end{split}$ (6.15)

Now, if $m-i-2\geq 1$, we apply formula (6.3) again to the term
$\det[\mathcal{F}_{m-i-2}(U)]_{[\alpha_{i+2}+1,n]}^{\check{\gamma}(I_{i+1})\cup
L_{i+2}}$, and so with
$\det[\mathcal{F}_{m-i-3}(U)]_{[\alpha_{i+3}+1,n]}^{\check{\gamma}(I_{i+2})\cup
L_{i+3}}$ in the resulting formula if $m-i-3\geq 1$. Continuing in this
fashion, we will finally reach the terms
$\det[\mathcal{F}_{0}(U)]_{[\alpha_{m}+1,n]}^{\check{\gamma}(I_{m-1})\cup
L_{m}}$, which are equal to $\det
U_{[\alpha_{m}+1,n]}^{\check{\gamma}(I_{m-1})\cup L_{m}}$. By Proposition 6.3,
the right-hand side will be equal (up to a sign) to $h_{\alpha_{i}+1,j}(U)$,
and the left-hand side will be the desired expansion in terms of minors of
$U$. ∎

######  Corollary 6.5.

The polynomial degree of $h_{\alpha_{i}+1,j}$ is given by

$\deg h_{\alpha_{i}+1,j}=n-j+1+\sum_{\ell=1}^{m-i}(n-\alpha_{i+\ell}).$ (6.16)

###  6.3 Duality of the frozen $h$-variables

Let $\mathbf{\Gamma}:=(\Gamma_{1},\Gamma_{2},\gamma)$ be a BD triple and
$\mathbf{\Gamma}^{\mathrm{op}}:=(\Gamma_{2},\Gamma_{1},\gamma^{*})$ be the
opposite BD triple. In this subsection, we prove a certain duality property
between the frozen $h$-variables in
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma})$ and
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma}^{\mathrm{op}})$. We will
subsequently use this property in two instances: 1) to prove Proposition 1.8
(semi-invariance of the frozen $h$-variables); 2) to show that the frozen
variables in $\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma})$ coincide with the
frozen variables in $\mathcal{GC}_{g}^{\dagger}(\mathbf{\Gamma})$ (Proposition
7.6).

######  Proposition 6.6.

Let $S^{\gamma}(\alpha_{0}):=\\{\alpha_{i}\\}_{i=0}^{m}$ and
$S^{\gamma^{*}}(\beta_{0}):=\\{\beta_{i}\\}_{i=0}^{m}$ be a $\gamma$\- and a
$\gamma^{*}$-string that start at $\alpha_{0}\in\Pi\setminus\Gamma_{2}$ and
$\beta_{0}\in\Pi\setminus\Gamma_{1}$, and such that $\beta_{0}=\alpha_{m}$.
Let $h_{\alpha_{0}+1,\alpha_{0}+1}$ be the frozen $h$-variable in
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma})$ associated with $\alpha_{0}$ and
$\bar{h}_{\beta_{0}+1,\beta_{0}+1}$ be the frozen $h$-variable in
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma}^{\mathrm{op}})$ associated with
$\beta_{0}$. Then

$\bar{h}_{\beta_{0}+1,\beta_{0}+1}(U^{T})=h_{\alpha_{0}+1,\alpha_{0}+1}(U).$
(6.17)

###### Proof.

We will compare the explicit expansions of the variables in terms of minors of
$U$. Consider the data (6.10) relative the BD triple
$\mathbf{\Gamma}^{\mathrm{op}}$ for the $\gamma^{*}$-string
$S^{\gamma^{*}}(\beta_{0})$; to distinguish it from the data for
$S^{\gamma}(\alpha_{0})$, let us decorate the former with a bar; for instance,
we set $\bar{K}_{i}:=[(\beta_{i})_{+}(\Gamma_{2})+1,n]$. Now, if $m=0$, the
statement is trivial; for $m>0$, it follows from Proposition 6.4 that both
variables can be expanded as

$\bar{h}_{\beta_{0}+1,\beta_{0}+1}(U)=\sum_{\begin{subarray}{c}(\bar{I}_{0},\bar{I}_{1},\ldots,\bar{I}_{m-1}):\\\
\bar{I}_{i}\in\bar{\mathcal{N}}_{i},\,0\leq i\leq m-1\end{subarray}}\det
U_{\bar{I}_{0}\cup\bar{K}_{0}}^{[\beta_{0},n]}\left[\prod_{i=1}^{m-1}\det
U_{\bar{I}_{i}\cup\bar{K}_{i}}^{\check{\gamma}^{*}(\bar{I}_{i-1})\cup\bar{L}_{i}}\right]\det
U^{\check{\gamma}^{*}(\bar{I}_{m-1})\cup\bar{L}_{m}}_{[\beta_{m},n]};$ (6.18)
$h_{\alpha_{0}+1,\alpha_{0}+1}(U)=\sum_{\begin{subarray}{c}(I_{0},I_{1},\ldots,I_{m-1}):\\\
I_{i}\in\mathcal{N}_{i},\,0\leq i\leq m-1\end{subarray}}\det
U^{[\alpha_{0},n]}_{I_{0}\cup K_{0}}\left[\prod_{i=1}^{m-1}\det
U^{\check{\gamma}(I_{i-1})\cup L_{i}}_{I_{i}\cup K_{i}}\right]\det
U_{[\alpha_{m},n]}^{\check{\gamma}(I_{m-1})\cup L_{m}}.$ (6.19)

Let us establish a connection between the data associated with the two
variables. We see that for each $i\in[0,m]$, $\beta_{i}=\alpha_{m-i}$,
$\bar{K}_{i}=L_{m-i}$, $\bar{L}_{i}=K_{m-i}$; as for $\mathcal{N}_{i}$, we see
that for each $i\in[0,m-1]$,
$\check{\gamma}|_{\mathcal{N}_{i}}:\mathcal{N}_{i}\rightarrow\bar{\mathcal{N}}_{m-i-1}$
is a bijection; therefore, in formula (6.18) we can substitute each
$\bar{I}_{i}$ with $\check{\gamma}(I_{m-i-1})$ and each
$\check{\gamma}^{*}(\bar{I}_{i-1})$ with $I_{m-i}$. Now, combining the latter
with the substitution $U\mapsto U^{T}$, formula (6.17) follows. ∎

