Seeds

To define a seed, we first define the following data:

• • Let $\tilde {B} = (b_{ij})$ be an $N\times (N+M)$ integer matrix whose principal part B is skew-symmetrizable (recall that the principal part of a matrix is its leading square submatrix). The matrices B and $\tilde {B}$ are called the exchange matrix and the extended exchange matrix, respectively;

• • Let $x_1,\ldots , x_N$ be an algebraically independent subset of $\mathcal {F}$ over $\mathbb {Q}$ such that the elements $x_1,\ldots ,x_N,\ldots ,x_{N+M}$ generate the field $\mathcal {F}$ . The elements $x_1,\ldots ,x_N$ are called cluster variables, and the tuples $\mathbf {x} := (x_1,\ldots ,x_N)$ and $\tilde {\mathbf {x}} := (x_1,\ldots ,x_{N+M})$ are called a cluster and an extended cluster, respectively;

• • For every $1 \leq i \leq N$ , let $d_i$ be a factor of $\gcd (b_{ij} \ | \ 1 \leq j \leq N)$ . The ith string $p_i$ is a tuple $p_i := (p_{ir})_{0 \leq r \leq d_i}$ , where each $p_{ir}$ is a monomial in the stable variables with an integer coefficient and such that $p_{i0} = p_{id_i} = 1$ . The ith string is called trivial if $d_i = 1$ . Set $\mathcal {P} := \{ p_i \ | \ 1 \leq i \leq N\}$ .

Now, a seed is $\Sigma := (\mathbf {x}, \tilde {B}, \mathcal {P})$ and an extended seed is $\tilde {\Sigma } := (\tilde {\mathbf {x}}, \tilde {B}, \mathcal {P})$ . In practice, one additionally names one of the seeds as the initial seed.

Generalized cluster mutations

Let $\Sigma = (\mathbf {x}, \tilde {B},\mathcal {P})$ be a seed constructed via the recipe from the previous paragraph. A generalized cluster mutation in direction k produces a seed $\Sigma ^{\prime } = (\mathbf {x}^{\prime }, \tilde {B}^{\prime }, \mathcal {P}^{\prime })$ that is constructed as follows.

• • Define cluster $\tau $ -monomials $u_{k;>}$ and $u_{k;<}$ , $1 \leq k \leq N$ , via

  $$\begin{align*}u_{k;>} := \prod_{\substack{1 \leq i \leq N, \\ b_{ki} > 0}} x_{i}^{b_{ki}/d_k}, \ \ u_{k;<} := \prod_{\substack{1 \leq i \leq N, \\ b_{ki} < 0}} x_{i}^{-b_{ki}/d_k}, \end{align*}$$
  and stable $\tau $ -monomials $v_{k;>}^{[r]}$ and $v_{k;<}^{[r]}$ , $1 \leq k \leq N$ , $0 \leq r \leq d_k$ , as
  $$\begin{align*}v_{k;>}^{[r]} := \prod_{\substack{N+1\leq i \leq N+M,\\ b_{ki} > 0}} x_i^{\lfloor rb_{ki}/d_k \rfloor}, \ \ v_{k;<}^{[r]} := \prod_{\substack{N+1 \leq i \leq N+M,\\ b_{ki} < 0}} x_i^{\lfloor-rb_{ki}/d_k\rfloor}, \end{align*}$$
  where the product over an empty set by definition equals $1$ and $\lfloor m \rfloor $ denotes the floor of a number $m \in \mathbb {Z}$ . Define $x_k^{\prime }$ via the generalized exchange relation
  (2.1) $$ \begin{align} x_k x_k^{\prime} := \sum_{r=0}^{d_k} p_{kr} u_{k;>}^{r} v_{k;>}^{[r]} u_{k;<}^{d_k-r} v_{k;<}^{[d_k-r]}, \end{align} $$
  and set $\mathbf {x}^{\prime } := (\mathbf {x} \setminus \{x_k\}) \cup \{x_k^{\prime }\}$ .

• • The matrix entries $b_{ij}^{\prime }$ of $\tilde {B}^{\prime }$ are defined as

  $$\begin{align*}b_{ij}^{\prime} := \begin{cases} -b_{ij} \ \ &\text{if } i=k \text{ or }j=k;\\ b_{ij} + \dfrac{|b_{ik}|b_{kj} + b_{ik} |b_{kj}|}{2} \ \ &\text{otherwise.} \end{cases} \end{align*}$$

• • The strings $p_i^{\prime } \in \mathcal {P}^{\prime }$ are given by the exchange coefficient mutation

  $$\begin{align*}p_{ir}^{\prime} := \begin{cases} p_{i,d_i-r} \ \ &\text{if }i = k; \\ p_{ir} \ \ &\text{otherwise.}\end{cases} \end{align*}$$

The seeds $\Sigma $ and $\Sigma ^{\prime }$ are also called adjacent. A few comments on the definition:

1. 1) Call a cluster variable $x_i$ isolated if $b_{ij} = 0$ for all $1 \leq j \leq N+M$ . The definition of $b_{ij}^{\prime }$ implies that a mutation preserves the property of being isolated;

2. 2) Since $\gcd \{b_{ij} \ | \ 1 \leq j \leq N\} = \gcd \{b_{ij}^{\prime } \ | \ 1 \leq j \leq N\}$ , the numbers $d_1,\ldots , d_N$ retain their defining property after a mutation is performed;

3. 3) If a string $p_k$ is trivial, then the generalized exchange relation in equation (2.1) becomes the exchange relation from the ordinary cluster theory of geometric type:

   (2.2) $$ \begin{align} x_k x_k^{\prime} = \prod_{\substack{1 \leq i \leq N+M\\ b_{ki>0}}} x_i^{b_{ki}} + \prod_{\substack{1 \leq i \leq N+M\\ b_{ki<0}}} x_i^{-b_{ki}}. \end{align} $$
   In fact, the generalized cluster structures studied in this paper have only one nontrivial string, hence all exchange relations except one are ordinary.

4. 4) The generalized exchange relation can also be written in the following form. For any i, denote $v_{i;>} := v_{i;>}^{[d_i]}$ , $v_{i;<} := v_{i;<}^{[d_i]}$ ; set

   $$\begin{align*}q_{ir} := \frac{v_{i;>}^r v_{i; < }^{d_i-r}}{(v_{i;>}^{[r]} v_{i;<}^{[d_i-r]})^{d_i}}, \ \ \hat{p}_{ir} := \frac{p_{ir}^{d_i}}{q_{ir}}, \ \ 1 \leq i \leq N,\ 0 \leq r \leq d_i. \end{align*}$$
   Note that the mutation rule for $\hat {p}_{ir}$ is the same as for $p_{ir}$ . Now, equation (2.1) becomes
   $$\begin{align*}x_k x_k^{\prime} = \sum_{r=0}^{d_k} (\hat{p}_{kr} v_{k;>}^r v_{k;<}^{d_k-r})^{1/d_k} u_{k;>}^r u_{k;<}^{d_k-r}. \end{align*}$$
   The expression $(\hat {p}_{kr} v_{k;>}^r v_{k;<}^{d_k-r})^{1/d_k}$ is a monomial in the stable variables.

Generalized cluster structure

Two seeds $\Sigma $ and $\Sigma ^{\prime }$ are called mutation equivalent if there’s a sequence $\Sigma _1, \ldots , \Sigma _m$ such that $\Sigma _1 = \Sigma $ , $\Sigma _m = \Sigma ^{\prime }$ and such that $\Sigma _{i+1}$ and $\Sigma _i$ are adjacent for each i. For a fixed seed $\Sigma $ , the set of all seeds that are mutation equivalent to $\Sigma $ is called the generalized cluster structure and is denoted as $\mathcal {GC}(\Sigma )$ or simply $\mathcal {GC}$ .

Generalized cluster algebra

Let $\mathcal {GC}$ be a generalized cluster structure constructed as above. Define $\mathbb {A} := \mathbb {Z}[x_{N+1},\ldots ,x_{N+M}]$ and $\bar {\mathbb {A}} := \mathbb {Z}[x_{N+1}^{\pm 1},\ldots , x_{N+M}^{\pm 1}]$ . Choose a ground ring $\hat {\mathbb {A}}$ , which is a subring of $\bar {\mathbb {A}}$ that contains $\mathbb {A}$ . The $\hat {\mathbb {A}}$ -subalgebra of $\mathcal {F}$ given by

(2.3) $$ \begin{align} \mathcal{A} := \mathcal{A(\mathcal{GC})} := \hat{\mathbb{A}}[\ \text{cluster variables from all seeds in }\mathcal{GC}\ ] \end{align} $$

is called the generalized cluster algebra. For any seed $\Sigma := ((x_1,\ldots ,x_{N}),\tilde {B},\mathcal {P})$ , set

(2.4) $$ \begin{align} \mathcal{L}(\Sigma) := \hat{\mathbb{A}}[x_1^{\pm 1},\ldots,x_N^{\pm 1}] \end{align} $$

to be the ring of Laurent polynomials associated with $\Sigma $ , and define

(2.5) $$ \begin{align} \bar{\mathcal{A}} :=\bar{\mathcal{A}}(\mathcal{GC}):= \bigcap_{\Sigma \in \mathcal{GC}}\mathcal{L}(\Sigma). \end{align} $$

The algebra $\bar {\mathcal {A}}$ is called the generalized upper cluster algebra. The generalized Laurent phenomenon states that $\mathcal {A} \subseteq \bar {\mathcal {A}}$ .

Upper bounds

Let $\mathbb {T}_N$ be a labeled N-regular tree. Associate with each vertex a seed so that adjacent seeds are adjacent in the tree,Footnote ² and if a seed $\Sigma ^{\prime }$ is adjacent to $\Sigma $ in direction k, label the corresponding edge in the tree with number k. A nerve $\mathcal {N}$ in $\mathbb {T}_N$ is a subtree on $N+1$ vertices such that all its edges have different labels (for instance, a star is a nerve). An upper bound $\bar {\mathcal {A}}(\mathcal {N})$ is defined as the algebra

(2.6) $$ \begin{align} \bar{\mathcal{A}}(\mathcal{N}) := \bigcap_{\Sigma \in V(\mathcal{N})}\mathcal{L}(\Sigma) \end{align} $$

where $V(\mathcal {N})$ stands for the vertex set of $\mathcal {N}$ . Upper bounds were first defined and studied in [Reference Berenstein, Fomin and Zelevinsky3]. Let L be the number of isolated variables in $\mathcal {GC}$ . For the ith nontrivial string in $\mathcal {P}$ , let $\tilde {B}(i)$ be a $(d_{i}-1) \times L$ matrix such that the rth row consists of the exponents of the isolated variables in $p_{ir}$ (recall that $p_{ir}$ is a monomial in the stable variables). The following result was proved in [Reference Gekhtman, Shapiro and Vainshtein18].

