2.1 Quantum cluster algebras

Fix a finite index set $\mathsf {K} = {\mathsf {K}_\mathrm {ex}} \sqcup {\mathsf {K}_\mathrm {fr}}$ with a decomposition into the set ${\mathsf {K}_\mathrm {ex}}$ of exchangeable indices and the set ${\mathsf {K}_\mathrm {fr}}$ of frozen indices. Let $L = (l_{ij})_{i,j \in \mathsf {K}}$ be a skew-symmetric integer-valued matrix and let $q$ be an indeterminate. We set $\mathbb {A} \mathbin {:=} \mathbb {Z}\mspace {1mu}[q^{\pm 1/2}]$ where $q^{1/2}$ denotes the formal square root of $q$.

Definition 2.1 We define the quantum torus $\mathcal {T}(L)$ to be the $\mathbb {A}$-algebra generated by a finite family of elements $\{X^{\pm 1}_k \}_{k \in \mathsf {K}}$ subject to the following defining relations:

\[ X_jX_j^{-1} = X_j^{-1}X_j=1 \quad\text{and}\quad X_iX_j = q^{l_{ij}}X_jX_i \quad \text{for $i,j \in \mathsf{K}$.} \]

For $\mathbf {a} = (\mathbf {a}_i)_{i \in \mathsf {K}} \in \mathbb {Z}\mspace {1mu}^\mathsf {K}$, we define the element $X^\mathbf {a}$ of $\mathcal {T}(L)$ as

\[ X^\mathbf{a} = q^{({1/2}) \sum_{i>j}\mathbf{a}_i\mathbf{a}_j l_{ij}} \prod^{\longrightarrow}_{i \in \mathsf{K}} X_i^{\mathbf{a}_i}. \]

Here $\overset {\longrightarrow }{\prod }_{i \in \mathsf {K}} X_i^{\mathbf {a}_i} \mathbin {:=} X_{i_1}^{\mathbf {a}_{i_1}} \cdots X_{i_r}^{\mathbf {a}_{i_r}}$, where $\mathsf {K}=\{ i_1,\ldots,i_r \}$ with a total order $i_1 < \cdots < i_r$. Note that $X^\mathbf {a}$ does not depend on the choice of a total order < on $\mathsf {K}$. Then $\{ X^\mathbf {a} \mid \mathbf {a} \in \mathbb {Z}\mspace {1mu}^{\mathsf {K}} \}$ forms an $\mathbb {A}$-basis of $\mathcal {T}(L)$. Since $\mathcal {T}(L)$ is an Ore domain, it is embedded into the skew field of fractions $\mathbb {F}(\mathcal {T}(L))$.

Let $\widetilde {B} = (b_{ij})_{i \in \mathsf {K},j\in {\mathsf {K}_\mathrm {ex}}}$ be an integer-valued $\mathsf {K} \times {\mathsf {K}_\mathrm {ex}}$-matrix whose principal part $B = (b_{ij})_{i,j\in {\mathsf {K}_\mathrm {ex}}}$ is skew-symmetrizable, i.e. there exists a diagonal matrix $D$ with a positive integer entries such that $DB$ is skew-symmetric. Such a matrix $\widetilde {B}$ is called an exchange matrix. We say that a pair $(L,B)$ is compatible if

\[ \sum_{k \in \mathsf{K}} b_{ki} l_{kj} = d_i\delta_{i,j}\quad \text{for any $i\in {\mathsf{K}_\mathrm{ex}}$ and $j\in\mathsf{K}$} \]

for some positive integers $\{ d_i \}_{i \in {\mathsf {K}_\mathrm {ex}}}$. We call the triple $\mathcal {S} = (\{ X_k \}_{k \in \mathsf {K}}, L , \widetilde {B} )$ a quantum seed in the quantum torus $\mathcal {T}(L)$ and $\{ X_k \}_{k \in \mathsf {K}}$ a quantum cluster.