######  Corollary 6.7.

Let $h_{ii}$ be a frozen $h$-variable in
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma})$, and let $P$ be a unipotent
upper triangular matrix. Then the following identity holds:

$h_{ii}(PU\tilde{\gamma}(P)^{-1})=h_{ii}(U).$ (6.20)

###### Proof.

By Proposition 6.6, there exists a frozen $h$-variable $\bar{h}_{jj}$ in
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma}^{\mathrm{op}})$ such that
$h_{ii}(U)=\bar{h}_{jj}(U^{T})$. Since $P^{T}$ is a unipotent lower triangular
matrix, applying formula (4.20), we see that

$h_{ii}(U)=\bar{h}_{jj}(U^{T})=\bar{h}_{jj}(\tilde{\gamma}(P^{-T})U^{T}P^{T})=h_{ii}(PU\tilde{\gamma}(P)^{-1}).$
(6.21)

Thus formula (6.20) holds. ∎

######  Corollary 6.8.

Let $h_{ii}$ be a frozen $h$-variable in
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma})$. Then for any
$U\in\operatorname{GL}_{n}$,

$U\nabla_{U}h_{ii}-\mathring{\gamma}^{*}\left(\nabla_{U}h_{ii}\cdot
U\right)\in\mathfrak{b}_{+}.$ (6.22)

###### Proof.

The statement follows from differentiating formula (6.20). ∎

###  6.4 Semi-invariance of the frozen $h$-variables

In this subsection, we prove Proposition 1.8 for
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma})$, and we also show that the
frozen $h$-variables do not vanish on $\operatorname{SL}_{n}^{\dagger}$. Let
$\mathbf{\Gamma}:=(\Gamma_{1},\Gamma_{2},\gamma)$ be a BD triple of type
$A_{n-1}$. We set $\mathcal{P}_{+}(\Gamma_{2})$ and
$\mathcal{P}_{-}(\Gamma_{1})$ to be the parabolic subgroups of
$\operatorname{GL}_{n}$ generated by the BD data $\mathbf{\Gamma}$. The
subgroup
$\mathcal{D}\subset\mathcal{P}_{+}(\Gamma_{1})\times\mathcal{P}_{-}(\Gamma_{2})$
is defined as

$\mathcal{D}:=\\{(g^{\prime},g)\in\mathcal{P}_{+}(\Gamma_{1})\times\mathcal{P}_{-}(\Gamma_{2})\
|\
{\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}(\mathring{\Pi}_{\Gamma_{1}}(g^{\prime}))=\mathring{\Pi}_{\Gamma_{2}}(g)\\}.$
(6.23)

The group $\mathcal{D}$ acts upon $\operatorname{GL}_{n}$ via

$(g^{\prime},g).U=g^{\prime}Ug^{-1},\ \ (g^{\prime},g)\in\mathcal{D},\
U\in\operatorname{GL}_{n}.$ (6.24)

######  Proposition.

Let $\psi$ be a frozen variable in
$\mathcal{GC}^{\dagger}_{h}(\mathbf{\Gamma},G)$,
$G\in\\{\operatorname{SL}_{n},\operatorname{GL}_{n}\\}$. Then the following
statements hold:

  1. 1)

The variable $\psi$ is semi-invariant with respect to the action (6.24);

  2. 2)

The variable $\psi$ is log-canonical with any $u_{ij}$;

  3. 3)

The zero locus of $\psi$ foliates into a union of symplectic leaves of
$(G,\pi_{\mathbf{\Gamma}}^{\dagger})$.

###### Proof.

The statements are evident for the $c$-variables, which are Casimirs of the
Poisson bracket. Part 3 for the $c$-variables follows from Proposition 2.5 and
the fact that the $c$-variables, as elements of
$\mathcal{O}(\operatorname{GL}_{n})$, are irreducible (see [20, Theorem
3.10]). From now on, we assume that $\psi$ is a frozen $h$-variable.

  1. 1)

Pick a pair $(g^{\prime},g)\in\mathcal{D}$ and decompose it as
$g^{\prime}=g^{\prime}_{+}g^{\prime}_{0}g^{\prime}_{-}$ and
$g=g_{+}g_{0}g_{-}$, where the subscript indicates if the matrix is unipotent
upper, diagonal or unipotent lower, respectively. The defining condition of
$\mathcal{D}$ implies that
${\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}^{*}(g_{-})=g^{\prime}_{-}$
and
${\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}(g^{\prime}_{+})=g_{+}$.
It follows from Corollary 4.10 and Corollary 6.7 that

$\psi(g^{\prime}Ug^{-1})=\psi\left(g^{\prime}_{+}g^{\prime}_{0}{\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}^{*}(g_{-})U\,\left.g_{-}\right.^{-1}\,\left.g_{0}\right.^{-1}{\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}(g^{\prime}_{+})^{-1}\right)=\chi_{\psi}(g^{\prime},g)\psi(U)$
(6.25)

where $\chi_{\psi}$ is a character on $\mathcal{D}$. Thus $\psi$ is semi-
invariant with respect to the action of $\mathcal{D}$.