Generalized cluster structures on varieties

Let V be a Zariski open subset of $\mathbb {C}^{N+M}$ , $\mathcal {O}(V)$ be the ring of regular functions, and let $\mathbb {C}(V)$ be the field of rational functions on V. As before, let $\mathcal {GC}$ be a generalized cluster structure, and assume that $f_1,\ldots , f_{N+M}$ is a transcendence basis of $\mathbb {C}(V)$ over $\mathbb {C}$ . Pick an extended cluster $(x_1,\ldots ,x_{N+M})$ in $\mathcal {GC}$ , and define a field isomorphism $\theta : \mathcal {F}_{\mathbb {C}} \rightarrow \mathbb C(V)$ via $\theta : x_i \mapsto f_i$ , $1 \leq i \leq N+M$ , where $\mathcal {F}_{\mathbb {C}} := \mathcal {F} \otimes \mathbb {C}$ is the extension by complex scalars of $\mathcal {F}$ . The pair $(\mathcal {GC}, \theta )$ (or sometimes just $\mathcal {GC}$ ) is called a generalized cluster structure on V. It’s called regular if $\theta (x)$ is a regular function for every variable x. Choose a ground ring as

$$\begin{align*}\hat{\mathbb{A}}:= \mathbb Z[x_{N+1}^{\pm 1},\ldots, x_{N+M^{\prime}}^{\pm 1}, x_{N+M^{\prime}+1},\ldots, x_{N+M}], \end{align*}$$

where $\theta (x_{N+i})$ does not vanish on V if and only if $1 \leq i \leq M^{\prime }$ . Set $\mathcal {A}_{\mathbb C} := \mathcal {A} \otimes \mathbb {C}$ and $\bar {\mathcal {A}}_{\mathbb C}:= \bar {\mathcal {A}}\otimes \mathbb C$ .

In the case of ordinary cluster structures, the proof of Proposition 2.2 is available in [Reference Gekhtman, Shapiro and Vainshtein15] (Proposition 3.37) and in a more general setup in [Reference Fomin, Williams and Zelevinsky13] (Proposition 6.4.1). As explained in [Reference Gekhtman, Shapiro and Vainshtein18], Proposition 2.2 is a direct corollary of a natural extension of Proposition 3.6 in [Reference Fomin and Pylyavskyy11] to the case of generalized cluster structures. When $\theta $ is an isomorphism between $\bar {\mathcal {A}}_{\mathbb {C}}(\mathcal {GC})$ and $\mathcal {O}(V)$ , these algebras are also said to be naturally isomorphic. A practical way of verifying Condition 2.2 of Proposition 2.2 is based on Proposition 2.1.

Poisson structures in $\mathcal {GC}$

Let $\{\cdot , \cdot \}$ be a Poisson bracket on $\mathcal {F}$ (or on $\mathcal {F}_{\mathbb {C}}$ ), and let $\tilde {\mathbf {x}}$ be any extended cluster in $\mathcal {GC}$ . We say that $\tilde {\mathbf {x}}$ is log-canonical if $\{x_i,x_j\} = \omega _{ij} x_i x_j$ for all $1 \leq i,j \leq N+M$ , where $\omega _{ij} \in \mathbb {Q}$ (or $\omega _{ij} \in \mathbb {C}$ for $\mathcal {F}_{\mathbb C}$ ). We call the generalized cluster structure compatible with the bracket if any extended cluster in $\mathcal {GC}$ is log-canonical. Let $\Omega :=(\omega _{ij})_{i,j=1}^{N+M}$ be the coefficient matrix of the bracket with respect to the extended cluster $\tilde {\mathbf {x}}$ . The following proposition is a natural generalization of Theorem 4.5 from [Reference Gekhtman, Shapiro and Vainshtein15].

Condition ii has the following interpretation, which is used in practice. For each $1\leq i \leq N$ , define $y_i := \prod _{j=1}^{N+M} x_j^{b_{ij}}$ . Then ii is equivalent to $\{\log y_i, \log x_j\} = \delta _{ij}\Delta _{ii}$ , where $\delta _{ij}$ is the Kronecker symbol. The variable $y_i$ is called the y-coordinate of the cluster variable $x_i$ . Note that Condition ii implies that $\tilde {B}$ has full rank.

Toric actions

Given an extended cluster $(x_1,\ldots ,x_{N+M})$ in $\mathcal {GC}$ , a local toric action (of rank s) is an action $\mathcal {F}_{\mathbb {C}} \curvearrowleft (\mathbb {C}^*)^s$ by field automorphisms given on the variables $x_i$ ’s as

$$\begin{align*}x_i.(t_1,\ldots,t_s) \mapsto x_i \prod_{j=1}^{s} t_j^{\omega_{ij}}, \ \ \ t_j \in \mathbb{C}^*, \end{align*}$$

where $W:= (\omega _{ij})$ is an integer-valued $(N+M) \times s$ matrix of rank s called the weight matrix of the action. We say that two local toric actions of rank s defined on some extended clusters $\tilde {\mathbf {x}}$ and $\tilde {\mathbf {x}}^{\prime }$ are compatible if the composition of mutations that takes $\tilde {\mathbf {x}}$ to $\tilde {\mathbf {x}}^{\prime }$ intertwines the actions. A collection of pairwise compatible local toric actions of rank s defined for every extended cluster is called a global toric action. We also say that a local toric action is $\mathcal {GC}$ -extendable if it belongs to some global toric action.

As noted in [Reference Gekhtman, Shapiro and Vainshtein18], this proposition is a natural extension of Lemma 5.3 in [Reference Gekhtman, Shapiro and Vainshtein15]. For the purposes of this paper, it suffices to assume that $\hat {p}_{ir}$ are invariant with respect to the action; however, in the case of ordinary cluster structures of geometric type, the statement of the proposition is if and only if.

Quasi-isomorphisms that arise from global toric actions

Let $\mathcal {GC}_1(\Sigma _1)$ and $\mathcal {GC}_2(\Sigma _2)$ be generalized cluster structures with initial extended seeds $\Sigma _1:=(\tilde {\mathbf {x}}, \tilde {B}_1, \mathcal {P}_1)$ and $\Sigma _2 := (\tilde {\mathbf {f}}, \tilde {B}_2, \mathcal {P}_2)$ , and let $\mathcal {F}_1$ and $\mathcal {F}_2$ be the corresponding ambient fields. Assume the following:

• • There is the same number of cluster and stable variables in $\tilde {\mathbf {x}}$ and $\tilde {\mathbf {f}}$ ;

• • The numbers $d_1,\ldots ,d_N$ from the definition of the generalized cluster structure are equal for both $\mathcal {GC}_1$ and $\mathcal {GC}_2$ ;

• • The strings $\mathcal {P}_1$ and $\mathcal {P}_2$ are the same in the following sense: If one picks $p_{ir}$ and substitutes all $x_i$ ’s with $f_i$ ’s, one obtains the rth component of the ith string from $\mathcal {P}_2$ , and vice versa;

• • The extended exchange matrices $\tilde {B}_1$ and $\tilde {B}_2$ are the same in all but the last column, which corresponds to a stable variable;

• • There are integer-valued vectors $u = (u_1,\ldots ,u_{n+m})^t$ and $v = (v_1,\ldots ,v_{n+m})^t$ that define local toric actions (of rank $1$ ) on $\tilde {\mathbf {x}}$ and $\tilde {\mathbf {f}}$ , respectively, and they are $\mathcal {GC}$ -extendable.

Proposition 2.5. Assume that $\frac {v_i-u_i}{u_{N+M}}$ is an integer for each $1 \leq i \leq N+M$ . Define a field isomorphism $\theta :\mathcal {F}_2 \rightarrow \mathcal {F}_1$ on the generators as $\theta (f_i):=x_i x_{N+M}^{\left (\frac {v_i-u_i}{u_{N+M}}\right )}$ , $1 \leq i \leq N+M$ . If $\tilde {\mathbf {x}}^{\prime } := (x_1^{\prime },\ldots ,x_{N+M}^{\prime })$ and $\tilde {\mathbf {f}}^{\prime } := (f_1^{\prime },\ldots ,f_{N+M}^{\prime })$ are two extended clusters obtained from $\tilde {\mathbf {x}}$ and $\tilde {\mathbf {f}}$ via the same sequence of mutations (i.e., the mutations followed the same indices), and $(u_1^{\prime },\ldots , u^{\prime }_{N+M})^t$ and $(v_1^{\prime },\ldots ,v^{\prime }_{N+M})^t$ are the weight vectors of the global toric actions in the extended clusters $\tilde {\mathbf {x}}^{\prime }$ and $\tilde {\mathbf {f}}^{\prime }$ , then

$$\begin{align*}\theta(f_i^{\prime}) = x_i^{\prime} x_{N+M}^{\left(\frac{v_i^{\prime} - u_i^{\prime}}{u_{N+M}}\right)}, \ \ 1 \leq i \leq N+M. \end{align*}$$

The above proposition is a generalization of Lemma 8.4 from [Reference Gekhtman, Shapiro and Vainshtein17] to the case of generalized cluster structures. The map $\theta $ is an instance of a quasi-isomorphism defined by Fraser in [Reference Fraser12].

Quiver

A quiver is a directed multigraph with no $1$ - and $2$ -cycles. Pick an extended seed $(\tilde {\mathbf {x}},\tilde {B},\mathcal {P})$ , and let $D:=\operatorname {\mathrm {diag}}(d_1^{-1},\ldots , d_N^{-1})$ be a diagonal matrix with $d_i$ ’s defined as above. Assume that $DB$ is skew-symmetric, where B is the principal part of $\tilde {B}$ . Then the matrix

$$\begin{align*}\hat{B}:= \begin{bmatrix} DB & \tilde{B}^{[N+1,N+M]} \\ -(\tilde{B}^{[N+1,N+M]})^T & 0 \end{bmatrix} \end{align*}$$

is the adjacency matrix of a quiver Q, in which each vertex i corresponds to a variable $x_i \in \tilde {\mathbf {x}}$ . The vertices that correspond to cluster variables are called mutable, the vertices that correspond to stable variables are called frozen and the vertices that correspond to isolated variables are called isolated. For each i, the number $d_i$ is called the multiplicity of the ith vertex. If one mutates the extended seed $(\tilde {x},\tilde {B},\mathcal {P})$ , then the quiver of the new seed can be obtained from the initial quiver via the following steps:

1. 1) For each path $i \rightarrow k \rightarrow j$ , add an arrow $i \rightarrow j$ ;

2. 2) If there is a pair of arrows $i \rightarrow j$ and $j \rightarrow i$ , remove both;

3. 3) Flip the orientation of all arrows going in and out of the vertex k.

The above process is also called a quiver mutation in direction k (or at vertex k). Instead of describing the matrix $\tilde {B}$ , we describe the corresponding quiver and multiplicities $d_i$ ’s.