For $k \in {\mathsf {K}_\mathrm {ex}}$, the mutation $\mu _k(L,\widetilde {B}) \mathbin {:=} (\mu _k(L),\mu _k(B))$ of a compatible pair $(L,\widetilde {B})$ in a direction $k$ is defined in a combinatorial way (see [Reference Berenstein and ZelevinskyBZ05]). Note that (i) the pair $(\mu _k(L),\mu _k(B))$ is also compatible with the same positive integers $\{d_i\}_{i \in \mathsf {K}}$ and (ii) the operation $\mu _k$ is an involution, i.e. $\mu _k(\mu _k(L,\widetilde {B})) = (L,\widetilde {B})$. We define an isomorphism of $\mathbb {Q}(q^{1/2})$-algebras $\mu _k^* \colon \mathbb {F}(\mathcal {T}(\mu _kL)) {\mathop\longrightarrow\limits^{\sim}} \mathbb {F}(\mathcal {T}(L))$ by

\[ \mu_k^*(X_j) \mathbin{:=} \begin{cases} X^{\mathbf{a}'} + X^{\mathbf{a}''} & \text{if } j =k, \\ X_j & \text{if } j \ne k, \end{cases} \]

where

\[ \mathbf{a}'_i = \begin{cases} -1 & \text{if } i =k, \\ \max(0,b_{ik}) & \text{if } i \ne k, \end{cases} \quad\text{and}\quad \mathbf{a}''_i = \begin{cases} -1 & \text{if } i =k, \\ \max(0,-b_{ik}) & \text{if } i \ne k. \end{cases} \]

Then the mutation $\mu _k(\mathcal {S})$ of the quantum seed $\mathcal {S}$ in a direction $k$ is defined to be the triple $( \{ X_i\}_{i \ne k} \sqcup \{ \mu _k^*(X_k) \}, \mu _k(L), \mu _k(\widetilde {B}))$.

For a quantum seed $\mathcal {S}=( \{X_k\}_{k\in \mathsf {K}},L,\widetilde {B})$, an element in $\mathbb {F}(\mathcal {T}(L))$ is called a quantum cluster variable (respectively, quantum cluster monomial) if it is of the form

\[ \mu_{k_1}^* \cdots \mu_{k_\ell}^*(X_j) \quad \text{$($respectively, } \mu_{k_1}^* \cdots \mu_{k_\ell}^*(X^\mathbf{a}) ) \]

for some finite sequence $(k_1,\ldots,k_\ell ) \in \mathsf {K}_\mathrm {ex}^\ell$ $(\ell \in \mathbb {Z}\mspace {1mu}_{\geqslant 0})$ and $j \in \mathsf {K}$ (respectively, $\mathbf {a} \in \mathbb {Z}\mspace {1mu}_{\geqslant 0}^\mathsf {K}$). For a quantum seed $\mathcal {S} = (\{ X_k \}_{k \in \mathsf {K}}, L,\widetilde {B} )$, the quantum cluster algebra $\mathscr {A}_q(\mathcal {S})$ is the $\mathbb {A}$-subalgebra of $\mathbb {F}(\mathcal {T}(L))$ generated by all the quantum cluster variables. Note that $\mathscr {A}_q(\mathcal {S})\simeq \mathscr {A}_q(\boldsymbol {\mu }(\mathcal {S}))$ for any sequence $\boldsymbol {\mu }$ of mutations.

The quantum Laurent phenomenon, proved by Berenstein and Zelevinsky in [Reference Berenstein and ZelevinskyBZ05], says that the quantum cluster algebra $\mathscr {A}_q(\mathcal {S})$ is indeed contained in $\mathcal {T}(L)$.