  2. 2)

For convenience, let us choose $\pi_{\mathbf{\Gamma}}^{\dagger}$ with a
ringed191919As we showed in Section 8.2 in [31], log-canonicity does not
depend on the choice of $R_{0}$. $R_{0}$ (see formula (4.8)). Let us set

$\displaystyle\mathring{\nu}(\psi)$ $\displaystyle:=\nabla_{U}\psi\cdot
U-\mathring{\gamma}(U\nabla_{U}\psi);$ (6.26)
$\displaystyle\mathring{\nu}^{*}(\psi)$
$\displaystyle:=U\nabla_{U}\psi-\mathring{\gamma}^{*}(\nabla_{U}\psi\cdot U).$
(6.27)

It follows from Corollary 4.11 and Corollary 6.8 that
$\mathring{\nu}(\psi)\in\mathfrak{b}_{-}$,
$\mathring{\nu}^{*}(\psi)\in\mathfrak{b}_{+}$,
$\pi_{0}(\mathring{\nu}(\log\psi))=\mathrm{const}$ and
$\pi_{0}(\mathring{\nu}^{*}(\log\psi))=\mathrm{const}$. Moreover, one can
write

$\displaystyle\mathring{\nu}(\psi)$
$\displaystyle=[\nabla_{U}\psi,U]+(1-\mathring{\gamma})(U\nabla_{U}\psi);$
(6.28) $\displaystyle\mathring{\nu}^{*}(\psi)$
$\displaystyle=-[\nabla_{U}\psi,U]+(1-\mathring{\gamma}^{*})(\nabla_{U}\psi\cdot
U).$

It follows that

$\displaystyle\frac{1}{1-\mathring{\gamma}}\pi_{>}[\nabla_{U}\psi,U]$
$\displaystyle=-\pi_{>}(U\nabla_{U}\psi);$ (6.29)
$\displaystyle\frac{\mathring{\gamma}^{*}}{1-\mathring{\gamma}^{*}}\pi_{<}[\nabla_{U}h,U]$
$\displaystyle=\pi_{<}(U\nabla_{U}\psi).$ (6.30)

Therefore,

$R([\nabla_{U}\psi,U])=-\pi_{>}(U\nabla_{U}\psi)-\pi_{<}(U\nabla_{U}\psi)+\mathring{R}_{0}\pi_{0}[\nabla_{U}\psi,U].$
(6.31)

Rewriting the last piece, we see that

$\begin{split}\mathring{R}_{0}\pi_{0}[\nabla_{U}\psi,U]=&\mathring{R}_{0}\pi_{0}\mathring{\nu}(\psi)-\mathring{R}_{0}(1-\mathring{\gamma})(U\nabla_{U}\psi)=\\\
=&\mathring{R}_{0}\pi_{0}\mathring{\nu}(\psi)-\pi_{0}\mathring{\pi}_{\Gamma_{1}}(U\nabla_{U}\psi)-\mathring{R}_{0}\pi_{0}\mathring{\pi}_{\hat{\Gamma}_{1}}(U\nabla_{U}\psi)=\\\
=&\mathring{R}_{0}\pi_{0}\mathring{\nu}(\psi)-\pi_{0}(U\nabla_{U}\psi)+\mathring{R}_{0}^{*}\pi_{0}\mathring{\pi}_{\hat{\Gamma}_{1}}(U\nabla_{U}\psi);\end{split}$
(6.32)

hence, formula (6.31) can be updated as

$R([\nabla_{U}\psi,U])=-U\nabla_{U}\psi+\mathring{R}_{0}\pi_{0}\mathring{\nu}(\psi)+\mathring{R}_{0}^{*}\pi_{0}\mathring{\pi}_{\hat{\Gamma}_{1}}(U\nabla_{U}\psi).$
(6.33)

Now, the Poisson bracket between $\psi$ and $u_{ij}$ is given by

$\begin{split}\\{\psi,u_{ij}\\}_{\mathbf{\Gamma}}^{\dagger}=&\langle
R([\nabla_{U}\psi,U]),[\nabla_{U}u_{ij},U]\rangle-\langle[\nabla_{U}\psi,U],\nabla_{U}u_{ij}U\rangle=\\\
=&-\langle
U\nabla_{U}\psi,[\nabla_{U}u_{ij},U]\rangle-\langle[\nabla_{U}\psi,U],\nabla_{U}u_{ij}U\rangle+\\\
+&\langle\mathring{R}_{0}\pi_{0}\mathring{\nu}(\psi)+\mathring{R}_{0}^{*}\pi_{0}\mathring{\pi}_{\hat{\Gamma}_{1}}(U\nabla_{U}\psi),\pi_{0}[\nabla_{U}u_{ij},U]\rangle.\end{split}$
(6.34)

The first two terms in the last identity cancel each other out. As for the
last term, notice that $\pi_{0}[\nabla_{U}u_{ij},U]=u_{ij}(e_{jj}-e_{ii})$,
and since $\mathring{\nu}(\log\psi)=\mathrm{const}$ and
$\mathring{\nu}^{*}(\log\psi)=\mathrm{const}$, we finally conclude that
$\\{\psi,u_{ij}\\}_{\mathbf{\Gamma}}^{\dagger}$ is proportional to the product
$\psi\cdot u_{ij}$.

  3. 3)

It is a consequence of Proposition 5.9 that $\psi$ is a prime element of
$\mathcal{O}(G)$, and it follows from Part 2 of the current proposition that
the principal ideal $(\psi)$ is Poisson. Now the statement follows from
Proposition 2.5.∎

######  Remark 6.9.

Let $\mathbf{\Gamma}$ be an aperiodic oriented BD triple and let
$\mathcal{GC}^{D}(\mathbf{\Gamma},\mathbf{\Gamma})$ be the generalized cluster
structure on $D(\operatorname{GL}_{n})$ constructed in [31]. Then any frozen
$g$\- or $h$-variable in $\mathcal{GC}^{D}(\mathbf{\Gamma},\mathbf{\Gamma})$
is semi-invariant with respect to both the right and the left action of
$\mathcal{D}$ upon $D(\operatorname{GL}_{n})$. Therefore, the frozen variables
in $\mathcal{GC}^{D}(\mathbf{\Gamma},\mathbf{\Gamma})$ and in
$\mathcal{GC}^{\dagger}_{h}(\mathbf{\Gamma},\operatorname{GL}_{n})$ are semi-
invariant with respect to the same group.

######  Proposition 6.10.

The following statements hold:

  1. 1)

For any choice of $r_{0}$, the frozen $h$-variables in
$\mathcal{GC}^{\dagger}_{h}(\mathbf{\Gamma},\operatorname{SL}_{n})$ do not
vanish on $\operatorname{SL}_{n}^{\dagger}$;

  2. 2)

If $r_{0}$ is ringed, then the frozen $h$-variables in
$\mathcal{GC}^{\dagger}(\mathbf{\Gamma},\operatorname{GL}_{n})$ do not vanish
on $\operatorname{GL}_{n}^{\dagger}$.