Poisson–Lie groups

A Poisson bracket $\{\cdot , \cdot \}$ on a commutative algebra is a Lie bracket that satisfies the Leibniz rule in each slot. Given a manifold M, a Poisson bivector field on M is a section $\pi \in \Gamma (M,\bigwedge ^2 TM)$ such that $\{f,g\}:=\pi (df\wedge dg)$ is a Poisson bracket on the space of smooth functions on M. A Lie group G endowed with a Poisson bivector field $\pi $ is called a Poisson–Lie group if for any $g,h \in G$ , $\pi _{gh} = (dL_g \otimes dL_g)\pi _h + (dR_h \otimes dR_h)\pi _g$ , where $L_g$ and $R_h$ are the left and right translations by g and h, respectively. Let $\mathfrak {g}$ be the Lie algebra of G and $r \in \mathfrak g \otimes \mathfrak g$ . If G is a connected Lie group, then the bivector fieldFootnote ³ $\pi _g := (dL_g\otimes dL_g) r - (dR_g \otimes dR_g)r$ defines the structure of a Poisson-Lie group on G if and only if the following conditions are satisfied:

1. 1) The symmetric part of r is $\operatorname {\mathrm {ad}}$ -invariant;

2. 2) The $3$ -tensor $[r,r]:=[r_{12},r_{13}] + [r_{12},r_{23}] + [r_{13},r_{23}]$ is $\operatorname {\mathrm {ad}}$ -invariant, where $(a\otimes b)_{12} = a \otimes b \otimes 1$ , $(a \otimes b)_{13} = a \otimes 1 \otimes b$ and $(a\otimes b)_{23} = 1 \otimes a \otimes b$ , $a,b \in \mathfrak {g}$ .

The classical Yang–Baxter equation (CYBE) is the equation $[r,r] = 0$ . For simple complex Lie algebras $\mathfrak {g}$ , Belavin and Drinfeld in [Reference Belavin and Drinfeld1, Reference Belavin and Drinfeld2] classified solutions of the CYBE that have a nondegenerate symmetric part. The classification was partially extended by Hodges in [Reference Hodges22] to the case of reductive complex Lie algebras (however, Hodges required the symmetric part of r to be a multiple of the Casimir element). A full classification of solutions of the CYBE with an arbitrary nondegenerate $\operatorname {\mathrm {ad}}$ -invariant symmetric part in the case of reductive complex Lie algebras was obtained by Delorme in [Reference Delorme7].

The Belavin–Drinfeld classification

Let $\mathfrak {g}$ be a reductive complex Lie algebra endowed with a nondegenerate symmetric invariant bilinear form $\langle \,,\,\rangle $ , and let $\Pi $ be a set of simple roots of $\mathfrak {g}$ . A Belavin–Drinfeld triple (for conciseness, a BD triple) is a triple $(\Gamma _1, \Gamma _2, \gamma )$ with $\Gamma _1, \Gamma _2 \subset \Pi $ and $\gamma : \Gamma _1 \rightarrow \Gamma _2$ a nilpotent isometry. The nilpotency condition means that for any $\alpha \in \Gamma _1$ there exists a number j such that $\gamma ^j(\alpha ) \notin \Gamma _1$ . Decompose $\mathfrak {g}$ as $\mathfrak {g} = \mathfrak {n}_+ \oplus \mathfrak {h} \oplus \mathfrak {n}_-$ , where $\mathfrak {n}_+ = \bigoplus _{\alpha> 0} \mathfrak {g}_{\alpha }$ and $\mathfrak {n}_- = \bigoplus _{\alpha> 0} \mathfrak {g}_{-\alpha }$ are nilpotent subalgebras, $\mathfrak {g}_{\alpha }$ are root subspaces and $\mathfrak {h}$ is a Cartan subalgebra. For every positive root $\alpha $ , choose $e_{\alpha } \in \mathfrak {g}_{\alpha }$ and $e_{-\alpha } \in \mathfrak {g}_{-\alpha }$ such that $\langle e_{\alpha },e_{-\alpha }\rangle = 1$ , and set $h_{\alpha }:=[e_{\alpha },e_{-\alpha }]$ . Let $\mathfrak {g}_{\Gamma _1}$ and $\mathfrak {g}_{\Gamma _2}$ be the simple Lie subalgebras of $\mathfrak {g}$ generated by $\Gamma _1$ and $\Gamma _2$ . Extend $\gamma $ to an isomorphism $\mathbb {Z}\Gamma _1 \xrightarrow {\sim } \mathbb {Z}\Gamma _2$ , and then define $\gamma : \mathfrak {g}_{\Gamma _1} \rightarrow \mathfrak {g}_{\Gamma _2}$ via $\gamma (e_{\alpha }) = e_{\gamma (\alpha )}$ and $\gamma (h_{\alpha }) = h_{\gamma (\alpha )}$ . Let $\gamma ^* : \mathfrak {g}_{\Gamma _2} \rightarrow \mathfrak {g}_{\Gamma _1}$ be the conjugate of $\gamma $ with respect to the form on $\mathfrak {g}$ . Extend both $\gamma $ and $\gamma ^*$ by zero to $[\mathfrak {g},\mathfrak {g}]$ . For an element $r \in \mathfrak {g}\otimes \mathfrak {g}$ , set $R_+, R_- :\mathfrak {g}\rightarrow \mathfrak {g}$ to be the linear transformations determined by $\langle R_+(x), y\rangle = \langle r, x\otimes y\rangle $ and $\langle R_-(y), x\rangle = -\langle r, x\otimes y\rangle $ , $x, y \in \mathfrak {g}$ . Let $\pi _{>}$ , $\pi _{<}$ and $\pi _0$ be the projections onto $\mathfrak {n}_+$ , $\mathfrak {n}_-$ and $\mathfrak {h}$ , respectively. In terms of $R_+$ and $R_-$ , the CYBE assumes the form

(2.7) $$ \begin{align} [R_+(x), R_+(y)] = R_+([R_+(x),y] + [x, R_-(y)]), \ \ x, y \in \mathfrak{g}. \end{align} $$

Let $R_0 : \mathfrak {h} \rightarrow \mathfrak {h}$ be a linear transformation that satisfies the following conditions:

(2.8) $$ \begin{align} R_0 + R_0^* = \operatorname{\mathrm{id}}_{\mathfrak{h}}; \end{align} $$

(2.9) $$ \begin{align} R_0(\alpha-\gamma(\alpha)) = \alpha, \ \ \alpha \in \Gamma_1, \end{align} $$

where $\operatorname {\mathrm {id}}_{\mathfrak {h}} : \mathfrak {h} \rightarrow \mathfrak {h}$ is the identity and $R_0^*$ is the adjoint of $R_0$ . If $\mathfrak {g}$ is simple, then the solutions $R_0$ of equations (2.8)–(2.9) form an affine subspace of $\hom (\mathfrak {h},\mathfrak {h})$ (linear maps) of dimension $k_{\mathbf {\Gamma }}(k_{\mathbf {\Gamma }}-1)/2$ .

Theorem 2.6. (Belavin, Drinfeld) Under the above setup, if

(2.10) $$ \begin{align} R_+ = \frac{1}{1-\gamma} \pi_{>} - \frac{\gamma^*}{1-\gamma^*}\pi_{<} + R_0 \pi_0, \end{align} $$

where $R_0$ is any solution of the system (2.8)–(2.9), then $R_+$ satisfies the CYBE (2.7). Moreover,

(2.11) $$ \begin{align} R_+ - R_- = \operatorname{\mathrm{id}}_{\mathfrak{g}}. \end{align} $$

Conversely, if $R_+ : \mathfrak {g} \rightarrow \mathfrak {g}$ is any linear transformation that satisfies equation (2.11), then $R_+$ assumes the form (2.10) for a suitable decomposition of $\mathfrak {g}$ , for some Belavin–Drinfeld triple and some choice of root vectors $e_{\alpha }$ .

The matrix $R_+$ from the theorem is called a classical R-matrix. In this form, the theorem follows from Theorem 6.3 in [Reference Hodges22]. It is important that the form on $\mathfrak {g}$ is fixed; however, if $\mathfrak {g}$ is simple, then all nondegenerate symmetric invariant bilinear forms are multiples of one another, so the theorem yields a full classification of solutions $r \in \mathfrak {g}\otimes \mathfrak {g}$ of the CYBE with nondegenerate $\operatorname {\mathrm {ad}}$ -invariant symmetric parts.

The Drinfeld double

Let G be a reductive complex connected Poisson–Lie group endowed with a nondegenerate symmetric invariant bilinear form on $\mathfrak {g}$ and with a Poisson bivector field defined as

$$\begin{align*}\pi_g := (dL_g \otimes dL_g)r - (dR_g\otimes dR_g)r\end{align*}$$

for some $r \in \mathfrak {g}\otimes \mathfrak {g}$ that satisfies the conditions of Theorem 2.6. Let $R_+$ and $R_-$ be defined from r as in the previous paragraph, and set $\mathfrak {d} := \mathfrak g \oplus \mathfrak g$ to be the direct sum of Lie algebras. Define a nondegenerate symmetric invariant bilinear form on $\mathfrak {d}$ as

$$\begin{align*}\langle (x_1,y_1), (x_2,y_2) \rangle = \langle x_1, x_2 \rangle - \langle y_1, y_2 \rangle, \ \ x_1, \,x_2,\, y_1,\, y_2 \in \mathfrak{g}. \end{align*}$$

As a vector space, $\mathfrak {d}$ splits into the direct sum of the following isotropic Lie subalgebras:

$$\begin{align*}\mathfrak{g}^{\delta} := \{(x, x) \ | \ x \in \mathfrak{g}\}, \ \ \mathfrak{g}^* := \{(R_+(x), R_-(x)) \ | \ x \in \mathfrak{g}\}. \end{align*}$$

Set $R^{\mathfrak {d}}_+ := P_{\mathfrak {g}^*}$ , where $P_{\mathfrak {g}^*}$ is the projection of $\mathfrak {d}$ onto $\mathfrak {g}^*$ , and let $r^{\mathfrak {d}} \in \mathfrak {d}\otimes \mathfrak {d}$ be the $2$ -tensor that corresponds to $R^{\mathfrak {d}}_+$ . Then $R_+^{\mathfrak {d}}$ yields the structure of a Poisson–Lie group on the Lie group $D(G):=G \times G$ via the Poisson bivector field $\pi ^{\mathfrak {d}}_{(g,h)} := (dL_{(g,h)}\otimes dL_{(g,h)}) r^{\mathfrak {d}} - (dR_{(g,h)}\otimes dR_{(g,h)})r^{\mathfrak {d}}$ , ${(g,h)} \in D(G)$ . The Poisson–Lie group $D(G)$ is called the Drinfeld double of G.