For a quantum seed $\mathcal {S}$ with a compatible pair $(L,\widetilde {B})$, an element $\mathbf {x} \in \mathcal {T}(L)$ is called pointed (respectively, copointed) if it is of the following form:

(2.1)\begin{equation} \mathbf{x} = q^{a} X^{\mathbf{g}^R} + \sum_{ \mathbf{c} \in \mathbb{Z}\mspace{1mu}_{\geqslant 0}^{{\mathsf{K}_\mathrm{ex}}} \setminus \{ 0 \} } p_\mathbf{c} X^{\mathbf{g}^R+\widetilde{B} \mathbf{c}} \quad \bigg(\text{respectively, } \mathbf{x} = q^{a} X^{\mathbf{g}^L} + \sum_{ \mathbf{c} \in \mathbb{Z}\mspace{1mu}_{\leqslant 0}^{{\mathsf{K}_\mathrm{ex}}} \setminus \{ 0 \} } p_\mathbf{c} X^{\mathbf{g}^L+\widetilde{B} \mathbf{c}}\bigg) \end{equation}

for some $a \in \frac {1}{2}\mathbb {Z}\mspace {1mu}$, $\mathbf {g}^R \in \mathbb {Z}\mspace {1mu}^\mathsf {K}$ (respectively, $\mathbf {g}^L \in \mathbb {Z}\mspace {1mu}^\mathsf {K}$) and $p_\mathbf {c} \in \mathbb {A}$. In this case, we call $\mathbf {g}^R$ the degree (respectively, codegree) of the pointed (respectively, copointed) element $\mathbf {x}$ and denote it by $\mathbf {deg}_{\mathcal {S}}(\mathbf {x})$ (respectively, $\mathbf {codeg}_{\mathcal {S}}(\mathbf {x})$). The degree (respectively, codegree) is often the called $g$-vector (respectively, dual $g$-vector) of $\mathbf {x}$ (see [Reference QinQin17, Definition 3.1.4] and [Reference QinQin20, Definition 3.1.3]). It is worth remarking that the notion of $g$-vector (respectively, dual $g$-vector) does depend on the compatible pair $(L,\widetilde {B})$ and, hence, on the seed $\mathcal {S}$. It is proved in [Reference TranTra11, Theorem 5.3] that every quantum cluster monomial in $\mathscr {A}_q(\mathcal {S})$ is pointed.

We say that an $\mathbb {A}$-algebra $R$ has a quantum cluster algebra structure if there exists a quantum seed $\mathcal {S}$ and an $\mathbb {A}$-algebra isomorphism $\Omega : \mathscr {A}_q(\mathcal {S}) {\mathop\longrightarrow\limits^{\sim}} R$. In the case, a quantum seed of $R$ refers to the image of a quantum seed in $\mathscr {A}_q(\mathcal {S})$, which is obtained by a sequence of mutations.

2.2 Quantum unipotent coordinate rings

Let $I$ be an index set. A Cartan datum $(\mathsf {A},\mathsf {P},\Pi,\mathsf {P}^\vee,\Pi ^\vee )$ consists of:

1. (a) a symmetrizable Cartan matrix $\mathsf {A}=(a_{i,j})_{i,j \in I}$, i.e. $\mathsf {D} \mathsf {A}$ is symmetric for a diagonal matrix $\mathsf {D} = {\rm diag}(\mathsf {d}_i \mid i \in I)$ with $\mathsf {d}_i\in \mathbb {Z}\mspace {1mu}_{>0}$;

2. (b) a free abelian group $\mathsf {P}$, called the weight lattice;

3. (c) $\Pi = \{{\mspace {1mu}\alpha }_i \mid i \in I \} \subset \mathsf {P}$, called the set of simple roots;

4. (d) $\Pi ^\vee = \{ h_i \mid i \in I \} \subset \mathsf {P}^\vee \mathbin {:=} \operatorname {Hom}(\mathsf {P},\mathbb {Z}\mspace {1mu})$, called the set of simple coroots;

5. (e) a $\mathbb {Q}$-valued symmetric bilinear form $( \cdot,\cdot )$ on $\mathsf {P}$;

satisfying the standard properties (see [Reference Kang, Kashiwara, Kim and OhKKKO18, § 1.1] for instance). Here we take $\mathsf {D} ={\rm diag}(\mathsf {d}_i \mid i \in I)$ such that $\mathsf {d}_i \mathbin {:=}({\mspace {1mu}\alpha }_i,{\mspace {1mu}\alpha }_i)/2\in \mathbb {Z}\mspace {1mu}_{>0}$ $(i\in I)$ in this paper.