###### Proof.

Both statements can be proved in the same way, so let us focus on the case of
$\operatorname{GL}_{n}$. Let $r_{0}$ be chosen as a ringed element (see
formula (4.8)), and let $\psi$ be a frozen $h$-variable in
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma},\operatorname{GL}_{n})$. We set
$\tilde{\psi}\in\mathcal{O}(D(\operatorname{GL}_{n}))$ to be a lift of $\psi$
given by

$\tilde{\psi}(X,Y):=\psi(X^{-1}Y),\ \ (X,Y)\in D(\operatorname{GL}_{n}).$
(6.35)

The choice of $r_{0}$ implies that
$\mathring{\gamma}R_{+}=\mathring{\pi}_{\Gamma_{2}}R_{-}$, hence
$\operatorname{GL}_{n}^{*}\subseteq\mathcal{D}$, where
$\operatorname{GL}_{n}^{*}$ is viewed as an immersed Lie subgroup of
$D(\operatorname{GL}_{n})$. We see that

$\tilde{\psi}(g^{\prime},g)=\psi((g^{\prime})^{-1}g)=\chi_{\psi}(g^{\prime},g)\psi(I)$
(6.36)

where $\chi_{\psi}$ is a character on $\mathcal{D}$ from Proposition 1.8.
Since $\mathcal{F}(I)=I$, it follows from the defining formula for the
$h$-functions (1.11) that $\psi(I)\neq 0$. Hence $\tilde{\psi}|_{\mathcal{D}}$
does not vanish at any point of $\mathcal{D}$, and thus $\psi$ is nonzero on
$\operatorname{GL}_{n}^{\dagger}$ and $\operatorname{SL}_{n}^{\dagger}$. ∎

######  Remark 6.11.

Proposition 6.10 justifies the choice of the ground ring for
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma},\operatorname{SL}_{n}^{\dagger})$
(see formula (1.19)). For
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma},\operatorname{GL}_{n}^{\dagger})$,
if $r_{0}$ is ringed, then one chooses

$\hat{\mathbb{A}}_{\mathbb{C}}(\mathbf{\Gamma},\operatorname{GL}_{n}^{\dagger}):=\mathbb{C}[h_{11}^{\pm
1},c_{1},\ldots,c_{n-1},h_{i+1,i+1}^{\pm 1}\ |\ i\in\Pi\setminus\Gamma_{2}].$
(6.37)

##  7 The $g$-convention

In this section, for any Belavin-Drinfeld triple $\mathbf{\Gamma}$ of type
$A_{n-1}$, we describe the generalized cluster structure
$\mathcal{GC}^{\dagger}_{g}(\mathbf{\Gamma})$ on $\operatorname{GL}_{n}$ (and
on the related varieties $\operatorname{SL}_{n}^{\dagger}$,
$\operatorname{SL}_{n}$, $\operatorname{GL}_{n}^{\dagger}$) in the _$g$
-convention_.

###  7.1 The map $\mathcal{F}^{\mathrm{op}}$

In this subsection, we recast the map $\mathcal{F}$ from Section 4.2 in the
$g$-convention. For a fixed set of simple roots $\Pi$ of type $A_{n-1}$, let
$\mathbf{\Gamma}:=(\Gamma_{1},\Gamma_{2},\gamma)$ be a BD triple,
$\mathcal{B}_{\pm}$, $\mathcal{N}_{\pm}$ be the Borel subgroups and their
unipotent radicals, $\mathcal{H}$ be the corresponding Cartan subgroup of
$\operatorname{GL}_{n}$. We decompose a generic element
$U\in\operatorname{GL}_{n}$ as $U=U_{\oplus}U_{-}=U_{+}U_{\ominus}$, where
$U_{\ominus}\in\mathcal{B}_{-}$, $U_{\oplus}\in\mathcal{B}_{+}$,
$U_{-}\in\mathcal{N}_{-}$, $U_{+}\in\mathcal{N}_{+}$. Define a sequence of
rational maps
$\mathcal{F}^{\mathrm{op}}_{k}:\operatorname{GL}_{n}\dashrightarrow\operatorname{GL}_{n}$
via202020To avoid confusion, let us emphasize that formula (7.1) defines
$\mathcal{F}_{k}^{\mathrm{op}}$ relative the BD triple $\mathbf{\Gamma}$ and
not $\mathbf{\Gamma}^{\mathrm{op}}$. If $\mathcal{F}_{k}^{\mathrm{op}}$ is
defined relative the opposite BD triple $\mathbf{\Gamma}^{\mathrm{op}}$, then
it satisfies the recurrence relation
$\mathcal{F}_{k}^{\mathrm{op}}(U)=U\tilde{\gamma}^{*}(\mathcal{F}_{k-1}(U)_{+})$.

$\mathcal{F}^{\mathrm{op}}_{0}(U):=U,\ \
\mathcal{F}^{\mathrm{op}}_{k}(U)=U\tilde{\gamma}(\mathcal{F}_{k-1}(U)_{+}),\ \
U\in\operatorname{GL}_{n},\ k\geq 1.$ (7.1)

The rational map
$\mathcal{F}^{\mathrm{op}}:\operatorname{GL}_{n}\dashrightarrow\operatorname{GL}_{n}$
is defined as the limit

$\mathcal{F}^{\mathrm{op}}(U):=\lim_{k\rightarrow\infty}\mathcal{F}^{\mathrm{op}}_{k}(U),\
\ U\in\operatorname{GL}_{n}.$ (7.2)

The sequence $\mathcal{F}_{k}^{\mathrm{op}}$ stabilizes at $k=\deg\gamma$. As
a corollary, $\mathcal{F}^{\mathrm{op}}(U)$ satisfies the relation

$\mathcal{F}^{\mathrm{op}}(U)=U\tilde{\gamma}(\mathcal{F}^{\mathrm{op}}(U)_{+}),$
(7.3)

and $\mathcal{F}^{\mathrm{op}}$ is a birational map with the inverse given by

$(\mathcal{F}^{\mathrm{op}})^{-1}(U)=U\tilde{\gamma}(U_{+})^{-1}.$ (7.4)

Let $T:\operatorname{GL}_{n}\rightarrow\operatorname{GL}_{n}$ be the
transposition map; that is, $T(U)=U^{T}$, $U\in\operatorname{GL}_{n}$.