The Poisson bracket on $D(G)$ can be written in the form

$$\begin{align*}\{f_1,f_2\} = \langle R_+(E_L f_1), E_Lf_2 \rangle - \langle R_+(E_R f_1), E_Rf_2 \rangle + \langle \nabla_X^R f_1, \nabla^R_Y f_2\rangle - \langle \nabla_X^L f_1, \nabla_Y^L f_2\rangle, \end{align*}$$

where $\nabla ^L f_i = (\nabla _X^L f_i, -\nabla _Y^L f_i)$ and $\nabla ^R f_i = (\nabla _X^R f_i, -\nabla _Y^R f_i)$ are the left and the right gradients, respectively, $E_Lf_i = \nabla ^L_X f_i+ \nabla ^L_Yf_i$ and $E_Rf_i = \nabla ^R_Xf_i + \nabla ^R_Yf_i$ . We define the gradients on G asFootnote ⁴

$$\begin{align*}\langle \nabla^L f|_g, x\rangle = \left. \frac{d}{dt}\right|{}_{t=0} f(g\exp (tx)), \ \ \ \langle \nabla^R f|_g, x\rangle = \left. \frac{d}{dt}\right|{}_{t=0} f(\exp(tx)g), \ \ g \in G, \ x \in \mathfrak{g}. \end{align*}$$

The group G can be identified with the connected Poisson–Lie subgroup $G^{\delta }$ of $D(G)$ that corresponds to the Lie subalgebra $\mathfrak {g}^{\delta }$ . The Poisson bracket $\{\cdot , \cdot \}_G$ on G can be expressed as

$$\begin{align*}\{f_1,f_2\}_G = \langle R_+(\nabla^L f_1), \nabla^L f_2 \rangle - \langle R_+(\nabla^R f_1), \nabla^R f_2 \rangle. \end{align*}$$

Additionally, the connected Poisson–Lie subgroup $G^*$ of $D(G)$ that corresponds to $\mathfrak {g}^*$ is called the dual Poisson–Lie group of G. The Poisson structure on $G^*$ (which is induced from $D(G)$ ) can be modeled locally in the group G via the map

$$\begin{align*}G^* \ni (g,h) \mapsto gh^{-1} \in G. \end{align*}$$

The image of this map is an open dense subset of G denoted as $G^{\dagger }$ (however, the map is not injective in general).

Following [Reference Gekhtman, Shapiro and Vainshtein20], we consider a more general Poisson bracket on $G\times G$ that is defined by a pair of classical R-matrices $R_+^c$ and $R_+^r$ (the meaning of the upper indices is unveiled later in the text). For such a pair, the Poisson bracket is defined as

(2.12) $$ \begin{align} \{f_1,f_2\} = \langle R_+^c(E_L f_1), E_Lf_2 \rangle - \langle R_+^r(E_R f_1), E_Rf_2 \rangle + \langle \nabla_X^R f_1, \nabla^R_Y f_2\rangle - \langle \nabla_X^L f_1, \nabla_Y^L f_2\rangle. \end{align} $$

We will frequently abuse the terminology and call $G\times G$ endowed with bracket (2.12) the Drinfeld double of G. However, bracket (2.12) yields the structure of a Poisson–Lie group on $G\times G$ if and only if $R^c_+ = R^r_+$ .

Symplectic foliation and Poisson submanifolds

Let $(M,\pi )$ be a Poisson manifold. An immersed submanfiold $S \subseteq M$ is called a Poisson submanifold if $\pi |_S \in \Gamma (S,\bigwedge ^2TS)$ . Examples of Poisson submanifolds include nonsingular parts of the zero loci of frozen variables (see Section 3.8). Let $\pi ^{\#} : TM^* \rightarrow TM$ be a morphism of vector bundles defined as $\langle \pi ^{\#}(\xi ),\eta \rangle := \langle \pi ,\xi \wedge \eta \rangle $ , $\xi ,\eta \in T_p^*M$ , $p \in M$ . The Poisson bivector $\pi $ is called nondegenerate if $\pi ^{\#}$ is an isomorphism of vector bundles. A symplectic leaf is a maximal (by inclusion) connected Poisson submanifold S of M for which $\pi |_S$ is nondegenerate. It is a theorem that any Poisson manifold M is a union of its symplectic leaves.

BD graph for a pair of BD triples

The graph $G_{(\mathbf {\Gamma }^r, \mathbf {\Gamma }^c)}$ is defined in the following way. The vertex set of the graph consists of two copies of $[1,n-1]$ , one of each is called the upper part and the other one is the lower part. We draw an edge between vertices i and $n-i$ if they belong to the same part (if $i = n-i$ , we draw a loop). If $\gamma _r(i) = j$ , draw a directed edge from i in the upper part to j in the lower part; if $\gamma _c(i) = j$ , draw a directed edge from j in the lower part to i in the upper part. The edges between vertices of the same part are called horizontal, between different parts vertical. Figure 1 provides two examples of BD graphs.

Figure 1

Examples of BD graphs. The vertical directed edges coming from $\mathbf {\Gamma }^r$ and $\mathbf {\Gamma }^c$ are painted in red and blue, respectively.

Paths in $G_{(\mathbf {\Gamma }^r,\mathbf {\Gamma }^c)}$ and aperiodicity

There is no orientation assigned to horizontal edges, hence we allow them to be traversed in both directions. An alternating path in $G_{(\mathbf {\Gamma }^r,\mathbf {\Gamma }^c)}$ is a path in which horizontal and vertical edges alternate. A path is a cycle if it starts where it ends. Now, we call the pair $(\mathbf {\Gamma }^r, \mathbf {\Gamma }^c)$ aperiodic if $G_{(\mathbf {\Gamma }^r, \mathbf {\Gamma }^c)}$ has no alternating cycles (equivalently, the map $\gamma _c^{-1} w_0 \gamma _r w_0$ is nilpotent, where $w_0$ is the longest Weyl group element of $[1,n-1]$ ). In examples in this paper, we denote alternating paths as $\cdots \rightarrow i \xrightarrow {X} i^{\prime } \xrightarrow {\gamma _r} j \xrightarrow {Y} j^{\prime } \xrightarrow {\gamma _c^*} i^{\prime } \rightarrow \cdots $ , where X and Y indicate whether an edge is in the upper or the lower part of $G_{(\mathbf {\Gamma }^r,\mathbf {\Gamma }^c)}$ , respectively, and $\gamma _r$ and $\gamma _c^*$ indicate a vertical edge directed downwards or upwards.

Oriented BD triples

Let $\mathbf {\Gamma } = (\Gamma _1,\Gamma _2,\gamma )$ be a BD triple. Since $\gamma $ is an isometry, if $\gamma (i) = j$ and $i+1 \in \Gamma _1$ , then $\gamma (i+1) = j \pm 1$ . We call the BD triple $\mathbf {\Gamma }$ oriented if $\gamma (i+1) = j+1$ for every $i \in \Gamma _1$ such that $i+1 \in \Gamma _1$ . A pair of BD triples $(\mathbf {\Gamma }^r, \mathbf {\Gamma }^c)$ is called oriented if both $\mathbf {\Gamma }^r$ and $\mathbf {\Gamma }^c$ are oriented.

Runs

Let $\mathbf {\Gamma } = (\Gamma _1,\Gamma _2,\gamma )$ be a BD triple. For an arbitrary $i \in [1,n]$ , set

$$\begin{align*}i_- := \max\{ j \in [0,n] \setminus \Gamma_1 \ | \ j < i\}, \ \ i_+ := \min \{j \in [1,n] \setminus \Gamma_1 \ | \ j \geq i \}. \end{align*}$$

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. For a pair of BD triples $(\mathbf {\Gamma }^r,\mathbf {\Gamma }^c)$ , the runs that correspond to $\mathbf {\Gamma }^r$ and $\mathbf {\Gamma }^c$ are called row and column runs, respectively. We will indicate with an upper index r or c whether a run is from $\mathbf {\Gamma }^r$ or $\mathbf {\Gamma }^c$ .

Properties of $\mathcal {L}$ -matrices

Observe the following:

• • The blocks are aligned in such a way that the indices in the blocks that correspond to the runs $\Delta ^r$ and $\bar {\Delta }^r$ (or $\Delta ^c$ and $\bar {\Delta }^c$ ) are in the same rows (columns);

• • For any variable $x_{ij}$ or $y_{ji}$ with $i> j$ , there is a unique $\mathcal {L}(X,Y)$ that contains it on the diagonal;

• • If $i_1 \rightarrow i_2 \rightarrow \cdots \rightarrow i_{2n}$ is the maximal alternating path that gives rise to $\mathcal {L}$ , then the size $N(\mathcal {L}) \times N(\mathcal {L})$ of $\mathcal {L}$ can be determined as

  $$\begin{align*}N(\mathcal{L}) = \sum_{k=1}^{n} i_{2k-1}. \end{align*}$$

Description of $\varphi $ -, f- and c-functions

Set $U:= X^{-1} Y$ and define

$$\begin{align*}F_{kl}(X,Y) := | X^{[n-k+1,n]} \ Y^{[n-l+1,n]}|_{[n-k-l+1,n]}, \ \ k, l \geq 1, \ k + l \leq n-1; \end{align*}$$

$$\begin{align*}\Phi_{kl}(X,Y) = [(U^0)^{[n-k+1,n]} \, U^{[n-l+1,n]} \, (U^2)^{[n]} \, \cdots \, (U^{n-k-l+1})^{[n]}], \ \ k,l \geq 1, \ k+l \leq n; \end{align*}$$

set $\tilde {\varphi }_{kl}(X,Y) := \det \Phi _{kl}(X,Y)$ and

(3.1) $$ \begin{align} f_{kl}(X,Y) := \det F_{kl}(X,Y), \ \ \varphi_{kl}(X,Y):=s_{kl} (\det X)^{n-k-l+1} \tilde{\varphi}_{kl}(X,Y), \end{align} $$

where

$$\begin{align*}s_{kl} = \begin{cases} (-1)^{k(l+1)} &n \text{ is even,}\\ (-1)^{(n-1)/2 + k(k-1)/2 + l(l-1)/2} &n \text{ is odd.} \end{cases} \end{align*}$$

All f- and $\varphi $ -functions are considered as cluster variables. The c-functions are defined via

$$\begin{align*}\det (X+\lambda Y) = \sum_{i=0}^{n} \lambda^{i} s_i c_i(X,Y), \end{align*}$$

where $s_i = (-1)^{i(n-1)}$ . Note that $c_{0} = \det X$ and $c_n = \det Y$ . The functions $c_1,\ldots ,c_{n-1}$ are considered as isolated stable variables, and the only nontrivial string, which is attached to $\varphi _{11}$ , is given by the tuple $(1,c_1,\ldots ,c_{n-1},1)$ .

Description of g- and h-functions

For $i> j$ , let $\mathcal {L}$ be an $\mathcal {L}$ -matrix such that $\mathcal {L}_{ss}(X,Y) = x_{ij}$ for some s, and let $N(\mathcal {L})$ be the size of $\mathcal {L}$ . Set $g_{ij} := \det \mathcal {L}^{[s,N(\mathcal {L})]}_{[s,N(\mathcal {L})]}$ . Similarly, if $\mathcal {L}$ is such that $\mathcal {L}_{ss}(X,Y) = y_{ji}$ , we set $h_{ji} := \det \mathcal {L}^{[s,N(\mathcal {L})]}_{[s,N(\mathcal {L})]}$ . In addition, we let $g_{ii} := \det X^{[i,n]}_{[i,n]}$ and $h_{ii} := \det Y^{[i,n]}_{[i,n]}$ , $1 \leq i \leq n$ . The functions $h_{11}$ and $g_{11}$ , as well as the determinants of the $\mathcal {L}$ -matrices, are considered as stable variables.