For $i \in I$, we choose $\varpi _i \in \mathsf {P}$ such that $\langle h_i,\varpi _j \rangle =\delta _{ij}$ for any $j \in I$ and call it the $i$th fundamental weight. The free abelian group $\mathsf {Q} \mathbin {:=} \bigoplus _{i \in I} \mathbb {Z}\mspace {1mu} {\mspace {1mu}\alpha }_i$ is called the root lattice and we set $\mathsf {Q}^+ = \sum _{i \in I}\mathbb {Z}\mspace {1mu}_{\geqslant 0} {\mspace {1mu}\alpha }_i \subset \mathsf {Q}$ and $\mathsf {Q}^- = \sum _{i \in I}\mathbb {Z}\mspace {1mu}_{\leqslant 0} {\mspace {1mu}\alpha }_i \subset \mathsf {Q}$. We denote by $\Delta$ the set of roots and by $\Delta ^\pm$ the set of positive roots (respectively, negative roots). For ${\mspace {1mu}\beta } \in \sum _{i \in I} m_i{\mspace {1mu}\alpha }_i \in \mathsf {Q}^+$, we set $|{\mspace {1mu}\beta }| \mathbin {:=} \sum _{i \in I} m_i$, ${\rm supp}({\mspace {1mu}\beta }) \mathbin {:=}\{ i \in I \mid m_i \ne 0 \}$ and $I^{\mspace {1mu}\beta } \mathbin {:=} \{ \nu = (\nu _1, \ldots,\nu _{|{\mspace {1mu}\beta }|}) \in I^{|{\mspace {1mu}\beta }|} \mid {\mspace {1mu}\alpha }_{\nu _1}+ \cdots + {\mspace {1mu}\alpha }_{\nu _{|{\mspace {1mu}\beta }|}} ={\mspace {1mu}\beta } \}$. Note that the symmetric group $\mathfrak {S}_{|{\mspace {1mu}\beta }|} = \langle r_{1},\ldots, r_{|{\mspace {1mu}\beta }|} \rangle$ acts on $I^{\mspace {1mu}\beta }$ by the place permutations.

Let $\mathfrak {g}$ be the Kac–Moody algebra associated with the Cartan datum $(\mathsf {A},\mathsf {P},\Pi,\mathsf {P}^\vee,\Pi ^\vee )$, and $\mathsf {W}$ the Weyl group of $\mathfrak {g}$. It is generated by the simple reflections $s_i\in \operatorname {Aut}(\mathsf {P})$ $(i \in I)$ defined by $s_i(\lambda ) = \lambda - \langle h_i,\lambda \rangle {\mspace {1mu}\alpha }_i$ for $\lambda \in \mathsf {P}$. For a sequence ${\boldsymbol {i}}=(i_1,\ldots,i_r) \in I^r$, we call it a reduced sequence of $w \in \mathsf {W}$ if $s_{i_1} \ldots s_{i_r}$ is a reduced expression of $w$. For $w,v \in \mathsf {W}$, we write $w \geqslant v$ if there is a reduced sequence of $v$ which appears in a reduced sequence of $w$ as a subsequence.

For $\lambda,\mu \in \mathsf {P}$, we write $\lambda \preccurlyeq \mu$ if there exists a sequence of real positive roots ${\mspace {1mu}\beta }_k$ $(1 \leqslant k \leqslant l)$ such that $\lambda = s_{{\mspace {1mu}\beta }_l} \cdots s_{{\mspace {1mu}\beta }_1} \mu$ and $({\mspace {1mu}\beta }_k,s_{{\mspace {1mu}\beta }_{k-1}} \cdots s_{{\mspace {1mu}\beta }_1}\mu )>0$ for $1 \leqslant k \leqslant l$. When $\Lambda \in \mathsf {P}^+$ and $\lambda,\mu \in \mathsf {W}\Lambda$ the relation $\lambda \preccurlyeq \mu$ holds if and only if there exist $w,v \in \mathsf {W}$ such that $\lambda = w\Lambda$, $\mu = v\Lambda$ and $v \leqslant w$.