######  Proposition 7.1.

Let $\mathcal{F}:\operatorname{GL}_{n}\dashrightarrow\operatorname{GL}_{n}$
and
$\mathcal{F}^{\mathrm{op}}:\operatorname{GL}_{n}\dashrightarrow\operatorname{GL}_{n}$
be constructed relative the BD triples $\mathbf{\Gamma}$ and
$\mathbf{\Gamma}^{\mathrm{op}}$, respectively. Then for any $k\geq 0$ the
following diagram commutes:

$\textstyle{\operatorname{GL}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T}$$\scriptstyle{\mathcal{F}_{k}}$$\textstyle{\operatorname{GL}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T}$$\textstyle{\operatorname{GL}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{F}_{k}^{\mathrm{op}}}$$\textstyle{\operatorname{GL}_{n}}$
(7.5)

###### Proof.

Indeed, the commutativity is evident for $k=0$. Assuming by induction that the
commutativity holds for $k$, we see that

$[\mathcal{F}_{k+1}(U^{T})]^{T}=\left[\tilde{\gamma}^{*}(\mathcal{F}_{k}(U^{T})_{-})U^{T}\right]^{T}=U\tilde{\gamma}^{*}\left[(\mathcal{F}_{k}(U^{T})^{T})_{+}\right]=U\tilde{\gamma}^{*}(\mathcal{F}^{\mathrm{op}}_{k}(U)_{+})=\mathcal{F}_{k+1}^{\mathrm{op}}(U).$
(7.6)

Thus the diagram commutes. ∎

######  Corollary 7.2.

For $P\in\mathcal{N}_{+}$ and $T\in\mathcal{H}$, the map
$\mathcal{F}^{\mathrm{op}}$ enjoys the following invariance properties:

$\displaystyle\mathcal{F}^{\mathrm{op}}(PU\tilde{\gamma}(P)^{-1})=P\cdot\mathcal{F}^{\mathrm{op}}(U);$
(7.7)
$\displaystyle\mathcal{F}^{\mathrm{op}}({\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}^{*}(T)UT^{-1})={\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}^{*}(T)\mathcal{F}^{\mathrm{op}}(U)T^{-1};$
(7.8)
$\displaystyle\mathcal{F}^{\mathrm{op}}(T^{-1}U{\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}(T))=T^{-1}\mathcal{F}^{\mathrm{op}}(U){\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}(T).$
(7.9)

###### Proof.

The identities follow from Corollary 4.9 and Proposition 7.1. ∎

###  7.2 Description of $\mathcal{GC}_{g}^{\dagger}(\mathbf{\Gamma})$

In this subsection, we describe the initial extended cluster of
$\mathcal{GC}_{g}^{\dagger}(\mathbf{\Gamma})$ for an arbitrary BD triple
$\mathbf{\Gamma}:=(\Gamma_{1},\Gamma_{2},\gamma)$ of type $A_{n-1}$. It
comprises three types of functions: $c$-functions, $\phi$-functions and
$g$-functions. The initial quiver is described in Appendix B.

##### Description of $\phi$\- and $c$-functions.

The $c$-functions are given by (1.7) (as in the $h$-convention). The
$\phi$-functions are similar to the $\varphi$-functions and are defined as
follows. For an element $U\in\operatorname{GL}_{n}$, let us set

$\Phi_{kl}^{\prime}(U):=\begin{bmatrix}(U^{0})^{[1,k]}&U^{[1,l]}&(U^{2})^{\\{1\\}}&\cdots&(U^{n-k-l+1})^{\\{1\\}}\end{bmatrix},\
\ k,l\geq 1,\ k+l\leq n;$ (7.10)

Then the $\phi$-functions are given by

$\phi_{kl}(U):=s_{kl}\det\Phi_{kl}^{\prime}(U)$ (7.11)

where $s_{kl}\in\\{1,-1\\}$ is a sign given by (1.4) (as in the
$h$-convention). For $P\in\mathcal{N}_{+}$ and $T\in\mathcal{H}$, the
$\phi$-functions enjoy the following invariance properties:

$\displaystyle\phi_{kl}(PUP^{-1})$ $\displaystyle=\phi_{kl}(U),$ (7.12)
$\displaystyle\phi_{kl}(TUT^{-1})$
$\displaystyle=\chi_{\phi_{kl}}(T)\phi_{kl}(U)$

where $\chi_{\phi_{kl}}$ is a character.

######  Remark 7.3.

The $\varphi$\- and $\phi$-functions are related as follows. Set
$W_{0}:=\sum_{i=1}^{n-1}(-1)^{i+1}e_{n-i+1,i}$ and consider the map
$w_{0}:\operatorname{GL}_{n}\rightarrow\operatorname{GL}_{n}$,
$w_{0}(U)=W_{0}^{-1}UW_{0}$. If $\phi_{kl}$ is a $\phi$-function, then
$\phi_{kl}(w_{0}(U))=\varphi_{kl}(U)$ is a $\varphi$-function. It follows that
the $\phi$-functions are irreducible as elements of
$\mathcal{O}(\operatorname{GL}_{n})$, and the mutations of $\phi_{kl}$ for
$2\leq k+l\leq n-1$ are regular and coprime with $\phi_{kl}$. Moreover, if
$\mathcal{O}(\operatorname{GL}_{n})^{\mathcal{N}_{\pm}}$ are the rings of
regular functions invariant with respect to the adjoint actions of
$\mathcal{N}_{+}$ and $\mathcal{N}_{-}$, then
$\omega_{0}^{*}:(\mathcal{O}(\operatorname{GL}_{n})^{\mathcal{N}_{+}},\\{\cdot,\cdot\\}_{\mathbf{\Gamma}}^{\dagger})\xrightarrow{\sim}(\mathcal{O}(\operatorname{GL}_{n})^{\mathcal{N}_{-}},\\{\cdot,\cdot\\}_{\mathbf{\Gamma}}^{\dagger})$
is a Poisson isomorphism for any $\mathbf{\Gamma}$. In particular, the
$\phi$-functions are log-canonical.