Conventions

The following identifications are frequently used in the text:

(3.2) $$ \begin{align} \begin{aligned} f_{n-l,l} &:= \varphi_{n-l,l},& &1 \leq l \leq n-1;\\ f_{0,l} &:= h_{n-l+1,n-l+1},& &1 \leq l \leq n-1;\\ f_{k,0} & := g_{n-k+1,n-k+1},& &1 \leq k \leq n-1. \end{aligned} \end{align} $$

The above equalities are set in concordance with the defining formulas for the variables, for which one simply extends the range of the allowed indices. Furthermore, for g-functions we set

(3.3) $$ \begin{align} g_{n+1,i+1} := \begin{cases} h_{1,j+1} &\text{if} \ \gamma_c^*(j) = i,\\ 1 &\text{otherwise;} \end{cases} \ \ \ g_{i,0} := \begin{cases} h_{jn} & \text{if} \ \gamma_r(i)=j,\\ 1 & \text{otherwise;} \end{cases} \end{align} $$

and for h-functions we set

(3.4) $$ \begin{align} h_{j+1,n+1} := \begin{cases} g_{i+1,1} &\text{if} \ \gamma_r(i) = j,\\ 1 &\text{otherwise;} \end{cases} \ \ \ h_{0,j} := \begin{cases} g_{ni} &\text{if}\ \gamma_c^*(j) = i,\\ 1 &\text{otherwise.} \end{cases} \end{align} $$

The meaning of these identifications follows from the following observation: If $g_{ni} = \det \mathcal {L}^{[s,N(\mathcal {L})]}_{[s,N(\mathcal {L})]}$ and $\gamma _c^*(j) = i$ , then $h_{1,j+1} = \det \mathcal {L}^{[s+1,N(\mathcal {L})]}_{[s+1,N(\mathcal {L})]}$ ; similarly, if $h_{jn} = \det \mathcal {L}^{[s,N(\mathcal {L})]}_{[s,N(\mathcal {L})]}$ and $\gamma _r(i) = j$ , then $g_{i+1,1} = \det \mathcal {L}^{[s+1,N(\mathcal {L})]}_{[s+1,N(\mathcal {L})]}$ .

Description of $\mathcal {GC}(\mathbf {\Gamma })$

The description of the initial quiver is given later in Section 3.6. The initial extended cluster is given by the union

$$\begin{align*}\begin{aligned} \{g_{ij}, \ h_{ji} \ | \ 1 \leq j \leq i \leq n\} \cup \{f_{kl} \ | \ k, l \geq 1&, \ k+l \leq n-1 \}\cup \\ &\cup \{ \varphi_{kl} \ | \ k, l \geq 1, \ k+l \leq n \} \cup \{ c_{i} \ | \ 1 \leq i \leq n-1\}. \end{aligned} \end{align*}$$

Let $\mathcal {L}_1(X,Y), \ldots , \mathcal {L}_m(X,Y)$ be the list of all $\mathcal {L}$ -matrices in $\mathcal {GC}(\mathbf {\Gamma })$ . The ground ring $\hat {\mathbb {A}} = \hat {\mathbb {A}}(\mathcal {GC}(\mathbf {\Gamma }))$ is set to be

$$\begin{align*}\hat{\mathbb{A}} := \mathbb{C}[c_1, \ldots, c_{n-1}, h_{11}^{\pm 1}, g_{11}^{\pm 1}, \det \mathcal{L}_1,\ldots, \det \mathcal{L}_m]. \end{align*}$$

All mutation relations are ordinary except the mutation at $\varphi _{11}$ . It is given by

(3.5) $$ \begin{align} \varphi_{11} \varphi_{11}^{\prime} = \sum_{r=0}^n c_r \varphi_{21}^r \varphi_{12}^{n-r}. \end{align} $$

A variable $\psi $ is frozen if and only if either $\psi = c_i$ for $0 \leq i \leq n$ , or $\psi = g_{i+1,1}$ for $i \in \Pi \setminus \Gamma _1^r$ , or $\psi = h_{1,j+1}$ for $j \in \Pi \setminus \Gamma _2^c$ .

The operators $\gamma , \gamma ^* : \operatorname {\mathrm {gl}}_n(\mathbb C) \rightarrow \operatorname {\mathrm {gl}}_n(\mathbb C)$

Let $\mathbf {\Gamma } := (\Gamma _1,\Gamma _2,\gamma )$ be an oriented BD triple. 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, where $\gamma (\Delta _i) = \bar {\Delta }_i$ , $1 \leq i \leq k$ . Set $\operatorname {\mathrm {gl}}(\Delta _i)$ to be a subalgebra of $\operatorname {\mathrm {gl}}_n(\mathbb {C})$ of the matrices that are zero outside of the block $\Delta _i \times \Delta _i$ (and similarly for $\operatorname {\mathrm {gl}}(\Delta _i)$ ). Define $\gamma _i : \operatorname {\mathrm {gl}}_n(\Delta _i) \rightarrow \operatorname {\mathrm {gl}}_n(\bar {\Delta }_i)$ to be the map that shifts the $\Delta _i\times \Delta _i$ block to $\bar {\Delta }_i \times \bar {\Delta }_i$ . Then the map $\gamma : \operatorname {\mathrm {gl}}_n(\mathbb {C}) \rightarrow \operatorname {\mathrm {gl}}_n(\mathbb {C})$ is defined as the direct sum $\gamma :=\bigoplus _{i=1}^k\gamma _i$ extended by zero to $\operatorname {\mathrm {gl}}_n(\mathbb {C})$ . Similarly, one sets $\gamma _i^* : \operatorname {\mathrm {gl}}_n(\bar {\Delta }_i)\rightarrow \operatorname {\mathrm {gl}}_n(\Delta _i)$ to be the map that shifts the $\bar {\Delta }_i \times \bar {\Delta }_i$ block to $\Delta _i \times \Delta _i$ . The map $\gamma ^* : \operatorname {\mathrm {gl}}_n(\mathbb {C}) \rightarrow \operatorname {\mathrm {gl}}_n(\mathbb {C})$ is obtained as the direct sum $\gamma ^*:=\bigoplus _{i=1}^k\gamma _i^*$ extended by zero to $\operatorname {\mathrm {gl}}_n(\mathbb {C})$ .

Example 3.4. Let us consider a BD pair defined by its BD graph below (note: $\Gamma _1^c = \emptyset $ ):

Similarly, the action of $\gamma ^*$ is given by

$$\begin{align*}\gamma^* \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} & a_{15}\\a_{21} & a_{22} & a_{23} & a_{24} & a_{25}\\ a_{31} & a_{32} & a_{33} & a_{34} & a_{35}\\ a_{41} & a_{42} & a_{43} & a_{44} & a_{45}\\a_{51} & a_{52} & a_{53} & a_{54} & a_{55}\end{bmatrix} = \begin{bmatrix}a_{33} & a_{34} & a_{35} & 0 & 0\\ a_{43} & a_{44} & a_{45} & 0 & 0 \\ a_{53} & a_{54} & a_{55} & 0 & 0 \\ 0 & 0 & 0 & a_{11} & a_{12} \\ 0 & 0 & 0 & a_{21} & a_{22}\end{bmatrix}. \end{align*}$$

The group homomorphisms $\tilde {\gamma }$ and $\tilde {\gamma }^*$

The maps $\gamma , \gamma ^* : \operatorname {\mathrm {gl}}_n(\mathbb C) \rightarrow \operatorname {\mathrm {gl}}_n(\mathbb C)$ are not Lie algebra homomorphisms; however, their restrictions to the Borel subalgebras $\mathfrak {b}_+$ and $\mathfrak {b}_-$ are Lie algebra homomorphisms, hence we can define group homomorphisms $\tilde {\gamma }, \tilde {\gamma }^* : \mathfrak {B}_{\pm } \rightarrow \mathfrak {B}_{\pm }$ , where $\mathfrak {B}_+$ and $\mathfrak {B}_-$ are the corresponding Borel subgroups. Notice that if the BD triple is oriented and $N_{\pm }$ is a unipotent (upper or lower) triangular matrix, then $\tilde {\gamma }(N_{\pm }) = \gamma (N_{\pm } - I) + I$ , where I is the identity matrix, and similarly for $\tilde {\gamma }^*$ . Likewise, let $\operatorname {\mathrm {GL}}(\Delta )\hookrightarrow \operatorname {\mathrm {GL}}_n$ be the group of invertible $|\Delta |\times |\Delta |$ matrices viewed as a block in $\operatorname {\mathrm {GL}}_n$ that occupies $\Delta \times \Delta $ ; since $\gamma : \operatorname {\mathrm {gl}}(\Delta ) \rightarrow \operatorname {\mathrm {gl}}(\bar {\Delta })$ is an isomorphism of Lie algebras, it can be integrated to an isomorphism of groups $\gamma : \operatorname {\mathrm {GL}}(\Delta ) \rightarrow \operatorname {\mathrm {GL}}(\bar {\Delta })$ (and similarly for $\gamma ^*$ ).

Differential operators

For a rational function $f \in \mathbb C(\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_n)$ , set

$$\begin{align*}\nabla_X f := \left( \frac{\partial f}{\partial x_{ji}} \right) _{i,j=1}^{n}, \ \ \nabla_Y f := \left( \frac{\partial f}{\partial y_{ji}} \right) _{i,j=1}^{n}. \end{align*}$$

Define

$$ \begin{align*} E_Lf &:= \nabla_X f \cdot X + \nabla_Y f \cdot Y, & E_R f &:= X \nabla_X f + Y \nabla_Y f, \\ \xi_L f &:= \gamma_c(\nabla_X f \cdot X) + \nabla_Y f \cdot Y, & \xi_R f &:= X\nabla_X f + \gamma_r^*(Y \nabla_Y f), \\ \eta_L f &:= \nabla_X f \cdot X + \gamma_c^*(\nabla_Y f \cdot Y), & \eta_R f&:= \gamma_r(X \nabla_X f) + Y \nabla_Y f. \end{align*} $$