Let $\mathcal {U}_q(\mathfrak {g})$ be the quantum group of $\mathfrak {g}$ over $\mathbb {Q}(q^{1/2})$, generated by $e_i,f_i$ $(i \in I)$ and $q^h$ $(h \in \mathsf {P}^\vee )$. We denote by $\mathcal {U}_q^+(\mathfrak {g})$ the subalgebra of $\mathcal {U}_q(\mathfrak {g})$ generated by $e_i$ and $\mathcal {U}_{\mathbb {A}}^+(\mathfrak {g})$ the $\mathbb {A}$-subalgebra of $\mathcal {U}_q(\mathfrak {g})^+$ generated by $e_i^n/[n]_i !$ $(i \in I, \; n \in \mathbb {Z}\mspace {1mu}_{>0})$, where

\begin{align*} q_i \mathbin{:=} q^{\mathsf{d}_i}, \quad [k]_i = \dfrac{q_i^k-q_i^{-k}}{q_i - q_i^{-1}} \quad\text{and}\quad [k]_i! = \prod_{s=1}^k [s]_i. \end{align*}

Set

\begin{align*} \mathcal{A}_q(\mathfrak{n}) = \oplus_{\beta \in \mathsf{Q}^-} \mathcal{A}_q(\mathfrak{n})_\beta \quad \text{where } \mathcal{A}_q(\mathfrak{n})_\beta \mathbin{:=} \operatorname{Hom}_{\mathbb{Q}(q^{1/2})}(\mathcal{U}^+_q(\mathfrak{g})_{-\beta}, \mathbb{Q}(q^{1/2})), \end{align*}

where $\mathcal {U}^+_q(\mathfrak {g})_{-\beta }$ denotes the $(-{\mspace {1mu}\beta })$-root space of $\mathcal {U}^+_q(\mathfrak {g})$. Then $\mathcal {A}_q(\mathfrak {n})$ also has an algebra structure and is called the quantum unipotent coordinate ring of $\mathfrak {g}$. We denote by $\mathcal {A}_{\mathbb {A}}(\mathfrak {n})$ the $\mathbb {A}$-submodule of $\mathcal {A}_q(\mathfrak {n})$ generated by $\psi \in \mathcal {A}_q(\mathfrak {n})$ such that $\psi (\mathcal {U}_{\mathbb {A}}^+(\mathfrak {g})) \subset \mathbb {A}$. Then, $\mathcal {A}_q(\mathfrak {n})$ is an $\mathbb {A}$-subalgebra with a $\mathcal {U}_{\mathbb {A}}^+(\mathfrak {g})$-bimodule structure.

For each $\lambda \in \mathsf {P}^+ \mathbin {:=} \sum _{i \in I} \mathbb {Z}\mspace {1mu}_{\geqslant 0} \varpi _i$ and Weyl group elements $w,w' \in \mathsf {W}$, we can define a specific homogeneous element $D(w\lambda,w'\lambda )$ of $\mathcal {A}_\mathbb {A}(\mathfrak {n})$, called a unipotent quantum minor (see, for example, [Reference Kang, Kashiwara, Kim and OhKKKO18, § 9]).

For $w \in \mathsf {W}$, we denote by $\mathcal {A}_\mathbb {A}(\mathfrak {n}(w))$ the $\mathbb {A}$-submodule of $\mathcal {A}_{\mathbb {A}}(\mathfrak {n})$ consisting of elements $\psi$ such that $e_{i_1} \cdots e_{i_{|{\mspace {1mu}\beta }|}}\psi =0$ for any ${\mspace {1mu}\beta } \in \mathsf {Q}^+ \setminus w \mathsf {Q}^-$ and $(\nu _{i_1},\ldots,\nu _{i_{|{\mspace {1mu}\beta }|}}) \in I^{\mspace {1mu}\beta }$. Then it is an $\mathbb {A}$-subalgebra and we call it the quantum unipotent coordinate ring associated with $w$.