######  Remark 7.4.

The $\phi$-functions have appeared before in [22]. They were used to define a
generalized cluster structure on the Drinfeld double
$D(\operatorname{GL}_{n})=\operatorname{GL}_{n}\times\operatorname{GL}_{n}$,
which is an alternative to the use of the $\varphi$-functions in [20]. It was
shown in [21] that the resulting constructions with $\phi$\- and
$\varphi$-functions (for the trivial BD triples) provide equivalent
generalized cluster structures in $n=2$ and $n=3$; however, they are not
equivalent in $n=4$ (in other words, the initial extended seeds are not
mutation equivalent). In the supplementary note [33], we show that
$\mathcal{GC}_{g}^{\dagger}(\mathbf{\Gamma})$ and
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma}^{\mathrm{op}})$ are equivalent for
$n=3$ and any BD triple $\mathbf{\Gamma}$. We conjecture that they are not
equivalent for $n\geq 4$.

##### Description of the $g$-functions.

Let
$\mathcal{F}^{\mathrm{op}}:\operatorname{GL}_{n}\dashrightarrow\operatorname{GL}_{n}$
be the birational map defined in (7.2). For each
$\alpha_{0}\in\Pi\setminus\Gamma_{1}$ and the associated $\gamma^{*}$-string
$S^{\gamma^{*}}(\alpha_{0}):=\\{\alpha_{i}\\}_{i=0}^{m}$ (see Definition 1.2),
for every $k\in[0,m]$ and $i\in[\alpha_{k}+1,n]$, set

$g_{i,\alpha_{k}+1}(U):=\det[\mathcal{F}^{\mathrm{op}}(U)]_{[i,n]}^{[\alpha_{k}+1,n-i+\alpha_{k}+1]}\prod_{t\geq
k+1}^{m}\det[\mathcal{F}^{\mathrm{op}}(U)]_{[\alpha_{t}+1,n]}^{[\alpha_{t}+1,n]}.$
(7.13)

It is evident from the above formula that the flag minors of
$\mathcal{F}^{\mathrm{op}}(U)$ can be expressed as

$\det[\mathcal{F}^{\mathrm{op}}(U)]_{[i,n]}^{[j,n-i+j]}=\begin{cases}\frac{g_{ij}(U)}{g_{\mu\mu}(U)}\
&\text{if}\ j-1\in\Gamma_{2}\\\ g_{ij}(U)\ &\text{otherwise}\end{cases}$
(7.14)

where $1\leq i\leq n$, $1\leq j\leq i$ and $\mu:=\gamma^{*}(j-1)+1$ if
$j-1\in\Gamma_{2}$.

######  Proposition 7.5.

Let $g_{ij}$ be a $g$-function from the initial extended cluster of
$\mathcal{GC}_{g}^{\dagger}(\mathbf{\Gamma})$ and $h_{ji}$ be an $h$-function
from the initial extended cluster of
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma}^{\mathrm{op}})$. Then
$g_{ij}(U)=(-1)^{\varepsilon_{ji}}h_{ji}(U^{T})$, where
$\varepsilon_{ji}=(i-j)(n-j)$.

###### Proof.

The statement is a consequence of Proposition 7.1 and the defining formulas
(1.11) and (7.13) for the $h$\- and $g$-functions. ∎

As a corollary, one can obtain an explicit expansion of the $g$-functions in
terms of minors of $U$, as in Section 6.2. One can also obtain invariance
properties of the $g$-functions from Corollary 7.2.

##### Frozen variables.

The frozen variables in
$\mathcal{GC}_{g}^{\dagger}(\mathbf{\Gamma},\operatorname{GL}_{n})$ are given
by the set

$\\{c_{1},c_{2},\ldots,c_{n-1}\\}\cup\\{g_{i+1,i+1}\ |\
i\in\Pi\setminus\Gamma_{1}\\}\cup\\{g_{11}\\}.$ (7.15)

In the case of
$\mathcal{GC}_{g}^{\dagger}(\mathbf{\Gamma},\operatorname{SL}_{n})$,
$g_{11}(U)=1$, so this variable is absent. As the next proposition shows, the
frozen variables in $\mathcal{GC}_{g}^{\dagger}(\mathbf{\Gamma})$ and
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma})$ are identical.

######  Proposition 7.6.

Let $\\{\alpha_{t}\\}_{t=0}^{m}$ be a $\gamma$-string for the BD triple
$(\Gamma_{1},\Gamma_{2},\gamma)$. Then

$h_{\alpha_{0}+1,\alpha_{0}+1}(U)=g_{\alpha_{m}+1,\alpha_{m}+1}(U)$ (7.16)

where $h_{\alpha_{0}+1,\alpha_{0}+1}$ is a frozen variable in
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma})$ and
$g_{\alpha_{m}+1,\alpha_{m}+1}$ is a frozen variable in
$\mathcal{GC}_{g}^{\dagger}(\mathbf{\Gamma})$.

###### Proof.

Let us denote by $\bar{h}$ the $h$-variables in
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma}^{\mathrm{op}})$. Since
$\alpha_{m}\notin\Gamma_{1}$, the variable
$\bar{h}_{\alpha_{m}+1,\alpha_{m}+1}$ is frozen in
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma}^{\mathrm{op}})$. It follows from
Proposition 7.5 that
$g_{\alpha_{m}+1,\alpha_{m}+1}(U)=\bar{h}_{\alpha_{m}+1,\alpha_{m}+1}(U^{T})$.
By Proposition 6.6,
$\bar{h}_{\alpha_{m}+1,\alpha_{m}+1}(U^{T})=h_{\alpha_{0}+1,\alpha_{0}+1}(U)$.
Thus the statement holds. ∎

The following proposition is a restatement of Proposition 1.8 in the case of
$\mathcal{GC}_{g}^{\dagger}(\mathbf{\Gamma})$.