Let $\ell $ denote r or c. Define subalgebras

$$\begin{align*}\mathfrak{g}_{\Gamma^{\ell}_1} := \bigoplus_{i=1}^{k} \operatorname{\mathrm{gl}}(\Delta_i^{\ell}), \ \ \ \ \mathfrak{g}_{\Gamma^{\ell}_2} := \bigoplus_{i=1}^{k} \operatorname{\mathrm{gl}}(\bar{\Delta}^{\ell}_i), \end{align*}$$

where $\operatorname {\mathrm {gl}}(\Delta _i^{\ell })$ and $\operatorname {\mathrm {gl}}(\bar {\Delta }_i^{\ell })$ are constructed above. Let $\pi _{\Gamma _1^{\ell }}$ and $\pi _{\Gamma _2^{\ell }}$ be the projections onto $\mathfrak {g}_{\Gamma ^{\ell }_1}$ and $\mathfrak {g}_{\Gamma ^{\ell }_2}$ , respectively; also, let $\pi _{\hat {\Gamma }_1^{\ell }}$ and $\pi _{\hat {\Gamma }_2^{\ell }}$ be the projections onto the orthogonal complements of $\mathfrak {g}_{\Gamma ^{\ell }_1}$ and $\mathfrak {g}_{\Gamma ^{\ell }_2}$ with respect to the trace form. There are numerous identities that relate the differential operators among each other and with the projections; they are easily derivable and extensively used in the paper. Let us mention some of them:

$$ \begin{align*} E_L &= \xi_L + (1-\gamma_c) (\nabla_X X), & E_R &= \xi_R + (1-\gamma_r^*)(Y\nabla_Y),\\ E_L &= \eta_L + (1-\gamma_c^*)(\nabla_Y Y), & E_R &= \eta_R + (1-\gamma_r)(X \nabla X),\\ \xi_L &= \gamma_c(\eta_L) + \pi_{\hat{\Gamma}_2^c}(\nabla_Y Y), & \xi_R &= \gamma_r^*(\eta_R) + \pi_{\hat{\Gamma}_1^r} (X \nabla_X),\\ \eta_L &= \gamma_c^*(\xi_L) + \pi_{\hat{\Gamma}_1^c} (\nabla_X X), & \eta_R &= \gamma_r(\xi_R) + \pi_{\hat{\Gamma}_2^r}(Y\nabla_Y). \end{align*} $$

The bracket and $R_0$

For any choice of $(R_0^r,R_0^c)$ on $\operatorname {\mathrm {SL}}_n\times \operatorname {\mathrm {SL}}_n$ , the variables $c_0,c_1,\ldots ,c_{n-1},c_n$ are Casimirs of the Poisson bracket. However, there is only one choice of $(R_0^r,R_0^c)$ for which these variables are Casimirs on $\operatorname {\mathrm {GL}}_n\times \operatorname {\mathrm {GL}}_n$ :

1. a) The functions $c_0, c_1,\ldots , c_{n-1},c_n$ are Casimirs if and only if the identity matrix is an eigenvector of both $R_0^r$ and $R_0^c$ (in this case, $R_0^r(I) = R_0^c(I) = (1/2)I$ from $R_0 + R_0^* = \operatorname {\mathrm {id}}_{\mathfrak h}$ ).

However, there is an important alternative choice of $(R_0^r,R_0^c)$ :

1. b) For a BD triple $(\Gamma _1,\Gamma _2,\gamma )$ , a solution $R_0$ of equations (2.8) and (2.9) is such that

   (3.6) $$ \begin{align}\begin{aligned} R_0 (1-\gamma) &= \pi_{\Gamma_1} + R_0 \pi_{\hat{\Gamma}_1} \qquad&\qquad R_0(1-\gamma^*) &= -\gamma^* +R_0 \pi_{\hat{\Gamma}_2}\\ R_0^*(1-\gamma) &= -\gamma + R_0^*\pi_{\hat{\Gamma_1}}\qquad&\qquad R_0^*(1-\gamma^*) &= \pi_{\Gamma_2} + R_0^*\pi_{\hat{\Gamma}_2} \end{aligned} \end{align} $$
   (the identities are viewed relative the Cartan subalgebra $\mathfrak {h}$ of $\operatorname {\mathrm {gl}}_n$ ).

Note that these conditions do not follow from the system (2.8) and (2.9). For instance, if $I_{\Delta }:=\sum _{i\in \Delta }e_{ii}$ , the first condition specifies the value of $R_0$ on $I_{\Delta }-I_{\bar {\Delta }}$ as

$$\begin{align*}R_0(I_{\Delta}-I_{\bar{\Delta}}) = I_{\Delta}. \end{align*}$$

Choosing $R_0^r$ and $R_0^c$ that satisfy equation (3.6) eases some of the computations with Poisson brackets, so this choice is employed in the proofs; however, in Section 8.2 we show that the results of the paper hold regardless of the choice of $(R_0^r,R_0^c)$ . Moreover, when $R_0:= R_0^r = R_0^c$ and $R_0$ satisfies equation (3.6), the connected Poisson dual $\operatorname {\mathrm {GL}}_n^*$ of $\operatorname {\mathrm {GL}}_n$ can be viewed as a subgroup of the direct product of certain parabolic subgroups modulo a relation (see Section 3.8 for details). Lastly, the Poisson bracket (2.12) attains the following form on $\operatorname {\mathrm {GL}}_n\times \operatorname {\mathrm {GL}}_n$ :

$$\begin{align*}\{f_1,f_2\} = \langle R_+^c (E_Lf_1), E_L f_2\rangle - \langle R_+^r(E_R f_1), E_R f_2\rangle + \langle X \nabla_X f_1, Y\nabla_Y f_2 \rangle - \langle \nabla_X f_1 \cdot X, \nabla_Y f_2 \cdot Y \rangle. \end{align*}$$

Invariance properties of f- and $\varphi $ -functions

Let f be any f-function and $\tilde {\varphi }$ be any $\tilde {\varphi }$ -function (recall that $\tilde {\varphi }$ differs from $\varphi $ by a factor of $\det X$ ; see equation (3.1)). Pick any unipotent upper triangular matrix $N_+$ , a pair of any unipotent lower triangular matrices $N_-$ and $N_-^{\prime }$ , and let A be any invertible matrix. Then

(3.7) $$ \begin{align} f(X,Y) = f(N_+X N_-, N_+ Y N_-^{\prime}), \ \ \ \tilde{\varphi}(X,Y) = \tilde{\varphi}(AXN_-, AYN_-). \end{align} $$

Let $\mathfrak {b}_+$ and $\mathfrak {b}_-$ be the subspaces of upper and lower triangular matrices. The infinitesimal version of equation (3.7) is

(3.8) $$ \begin{align}\begin{aligned} \nabla_X f \cdot X, \ \nabla_Y f\cdot Y \in &\,\mathfrak{b}_-, \qquad&\qquad &E_Rf \in \mathfrak{b}_+;\\ E_L\tilde{\varphi} \in &\,\mathfrak{b}_-, \qquad&\qquad &E_R\tilde{\varphi} = 0. \end{aligned}\end{align} $$

Moreover,

(3.9) $$ \begin{align}\begin{aligned} \pi_0 E_L \log f &= \text{const}, \qquad&\qquad \pi_0 E_R \log f &= \text{const}, \\ \pi_0 E_L \log \varphi &= \text{const}, \qquad&\qquad \pi_0 E_R \log \varphi &= \text{const}, \end{aligned}\end{align} $$

where $\pi _0$ is the projection onto the space of diagonal matrices; by const we mean that the left-hand sides (LHS) of the formulas do not depend on $(X,Y)$ . For the c-functions,

$$\begin{align*}\pi_0 E_L \log c_i = \pi_0 E_R \log c_i = I, \ \ \ 0 \leq i \leq n, \end{align*}$$

where I is the identity matrix.

Invariance properties of g- and h-functions

Let $\psi $ be any g- or h-function, and let $N_+$ and $N_-$ be any unipotent upper and lower triangular matrices. Then

(3.10) $$ \begin{align} \psi(N_+ X, \tilde{\gamma}_r(N_+)Y) = \psi(X\tilde{\gamma}_c^*(N_-),YN_-) = \psi(X,Y). \end{align} $$

Let T be any diagonal matrix; then we also have

(3.11) $$ \begin{align} \begin{aligned} \psi(X\tilde{\gamma}_c^*(T),YT) &= \hat{\xi}_L(T)\psi(X,Y), & \psi(TX,\tilde{\gamma}_r(T)Y) &= \hat{\xi}_R(T) \psi(X,Y),\\ \psi(XT,Y\tilde{\gamma}_c(T)) &= \hat{\eta}_L(T) \psi(X,Y), & \psi(\tilde{\gamma}_r^*(T)X,TY) &= \hat{\eta}_R(T) \psi(X,Y), \end{aligned}\end{align} $$

where $\hat {\xi }_R$ , $\hat {\xi }_L$ , $\hat {\eta }_R$ and $\hat {\eta }_L$ are constants that depend only on T and $\psi $ (they can be viewed as characters on the group of invertible diagonal matrices). The infinitesimal version of equation (3.10) is

(3.12) $$ \begin{align} \xi_L \psi \in \mathfrak{b}_-,\ \ \ \xi_R \psi \in \mathfrak{b}_+, \end{align} $$

and the infinitesimal version of equation (3.11) is

(3.13) $$ \begin{align} \begin{aligned} \pi_0 \xi_L \log \psi &= \text{const}, \qquad&\qquad \pi_0 \xi_R \log \psi &= \text{const},\\ \pi_0 \eta_L \log \psi &= \text{const}, \qquad&\qquad \pi_0 \eta_R \log \psi &= \text{const}. \end{aligned} \end{align} $$

Finally, let us mention the results of Lemma 4.4 and Corollary 4.6 from [Reference Gekhtman, Shapiro and Vainshtein20]. If $\Delta ^r$ , $\Delta ^c$ , $\bar {\Delta }^r$ and $\bar {\Delta }^c$ are any X- and Y- row and column runs (trivial or not), then

(3.14) $$ \begin{align}\begin{aligned} \operatorname{\mathrm{tr}}( (\nabla_X\log \psi\cdot X)_{\Delta^c}^{\Delta^c}) &= \text{const}, \qquad&\qquad \operatorname{\mathrm{tr}}( (X \nabla_X\log \psi)_{\Delta^r}^{\Delta^r}) &= \text{const}, \\ \operatorname{\mathrm{tr}}( (\nabla_Y \log \psi\cdot Y)_{\bar{\Delta}^c}^{\bar{\Delta}^c}) & = \text{const}, \qquad&\qquad \operatorname{\mathrm{tr}}( (Y\nabla_Y \log \psi)_{\bar{\Delta}^r}^{\bar{\Delta}^r}) & = \text{const}; \end{aligned}\end{align} $$

also,

(3.15) $$ \begin{align}\begin{aligned} \operatorname{\mathrm{tr}}( \nabla_X\log \psi\cdot X) &= \text{const}, \qquad&\qquad \operatorname{\mathrm{tr}}(X \nabla_X\log \psi) &= \text{const}, \\ \operatorname{\mathrm{tr}}( \nabla_Y \log \psi\cdot Y) & = \text{const}, \qquad&\qquad \operatorname{\mathrm{tr}}(Y\nabla_Y \log \psi) & = \text{const}. \end{aligned} \end{align} $$

3.6.1 The quiver for the trivial BD pair

Below one can find pictures of the neighborhoods of all variables in the initial quiver in the case of the trivial BD pair. A few of remarks beforehand:

• • The circled vertices are mutable (in the sense of ordinary exchange relations (2.2)), the square vertices are frozen, the rounded square vertices may or may not be mutable depending on the indices and the hexagon vertex is a mutable vertex with a generalized mutation relation (see equations (2.1) and (3.5));

• • Since $c_1,\ldots ,c_{n-1}$ are isolated variables, they are not shown on the resulting quiver;

• • For $k=2$ and $n> 3$ , the vertices $\varphi _{1k}$ and $\varphi _{k-1,2}$ coincide; hence, the pictures provided below suggest that there are two edges pointing from $\varphi _{21}$ to $\varphi _{12}$ (however, there is only one arrow in $n = 3$ ).