For a reduced sequence ${\boldsymbol {i}}=(i_1,\ldots,i_r)$ of $w \in \mathsf {W}$ and $1 \leqslant k \leqslant r$, define $w^{{\boldsymbol {i}}}_{\leqslant k}=s_{i_1} \cdots s_{i_k}$ and $w^{\boldsymbol {i}}_{< k}=s_{i_1}\cdots s_{i_{k-1}}$. Then $\mathcal {A}_\mathbb {A}(\mathfrak {n}(w))$ is generated by the set of unipotent quantum minors $\{ D(w^{\boldsymbol {i}}_{\leqslant k}\varpi _{i_{k}}, w^{\boldsymbol {i}}_{< k}\varpi _{i_{k}}) \mid 1 \leqslant k \leqslant r \}$ as an $\mathbb {A}$-algebra.

It is proved in [Reference Geiss, Leclerc and SchröerGLS13a, Reference Goodearl and YakimovGY17, Reference Kang, Kashiwara, Kim and OhKKKO18, Reference QinQin20] that $\mathcal {A}_\mathbb {A}(\mathfrak {n}(w))$ has a quantum cluster algebra structure, one of whose quantum seeds $\mathcal {S}^{\boldsymbol {i}}$ can be obtained from a reduced sequence ${\boldsymbol {i}}=(i_1,\ldots,i_r)$ of $w$. To introduce $\mathcal {S}^{\boldsymbol {i}}$, we need preparations.

Let ${\boldsymbol {j}}=(j_1,\ldots,j_l)$ be a sequence of indices in $I$. For $1 \leqslant k \leqslant l$ and $j \in I$, we set

\begin{align*} k^{\boldsymbol{j}}_+&\mathbin{:=} \min(\{u\mid k< u \leqslant l,\; j_u=j_k\} \cup \{l+1\}), \\ k^{\boldsymbol{j}}_-&\mathbin{:=} \max(\{u\mid 1\leqslant u < k,\;j_u=j_k\} \cup \{0\}). \end{align*}

We also set

\[ k^{\boldsymbol{j}}_{\min} \mathbin{:=} \min\{u\mid 1\leqslant u\leqslant k ,\ j_u=j_k\} \quad\text{and}\quad k^{\boldsymbol{j}}_{\max} \mathbin{:=} \max\{u\mid k\leqslant u\leqslant l,\ j_u=j_k\}. \]

We sometimes drop $^{\boldsymbol {j}}$ in the above notation if there is no danger of confusion.

Take $\mathsf {K}=[1,r]$ as an index set and decompose $\mathsf {K}$ into

\[ {\mathsf{K}_\mathrm{fr}} = \{ k \mid 1\leqslant k \leqslant r, \; k^{\boldsymbol{i}}_+ = r+1 \} \quad\text{and}\quad {\mathsf{K}_\mathrm{ex}} \mathbin{:=} \mathsf{K} \setminus {\mathsf{K}_\mathrm{fr}}. \]

We define the $\mathbb {Z}\mspace {1mu}$-valued $\mathsf {K}\times {\mathsf {K}_\mathrm {ex}}$ matrix $\widetilde {B}^{\boldsymbol {i}} =(b^{\boldsymbol {i}}_{st})_{s\in \mathsf {K},t\in {\mathsf {K}_\mathrm {ex}}}$ and the $\mathbb {Z}\mspace {1mu}$-valued skew-symmetric $\mathsf {K}\times \mathsf {K}$ matrix $L^{\boldsymbol {i}}=(l^{\boldsymbol {i}}_{st})_{s,t \in \mathsf {K}}$ as follows:

\begin{align*} b^{\boldsymbol{i}}_{st} &= \begin{cases} \pm 1 & \text{if $s=t^{\boldsymbol{i}}_\pm$,} \\ -a_{i_s,i_t} & \text{if $s< t< s^{\boldsymbol{i}}_+< t^{\boldsymbol{i}}_+$,}\\ a_{i_s,i_t} & \text{if $t< s< t^{\boldsymbol{i}}_+< s^{\boldsymbol{i}}_+$,} \\ 0 & \text{otherwise}, \end{cases} \\ l^{\boldsymbol{i}}_{st} & = (\varpi_{i_s}-w^{\boldsymbol{i}}_{\leqslant s}\varpi_{i_s}, \varpi_{i_t}+w^{\boldsymbol{i}}_{\leqslant t}\varpi_{i_t} )\quad\text{for } s< t. \end{align*}