######  Proposition.

Let $\psi$ be a frozen variable in
$\mathcal{GC}^{\dagger}_{g}(\mathbf{\Gamma},G)$,
$G\in\\{\operatorname{SL}_{n},\operatorname{GL}_{n}\\}$. Then the following
statements hold:

  1. 1)

The variable $\psi$ is semi-invariant with respect to the action (1.29);

  2. 2)

The variable $\psi$ is log-canonical with any $u_{ij}$;

  3. 3)

The zero locus of $\psi$ foliates into a union of symplectic leaves of
$(G,\pi_{\mathbf{\Gamma}}^{\dagger})$.

###### Proof.

It is a consequence of Proposition 1.8 for
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma})$ (see Section 6.4) and
Proposition 7.6. ∎

##### Initial extended cluster.

The initial extended cluster $\Psi_{0}$ of
$\mathcal{GC}^{\dagger}_{g}(\mathbf{\Gamma},\operatorname{GL}_{n})$ is given
by the set

$\\{g_{ij}\ |\ 2\leq j\leq i\leq n\\}\cup\\{\phi_{kl}\ |\ k,l\geq 1,k+l\leq
n\\}\cup\\{c_{1},\ldots,c_{n-1}\\}\cup\\{g_{11}\\}$ (7.17)

where $g_{11}(U):=\det(U)$. In the case of $\operatorname{SL}_{n}$ and
$\operatorname{SL}_{n}^{\dagger}$, the initial extended cluster is the same
except the variable $g_{11}$ is removed.

##### Ground rings.

The ground rings are defined as follows.

$\displaystyle\hat{\mathbb{A}}_{\mathbb{C}}(\mathbf{\Gamma},\operatorname{GL}_{n}):=$
$\displaystyle\mathbb{C}[g_{11}^{\pm 1},c_{1},\ldots,c_{n-1},g_{i+1,i+1}\ |\
i\in\Pi\setminus\Gamma_{1}];$ (7.18)
$\displaystyle\hat{\mathbb{A}}_{\mathbb{C}}(\mathbf{\Gamma},\operatorname{SL}_{n}):=$
$\displaystyle\mathbb{C}[c_{1},\ldots,c_{n-1},g_{i+1,i+1}\ |\
i\in\Pi\setminus\Gamma_{1}];$ (7.19)
$\displaystyle\hat{\mathbb{A}}_{\mathbb{C}}(\mathbf{\Gamma},\operatorname{SL}_{n}^{\dagger}):=$
$\displaystyle\mathbb{C}[c_{1},\ldots,c_{n-1},g_{i+1,i+1}^{\pm 1}\ |\
i\in\Pi\setminus\Gamma_{1}].$ (7.20)

In the case of $\operatorname{GL}_{n}^{\dagger}$, if $r_{0}$ is ringed, then
one chooses

$\hat{\mathbb{A}}_{\mathbb{C}}(\mathbf{\Gamma},\operatorname{GL}_{n}^{\dagger}):=\mathbb{C}[g_{11}^{\pm
1},c_{1},\ldots,c_{n-1},g_{i+1,i+1}^{\pm 1}\ |\ i\in\Pi\setminus\Gamma_{1}].$
(7.21)

The choice of the ground rings is justified by Proposition 6.6 and Proposition
6.10.

##### A generalized cluster mutation.

Only the variable $\phi_{11}$ is equipped with a nontrivial string, which is
given by $(1,c_{1},\ldots,c_{n-1},1)$. The generalized mutation relation for
$\phi_{11}$ reads

$\phi_{11}\phi_{11}^{\prime}=\sum_{r=0}^{n}c_{r}\phi_{21}^{r}\phi_{12}^{n-r},$
(7.22)

which is similar to the relation (1.20) in the $h$-convention. The mutations
of all the other variables follow the usual pattern (2.23) from the theory of
cluster algebras of geometric type.

###  7.3 Birational quasi-isomorphisms

In this subsection, we recast the birational quasi-isomorphisms from Section 4
in the $g$-convention. Let us recall from Section 3.5 that the rational map
$\mathcal{Q}^{\mathrm{op}}:(\operatorname{GL}_{n},\pi_{\text{std}}^{\dagger})\dashrightarrow(\operatorname{GL}_{n},\pi_{\mathbf{\Gamma}}^{\dagger})$
is given by

$\mathcal{Q}^{\mathrm{op}}(U):=\rho^{\mathrm{op}}(U)U\rho^{\mathrm{op}}(U)^{-1},\
\ \rho^{\mathrm{op}}(U):=\prod_{i=1}^{\leftarrow}(\tilde{\gamma})^{i}(U_{+}),\
U\in\operatorname{GL}_{n}.$ (7.23)

We showed in Proposition 3.15 that $\mathcal{Q}^{\mathrm{op}}$ is a Poisson
isomorphism provided the $r_{0}$ parts of $\pi_{\text{std}}^{\dagger}$ and
$\pi_{\mathbf{\Gamma}}^{\dagger}$ coincide. Define the rational map
$\mathcal{F}^{\mathrm{op},c}:(\operatorname{GL}_{n},\pi_{\mathbf{\Gamma}}^{\dagger})\dashrightarrow(\operatorname{GL}_{n},\pi_{\text{std}}^{\dagger})$
as

$\mathcal{F}^{\mathrm{op},c}(U):=[\mathcal{F}^{\mathrm{op}}(U)]^{-1}U\mathcal{F}^{\mathrm{op}}(U)=\tilde{\gamma}(\mathcal{F}^{\mathrm{op}}(U)_{+})^{-1}\mathcal{F}^{\mathrm{op}}(U).$
(7.24)

######  Proposition 7.7.

The rational maps $\mathcal{Q}^{\mathrm{op}}$ and
$\mathcal{F}^{\mathrm{op},c}$ are inverse to each other.

###### Proof.

The statement follows from the commutative diagram (3.55) and Proposition
4.13. ∎

Similarly to the $h$-convention, if
$\tilde{\mathbf{\Gamma}}\prec\mathbf{\Gamma}$, then the $g$-variables in
$\mathcal{GC}_{g}^{\dagger}(\tilde{\mathbf{\Gamma}})$ are decorated with a
tilde, and the marker for the pair
$(\mathcal{GC}_{g}^{\dagger}(\mathbf{\Gamma}),\mathcal{GC}_{g}^{\dagger}(\tilde{\mathbf{\Gamma}}))$
is chosen in such a way that each $\phi$\- and $c$-function is related to
itself, and each $g_{ij}$ is related to $\tilde{g}_{ij}$.