Figure 2

The neighborhood of $\varphi _{kl}$ for $k,l \neq 1$ , $k+l < n$ .

Figure 3

The neighborhood of $\varphi _{1l}$ for $2 \leq l \leq n-1$ .

Figure 4

The neighborhood of $\varphi _{k1}$ for $2 \leq k \leq n-1$ .

Figure 5

The neighborhood of $\varphi _{kl}$ for (a) $k=l=1$ and (b) $k+l=n$ .

Figure 6

The neighborhood of $f_{kl}$ for $k+l <n$ (convention (3.2) is in place.).

Figure 7

The neighborhood of $g_{ij}$ for $1 < j \leq i\leq n$ (convention (3.3) is in place).

Figure 8

The neighborhood of $g_{i1}$ for $1 \leq i \leq n$ .

Figure 9

The neighborhood of $h_{ij}$ for $1 < i < j \leq n$ .

Figure 10

The neighborhood of $h_{ij}$ for $1 < i = j \leq n$ .

Figure 11

The neighborhood of $h_{1j}$ .

3.6.2 The quiver for a nontrivial BD pair (algorithm)

If $\mathbf {\Gamma } = (\mathbf {\Gamma }^r, \mathbf {\Gamma }^c)$ is nontrivial, one proceeds as follows. First, draw the quiver for the case of the trivial BD pair, employing the neighborhoods as described above. Second, add new arrows as prescribed by the following algorithm:

1. 1) If $i \in \Gamma _1^r$ , unfreeze $g_{i+1,1}$ and add additional arrows, as indicated in Figure 12(a);

   Figure 12

   Additional arrows for $g_{i+1,1}$ and $h_{1,j+1}$ .

   

2. 2) If $j \in \Gamma _2^c$ , unfreeze $h_{1,j+1}$ and add additional arrows, as indicated in Figure 12(b);

3. 3) Repeat for all roots in $\Gamma _1^r$ and $\Gamma _2^c$ .

Note that the algorithm does not depend on the order of the roots of $\Gamma _1^r$ and $\Gamma _2^c$ . Indeed, adding new arrows corresponds to adding a certain matrix (determined by the figure) to the current adjacency matrix of the quiver; since addition of matrices is commutative, the order of the roots is irrelevant.

Figure 13

The neighborhood of $g_{11}$ .

Figure 14

The neighborhood of $g_{i1}$ for $1 < i < n$ .

Figure 15

The neighborhood of $g_{n1}$ .

Figure 16

The neighborhood of $g_{nj}$ for $2 \leq j \leq n$ .

Figure 17

The neighborhood of $h_{nn}$ .

Figure 18

The neighborhood of $h_{in}$ for $2 \leq j \leq n-1$ .

Figure 19

The neighborhood of $h_{1n}$ .

Figure 20

The neighborhood of $h_{1j}$ for $1<j<n$ .

Figure 21

The neighborhood of $h_{11}$ .

3.6.3 The quiver for a nontrivial BD pair (explicit)

As an alternative to the algorithm described in the previous paragraph, we provide explicit neighborhoods of the variables $g_{i1}$ , $h_{1i}$ , $g_{ni}$ , $h_{in}$ , $1 \leq i \leq n$ in the case of a nontrivial BD pair. All the other neighborhoods are the same as in the case of the trivial BD pair.

Notation

We denote by $(X,Y)$ the standard coordinates on $D(\operatorname {\mathrm {GL}}_n)$ (regardless of the associated BD pair). If $\psi $ is a cluster or stable variable in $\mathcal {GC}(\mathbf {\Gamma })$ , then by $\tilde {\psi }$ we denote the corresponding variable in $\mathcal {GC}(\tilde {\mathbf {\Gamma }})$ ; that is, $\psi $ and $\tilde {\psi }$ are either the variables attached to the same vertices in the initial quivers or in the quivers that are obtained via the same sequences of mutations. All g-, h-, f-, $\varphi $ - and c- functions in the initial extended cluster of $\mathcal {GC}(\tilde {\mathbf {\Gamma }})$ are marked with a tilde as well.

Removing the rightmost root from a row run

Let $\Delta ^r = [p+1,p+k]$ be a nontrivial row X-run in $\mathbf {\Gamma }$ , and let $\bar {\Delta }^r = [q+1,q+k] := \gamma _r(\Delta ^r)$ be the corresponding row Y-run. Define $\tilde {\mathbf {\Gamma }} = (\tilde {\mathbf {\Gamma }}^r, \mathbf {\Gamma }^c)$ with $\tilde {\mathbf {\Gamma }}^r = (\tilde {\Gamma }_1^r, \tilde {\Gamma }_2^r, \gamma _r|_{\tilde {\Gamma }_1^r})$ given by $\tilde {\Gamma }_1^r = \Gamma _1^r \setminus \{p+k-1\}$ and $\tilde {\Gamma }_2^r = \Gamma _2^r \setminus \{q+k-1\}$ . Let us examine the difference between the $\mathcal {L}$ -matrices in $\mathcal {GC}(\mathbf {\Gamma })$ and $\mathcal {GC}(\tilde {\mathbf {\Gamma }})$ . For any $\mathcal {L}$ -matrix $\mathcal {L}(X,Y)$ in $\mathcal {GC}(\mathbf {\Gamma })$ , let $\tilde {\mathcal {L}}(X,Y)$ be a matrix obtained from $\mathcal {L}(X,Y)$ via removing the last row of each Y-block of the form $Y^{J}_{[1,q+k]}$ . If $\mathcal {L}(X,Y)$ arises from a maximal alternating path in $G_{\mathbf {\Gamma }}$ that does not pass through the edge $(p+k-1)\xrightarrow {\gamma _r}(q+k-1)$ , then $\tilde {\mathcal {L}}(X,Y)$ is an $\mathcal {L}$ -matrix in $\mathcal {GC}(\tilde {\mathbf {\Gamma }})$ that arises from the same path in $G_{\tilde {\mathbf {\Gamma }}}$ . However, if $\mathcal {L}^*(X,Y):=\mathcal {L}(X,Y)$ is constructed from a path that does pass through $(p+k-1)\xrightarrow {\gamma _r}(q+k-1)$ , then $\tilde {\mathcal {L}}^*(X,Y)$ is a reducible matrix with blocks that correspond to the remaining two $\mathcal {L}$ -matrices in $\mathcal {GC}(\tilde {\mathbf {\Gamma }})$ . Let us set $s_0$ to be the number such that $\mathcal {L}^*_{s_0s_0}(X,Y) = x_{p+k,1}$ . Define a rational map $\mathcal {U} : D(\operatorname {\mathrm {GL}}_n)_{\tilde {\mathbf {\Gamma }}} \dashrightarrow D(\operatorname {\mathrm {GL}}_n)_{\mathbf {\Gamma }}$ via the following data:

(5.1) $$ \begin{align} \alpha_i(X,Y) := \frac{1}{\tilde{g}_{p+k,1}(X,Y)} {\det \tilde{\mathcal{L}}^*}^{[s_0, N(\tilde{\mathcal{L}}^*)]}_{\{s_0-k+i\}\cup [s_0+1,N(\tilde{\mathcal{L}}^*)]}(X,Y), \ \ i=1,\ldots,k-1; \end{align} $$

(5.2) $$ \begin{align} U_0(X,Y) = I + \sum_{i = 1}^{k-1} \alpha_i(X,Y) e_{q+i,q+k}; \end{align} $$

(5.3) $$ \begin{align} U_+(X,Y) := \left(\prod_{k\geq 1}^{\leftarrow} \tilde{\gamma}_r^k(U_0)\right)U_0; \end{align} $$

(5.4) $$ \begin{align} \mathcal{U}(X,Y) := \left(U_+(X,Y)X,U_+(X,Y)Y\right). \end{align} $$

Proposition 5.1. Let $\mathcal {U}:D(\operatorname {\mathrm {GL}}_n)_{\tilde {\mathbf {\Gamma }}} \dashrightarrow D(\operatorname {\mathrm {GL}}_n)_{\mathbf {\Gamma }}$ be the rational map given by equation (5.4). Then the map $\mathcal {U}$ acts on the cluster and stable variables via the following formulas:

(5.5) $$ \begin{align} \mathcal{U}^*(g_{ij}(X,Y)) = \begin{cases} \tilde{g}_{ij}(X,Y)\tilde{g}_{p+k,1}(X,Y) \ &\text{if}\ \mathcal{L}^*_{ss}(X,Y) = x_{ij} \ \text{for} \ s < s_0;\\ \tilde{g}_{ij}(X,Y) \ &\text{otherwise}; \end{cases} \end{align} $$

(5.6) $$ \begin{align} \mathcal{U}^*(h_{ij}(X,Y)) = \begin{cases} \tilde{h}_{ij}(X,Y)\tilde{g}_{p+k,1}(X,Y) \ &\text{if}\ \mathcal{L}^*_{ss}(X,Y) = y_{ij} \ \text{for} \ s < s_0;\\ \tilde{h}_{ij}(X,Y) \ &\text{otherwise}; \end{cases} \end{align} $$

if $\psi $ is any $\varphi $ -, f- or c- function in the initial extended cluster, then

(5.7) $$ \begin{align} \mathcal{U}^*(\psi(X,Y)) = \tilde{\psi}(X,Y). \end{align} $$

Note that the first lines in equations (5.5) and (5.6) reflect the fact that $\tilde {\mathcal {L}}^*(X,Y)$ is a reducible matrix with blocks equal to a pair of $\mathcal {L}$ -matrices from $\mathcal {GC}(\tilde {\mathbf {\Gamma }})$ . The proof of the above proposition is exactly the same as in [Reference Gekhtman, Shapiro and Vainshtein20].

Motivation of formulas (5.1)–(5.4)

Though the formulas are complicated, they are designed in concordance with the invariance properties of the variables. The $\varphi $ -, f- and c-variables are the same in the initial extended clusters of $\mathcal {GC}(\mathbf {\Gamma })$ and $\mathcal {GC}(\tilde {\mathbf {\Gamma }})$ , and they are all invariant with respect to the left action $N_+.(X,Y) = (N_+X,N_+Y)$ by unipotent upper triangular matrices. Since $U_+$ is such, we see that formula (5.7) holds. Now, if $\psi $ is a g- or h-function, recall that one of its invariance properties reads

$$\begin{align*}\psi(N_+X,\tilde{\gamma}_r(N_+)Y) = \psi(X,Y). \end{align*}$$

Notice that $\tilde {\gamma }_r(U_+)\cdot U_0 = U_+$ ; therefore,

$$\begin{align*}\mathcal{U}^*(\psi(X,Y)) = \psi(U_+X,U_+Y) = \psi(U_+X,\tilde{\gamma}_r(U_+)U_0Y) = \psi(X,U_0Y); \end{align*}$$

hence, the main part of the proof of Proposition 5.1 is to show that

$$\begin{align*}\psi(X,U_0Y) = \tilde{\psi}(X,Y)\tilde{g}_{p+k,1}^{\varepsilon}(X,Y) \end{align*}$$

for some $\varepsilon \geq 0$ . A similar reasoning explains formulas for $\mathcal {U}$ for other choices of roots below.