Then the quantum seed $\mathcal {S}^{\boldsymbol {i}}$ of $\mathcal {A}_\mathbb {A}(\mathfrak {n}(w))$ is given as follows:

(2.2)\begin{equation} \mathcal{S}^{\boldsymbol{i}} \mathbin{:=} \Big( \{ q^{c^{\boldsymbol{i}}_k} D(w^{\boldsymbol{i}}_{\leqslant k}\varpi_{i_k},\varpi_{i_k} ) \}_{k \in \mathsf{K}}, L^{\boldsymbol{i}}, \widetilde{B}^{\boldsymbol{i}} \Big), \end{equation}

where $c^{\boldsymbol {i}}_s = \frac {1}{4} (\varpi _{i_s}-w^{\boldsymbol {i}}_{\leqslant s}\varpi _{i_s} ,\varpi _{i_s}-w^{\boldsymbol {i}}_{\leqslant s}\varpi _{i_s} ) \in \mathbb {Z}\mspace {1mu}/2$. Note that $(L^{\boldsymbol {i}}\widetilde {B}^{\boldsymbol {i}})_{ab} = -2 \mathsf {d}_{i_a} \times \delta _{a,b}$ for $(a,b)\in \mathsf {K}\times {\mathsf {K}_\mathrm {ex}}$, ${\rm wt}(D(w^{\boldsymbol {i}}_{\leqslant k}\varpi _{i_k},\varpi _{i_k} )) = - \varpi _{i_s} + w^{\boldsymbol {i}}_{\leqslant s}\varpi _{i_s}$, and

\[ \big\{ q^{c^{\boldsymbol{i}}_k} D(w^{\boldsymbol{i}}_{\leqslant k}\varpi_{i_k},\varpi_{i_k}) = q^{c^{\boldsymbol{i}}_k} D(w\varpi_{i_k},\varpi_{i_k} ) \mid k \in {\mathsf{K}_\mathrm{fr}}\!\big\} \]

forms the set of frozen variables of the quantum cluster algebra $\mathcal {A}_\mathbb {A}(\mathfrak {n}(w))$. We call $\mathcal {S}^{\boldsymbol {i}}$ the GLS seed (associated with ${\boldsymbol {i}}$).

We set $\mathcal {D}(w) \mathbin {:=} \{ q^m D(w\varpi,\varpi ) \mid m \in \mathbb {Z}\mspace {1mu}/2, \; \varpi \in \mathsf {P}^+ \}$. Then it is well-known that $\mathcal {D}(w)$ consists of $q$-central elements of $\mathcal {A}_\mathbb {A}(\mathfrak {n}(w))$ and, hence, forms an Ore set. We denote by $\mathcal {A}_\mathbb {A}(\mathfrak {n}^w)$ the quotient ring of $\mathcal {A}_\mathbb {A}(\mathfrak {n}(w))$ by the Ore set $\mathcal {D}(w)$. Then $\mathcal {A}_\mathbb {A}(\mathfrak {n}^w)$ has also the quantum cluster algebra structure with the invertible frozen variables $\{ q^{c^{\boldsymbol {i}}_k} D(w^{\boldsymbol {i}}_{\leqslant k}\varpi _{i_k},\varpi _{i_k}) \}_{k \in {\mathsf {K}_\mathrm {fr}}}$ in the sense of [Reference Berenstein and ZelevinskyBZ05].