######  Proposition 7.8.

Let $\psi$ be any variable from the initial extended cluster of
$\mathcal{GC}^{\dagger}_{g}(\mathbf{\Gamma})$. Then the following identities
hold:

  1. 1)

If $\psi$ is a $\phi$\- or $c$-function, then
$(\mathcal{Q}^{\mathrm{op}})^{*}(\psi(U))=\psi(U)$;

  2. 2)

If $\psi=g_{ij}$ for some $g$-function $g_{ij}(U)$, let
$\\{\alpha_{t}\\}_{t=0}^{m}$ be the $\gamma^{*}$-string such that
$j-1=\alpha_{k}$ for some $k$; then,

$(\mathcal{Q}^{\mathrm{op}})^{*}(g_{ij}(U))=\tilde{g}_{ij}(U)\prod_{t\geq
k+1}\tilde{g}_{\alpha_{t}+1,\alpha_{t}+1}(U).$ (7.25)

Moreover,
$\mathcal{Q}^{\mathrm{op}}:(G,\mathcal{GC}_{g}^{\dagger}(\mathbf{\Gamma}_{\text{std}}))\dashrightarrow(G,\mathcal{GC}_{g}^{\dagger}(\mathbf{\Gamma}))$
is a birational quasi-isomorphism for
$G\in\\{\operatorname{SL}_{n},\operatorname{GL}_{n}\\}$.

###### Proof.

Part 1 follows from the invariance properties of $\phi$\- and $c$-functions.
Part 2 follows from the commutative diagram (3.55), Proposition 4.15 and
Proposition 7.5. To show that $\mathcal{Q}^{\mathrm{op}}$ is a birational
quasi-isomorphism, one applies Proposition 2.25 and proceeds along the same
lines as in the proof of Proposition 4.15. ∎

Let $\tilde{\mathcal{Q}}^{\mathrm{op}}$ denote the rational map
$\tilde{\mathcal{Q}}^{\mathrm{op}}:(\operatorname{GL}_{n},\pi_{\text{std}}^{\dagger})\dashrightarrow(\operatorname{GL}_{n},\pi_{\tilde{\mathbf{\Gamma}}}^{\dagger})$
given by formula (7.23) relative $\tilde{\mathbf{\Gamma}}$. The following
proposition is a restatement of Proposition 4.17 for
$\mathcal{GC}_{g}^{\dagger}(\mathbf{\Gamma})$, which in turn is a refinement
of Proposition 1.7.

######  Proposition 7.9.

Given any pair $\mathbf{\Gamma}$ and $\tilde{\mathbf{\Gamma}}$ of BD triples
of type $A_{n-1}$, set
$\mathcal{G}^{\mathrm{op}}:=\mathcal{Q}^{\mathrm{op}}\circ(\tilde{\mathcal{Q}}^{\mathrm{op}})^{-1}$.
Then $\mathcal{G}^{\mathrm{op}}$ is a birational map given by the formula:

$\displaystyle\mathcal{G}^{\mathrm{op}}(U)=G^{\mathrm{op}}(U)\cdot U\cdot
G^{\mathrm{op}}(U)^{-1},\ \ $ $\displaystyle
G^{\mathrm{op}}(U):=\left[\prod_{i\geq
1}^{\leftarrow}(\tilde{\gamma})^{i}(G_{0}^{\mathrm{op}}(U))\right]\cdot
G_{0}^{\mathrm{op}}(U),$ (7.26) $\displaystyle
G_{0}^{\mathrm{op}}(U):=\tilde{\gamma}(\tilde{\mathcal{F}}^{\mathrm{op}}(U)_{+})\tilde{\theta}(\tilde{\mathcal{F}}^{\mathrm{op}}(U)_{+})^{-1}.$
(7.27)

Moreover, for $\tilde{\mathbf{\Gamma}}\prec\mathbf{\Gamma}$ and
$G\in\\{\operatorname{SL}_{n},\operatorname{GL}_{n}\\}$,
$\mathcal{G}^{\mathrm{op}}:(G,\mathcal{GC}_{g}^{\dagger}(\tilde{\mathbf{\Gamma}}))\dashrightarrow(G,\mathcal{GC}_{g}^{\dagger}(\mathbf{\Gamma}))$
is a birational quasi-isomorphism, and if the $r_{0}$ parts of
$\pi_{\tilde{\mathbf{\Gamma}}}^{\dagger}$ and
$\pi_{\mathbf{\Gamma}}^{\dagger}$ coincide, then
$\mathcal{G}^{\mathrm{op}}:(G,\pi_{\tilde{\mathbf{\Gamma}}}^{\dagger})\dashrightarrow(G,\pi_{\mathbf{\Gamma}}^{\dagger})$
is a Poisson isomorphism.

###### Proof.

Formula (7.26) follows from Proposition 4.17 and the commutative diagram
(3.55). The proof of the fact that $\mathcal{G}^{\mathrm{op}}$ is a birational
quasi-isomorphism follows along the same lines as the proof of Proposition
4.17. ∎

Formula (7.26) also determines the inverse of $\mathcal{G}^{\mathrm{op}}$, as
in Remark 4.18. In the case $\tilde{\mathbf{\Gamma}}\prec\mathbf{\Gamma}$ and
$|\Gamma_{1}\setminus\tilde{\Gamma}_{1}|=1$, one can also derive a more
explicit expression for $G_{0}^{\mathrm{op}}(U)$ via Proposition 4.19.

###  7.4 Proofs of Theorems 1.5 and 1.6 for
$\mathcal{GC}_{g}^{\dagger}(\mathbf{\Gamma})$

The proofs of Parts 2, 3 and 4 of the theorems are subdivided into the three
propositions below (Part 1 is then a consequence of Part 4). We will transfer
the results from the case of
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma}^{\mathrm{op}})$ to