The inverse of $\mathcal {U}$

Though we do not need formulas for the inverse of $\mathcal {U}$ in the proofs (except in some simple cases), let us state them for completeness. Let $\theta _r:=\gamma _r|_{\tilde {\Gamma }_1^r}$ be the BD map for the triple $\tilde {\mathbf {\Gamma }}^r$ . The verification of the formulas is similar to the proof of Proposition 5.1 and is based on an application of a series of long Plücker relations.

(5.8) $$ \begin{align} \beta_i(X,Y) := -\frac{1}{g_{p+k,1}(X,Y)} {\det (\mathcal{L}}^*)^{[s_0, N(\tilde{\mathcal{L}}^*)]}_{\{s_0-k+i\}\cup [s_0+1,N(\tilde{\mathcal{L}}^*)]}(X,Y), \ \ i=1,\ldots,k-1; \end{align} $$

(5.9) $$ \begin{align} \tilde{U}_0(X,Y) = I + \sum_{i = 1}^{k-1} \beta_i(X,Y) e_{q+i,q+k}; \end{align} $$

(5.10) $$ \begin{align} \tilde{U}_+(X,Y) := \left(\prod_{k\geq 1}^{\leftarrow} \tilde{\theta}_r^k(\tilde{U}_0)\right)\tilde{U}_0; \end{align} $$

(5.11) $$ \begin{align} \mathcal{U}^{-1}(X,Y) := \left(\tilde{U}_+(X,Y)X,\tilde{U}_+(X,Y)Y\right). \end{align} $$

The formulas for the inverse of $\mathcal {U}$ in the other cases below are obtained via the same scheme: 1) add an extra negative sign in front of the coefficients; 2) substitute $\tilde {\mathcal {L}}^*$ with $\mathcal {L}^*$ and the frozen variable in the denominator with the corresponding cluster variable from $\mathcal {GC}(\mathbf {\Gamma })$ ; 3) substitute $\tilde {\gamma }$ with $\tilde {\theta }$ .

Removing the leftmost root from a row run

As before, let $\Delta ^r = [p+1,p+k]$ be a nontrivial row X-run in $\mathbf {\Gamma }$ and let $\bar {\Delta }^r = [q+1,q+k] := \gamma _r(\Delta ^r)$ be the corresponding row Y-run. Define $\tilde {\mathbf {\Gamma }} = (\tilde {\mathbf {\Gamma }}^r, \mathbf {\Gamma }^c)$ with $\tilde {\mathbf {\Gamma }}^r = (\tilde {\Gamma }_1^r, \tilde {\Gamma }_2^r, \gamma _r|_{\tilde {\Gamma }_1^r})$ given by $\tilde {\Gamma }_1^r = \Gamma _1^r \setminus \{p+1\}$ and $\tilde {\Gamma }_2^r = \Gamma _2^r \setminus \{q+1\}$ . For an $\mathcal {L}$ -matrix $\mathcal {L}(X,Y)$ in $\mathcal {GC}(\mathbf {\Gamma })$ , let $\tilde {\mathcal {L}}(X,Y)$ be a matrix that is obtained from $\mathcal {L}(X,Y)$ by removing the first row of each X-block of the form $X^{J}_{[p+1,n]}$ . If $\mathcal {L}(X,Y)$ arises from a path that does not traverse the edge $(p+1) \xrightarrow {\gamma _r} (q+1)$ , then $\tilde {\mathcal {L}}(X,Y)$ is an $\mathcal {L}$ -matrix in $\mathcal {GC}(\tilde {\mathbf {\Gamma }})$ ; if it does traverse the latter edge, $\tilde {\mathcal {L}}(X,Y)$ is a reducible matrix with blocks that are $\mathcal {L}$ -matrices in $\mathcal {GC}(\tilde {\mathbf {\Gamma }})$ . Let us denote the $\mathcal {L}$ -matrix that corresponds to the latter path as $\mathcal {L}^*(X,Y)$ , and let us denote by $s_0$ the number for which $\mathcal {L}^*_{s_0s_0}(X,Y) = x_{p+2,2}$ . We define the rational map $\mathcal {U}:D(\operatorname {\mathrm {GL}}_n)_{\tilde {\mathbf {\Gamma }}} \dashrightarrow D(\operatorname {\mathrm {GL}}_n)_{\mathbf {\Gamma }}$ via the following data:

(5.12) $$ \begin{align} \alpha_i(X,Y) := (-1)^{i-1}\frac{1}{\tilde{g}_{p+2,1}(X,Y)} {\det \tilde{\mathcal{L}}^*}^{[s_0, N(\tilde{\mathcal{L}}^*)]}_{[s_0-1,N(\tilde{\mathcal{L}}^*)]\setminus\{s_0+i-1\}}(X,Y), \ \ i = 1,\ldots,k-1; \end{align} $$

(5.13) $$ \begin{align} U_0 := I + \sum_{i=1}^{k-1}\alpha_{i}(X,Y)e_{q+1,q+i+1}; \end{align} $$

(5.14) $$ \begin{align} U_+:=\left(\prod_{k\geq 1}^{\leftarrow} \tilde{\gamma}_r^k(U_0)\right)U_0; \end{align} $$

(5.15) $$ \begin{align} \mathcal{U}(X,Y) := \left(U_+(X,Y)X,U_+(X,Y)Y\right). \end{align} $$

The next proposition corresponds to Theorem 7.3 in [Reference Gekhtman, Shapiro and Vainshtein20] and can be proved in exactly the same way:

Proposition 5.2. Let $\mathcal {U}:D(\operatorname {\mathrm {GL}}_n)_{\tilde {\mathbf {\Gamma }}} \dashrightarrow D(\operatorname {\mathrm {GL}}_n)_{\mathbf {\Gamma }}$ be the rational map given by equation (5.15). Then the map $\mathcal {U}$ acts on the cluster and stable variables via the following formulas:

(5.16) $$ \begin{align} \mathcal{U}^*(g_{ij}(X,Y)) = \begin{cases} \tilde{g}_{ij}(X,Y)\tilde{g}_{p+2,1}(X,Y) \ &\text{if}\ \mathcal{L}^*_{ss}(X,Y) = x_{ij} \ \text{for} \ s < s_0;\\ \tilde{g}_{ij}(X,Y) \ &\text{otherwise}; \end{cases} \end{align} $$

(5.17) $$ \begin{align} \mathcal{U}^*(h_{ij}(X,Y)) = \begin{cases} \tilde{h}_{ij}(X,Y)\tilde{g}_{p+2,1}(X,Y) \ &\text{if}\ \mathcal{L}^*_{ss}(X,Y) = y_{ij} \ \text{for} \ s < s_0;\\ \tilde{h}_{ij}(X,Y) \ &\text{otherwise}; \end{cases} \end{align} $$

if $\psi $ is any $\varphi $ -, f- or c- function in the initial extended cluster, then

(5.18) $$ \begin{align} \mathcal{U}^*(\psi(X,Y)) = \tilde{\psi}(X,Y). \end{align} $$

Removing roots from column runs

For a BD triple $\mathbf {\Gamma } = (\Gamma _1,\Gamma _2,\gamma )$ , let us define the opposite BD triple $\mathbf {\Gamma }^{\text {op}}$ as $\mathbf {\Gamma }^{\text {op}}:= (\Gamma _2,\Gamma _1,\gamma ^*)$ ; likewise, if $\mathbf {\Gamma } = (\mathbf {\Gamma }^r,\mathbf {\Gamma }^c)$ is a BD pair, we call $\mathbf {\Gamma }^{\text {op}} := ((\mathbf {\Gamma }^c)^{\text {op}}, (\mathbf {\Gamma }^r)^{\text {op}})$ the opposite BD pair. As explained in [Reference Gekhtman, Shapiro and Vainshtein20], the $\mathcal {L}$ -matrices in $\mathcal {GC}(\mathbf {\Gamma })$ and $\mathcal {GC}(\mathbf {\Gamma }^{\text {op}})$ are related via the involution

$$\begin{align*}\mathcal{L}(X,Y) \mapsto \mathcal{L}(Y^t, X^t)^t. \end{align*}$$

In particular, the involution $(X,Y)\mapsto (Y^t, X^t)$ maps g- and h-functions from $\mathcal {GC}(\mathbf {\Gamma })$ to h- and g-functions from $\mathcal {GC}(\mathbf {\Gamma }^{\text {op}})$ . This allows one to translate the construction of the rational maps from the case of removing a pair of roots from row runs to the case of removing a pair of roots from column runs. In the latter case, for some unipotent lower triangular matrix $U_0:=U_0(X,Y)$ , we set

$$\begin{align*}U_- := U_0 \prod_{k \geq 1}^{\rightarrow} (\tilde{\gamma}_c^*)^k(U_0) \end{align*}$$

and define the rational map $\mathcal {U} : D(\operatorname {\mathrm {GL}}_n)_{\tilde {\mathbf {\Gamma }}} \dashrightarrow D(\operatorname {\mathrm {GL}}_n)_{\mathbf {\Gamma }}$ via

$$\begin{align*}\mathcal{U}(X,Y) := (XU_-(X,Y),YU_-(X,Y)). \end{align*}$$

As one can observe in the previous cases, the entries of the matrix $U_0$ belong to the localization $\mathcal {O}(D(\operatorname {\mathrm {GL}}_n))[\tilde {\psi }^{\pm }_{\square }]$ , where the variable $\tilde {\psi _{\square }}$ is a stable variable in $\mathcal {GC}(\tilde {\mathbf {\Gamma }})$ such that $\psi _{\square }$ is a cluster variable in $\mathcal {GC}(\mathbf {\Gamma })$ (see the paragraph on notation above). Let $\Delta ^c = [p+1,p+k]$ be a nontrivial column X-run and $\gamma (\Delta ^c) = [q+1,q+k]$ be the corresponding column Y-run. Then, if $p+1$ and $q+1$ are removed, the variable $\psi _{\square }$ is $\psi _{\square }=h_{1,q+2}$ ; if $p+k-1$ and $q+k-1$ are removed, then $\psi _{\square } = h_{1,q+k}$ .

The action of $\mathcal {U}$ upon other clusters

Let $\tilde {\mathbf {\Gamma }}$ be obtained from $\mathbf {\Gamma }$ in one of the four ways described above (i.e., by removing a pair of rightmost or leftmost roots from row or column runs), and let $\psi _{\square }$ be the variable that is cluster in $\mathcal {GC}(\mathbf {\Gamma })$ and such that the corresponding variable $\tilde {\psi }_{\square }$ is stable in $\mathcal {GC}(\tilde {\mathbf {\Gamma }})$ . The following proposition corresponds to Proposition 7.4 in [Reference Gekhtman, Shapiro and Vainshtein20] and describes the action of the maps $\mathcal {U}$ defined above upon an extended cluster other than the initial one.