2.3 Quiver Hecke algebras and categorifications

Let $\mathbf {k}$ be a base field. For $i,j \in I$, we choose polynomials $\mathcal {Q}_{i,j}(u,v) \in \mathbf {k}[u,v]$ such that (a) $\mathcal {Q}_{i,j}(u,v) =\mathcal {Q}_{j,i}(v,u)$ and (b) each $\mathcal {Q}_{i,j}(u,v)$ is of the following form:

\[ \mathcal{Q}_{i,j}(u,v) = \delta(i\ne j) \hspace{-6ex} \sum_{p({\mspace{1mu}\alpha}_i,{\mspace{1mu}\alpha}_i)+q({\mspace{1mu}\alpha}_j,{\mspace{1mu}\alpha}_j)=-2({\mspace{1mu}\alpha}_i,{\mspace{1mu}\alpha}_j)} \hspace{-6ex} t_{i,j;p,q} u^p v^q \quad \text{where} \ t_{i,j;-a_{i,j},0} \in \mathbf{k}^\times. \]

For a Cartan datum $(\mathsf {A},\mathsf {P},\Pi,\mathsf {P}^\vee,\Pi ^\vee )$ and ${\mspace {1mu}\beta } \in \mathsf {Q}^+$, the quiver Hecke algebra $R({\mspace {1mu}\beta })$ associated with $(\mathcal {Q}_{i,j})_{i,j \in I}$ is the $\mathbb {Z}\mspace {1mu}$-graded algebra over $\mathbf {k}$ generated by the elements

\[ \{ e(\nu) \}_{\nu \in I^{\mspace{1mu}\beta}} , \quad \{ x_k \}_{1 \leqslant k \leqslant |{\mspace{1mu}\beta}|}, \quad\{ \tau_m \}_{1 \leqslant m <|{\mspace{1mu}\beta}|} \]

subject to certain defining relations (see [Reference Kashiwara, Kim, Oh and ParkKKOP21, Definition 1.1] for instance). Note that the $\mathbb {Z}\mspace {1mu}$-grading of $R({\mspace {1mu}\beta })$ is determined by the degrees of following elements:

\[ \deg(e(\nu))=0, \quad \deg(x_ke(\nu)) = ({\mspace{1mu}\alpha}_{\nu_k},{\mspace{1mu}\alpha}_{\nu_k}), \quad\text{and}\quad \deg(\tau_me(\nu)) = -({\mspace{1mu}\alpha}_{\nu_m},{\mspace{1mu}\alpha}_{\nu_{m+1}}). \]

We say that $R({\mspace {1mu}\beta })$ is symmetric if $\mathcal {Q}_{i,j}(u,v) \in \mathbf {k}[u-v]$ for $i,j \in {\rm supp}({\mspace {1mu}\beta })$.

We denote by $R({\mspace {1mu}\beta })\text {-}\mathrm {gmod}$ the category of finite-dimensional graded $R({\mspace {1mu}\beta })$-modules with homomorphisms of degree $0$. For $M \in R({\mspace {1mu}\beta })\text {-}\mathrm {gmod}$, we set ${\rm wt}(M) \mathbin {:=} -{\mspace {1mu}\beta } \in \mathsf {Q}^-$. Note that there exists the degree shift functor, denoted by $q$, such that $(qM)_n = M_{n-1}$ for $M=\oplus_{k \in \mathbb {Z}\mspace {1mu}} M_k \in R({\mspace {1mu}\beta })\text {-}\mathrm {gmod}$.

Throughout this paper, we usually deal with graded $R({\mspace {1mu}\beta })$-modules $({\mspace {1mu}\beta } \in \mathsf {Q}^+)$ and sometimes skip grading shifts. Thus, we usually say that $M$ is an $R$-module instead of saying that $M$ is a graded $R({\mspace {1mu}\beta })$-module and $f\colon M \to N$ is a homomorphism if $f\colon q^aM \to N$ is a morphism in $R({\mspace {1mu}\beta })\text {-}\mathrm {gmod}$. We set

\[ \mathrm{H{\scriptstyle OM}}_{R({\mspace{1mu}\beta})}(M,N) \mathbin{:=} \oplus_{a \in \mathbb{Z}\mspace{1mu}} \mathrm{H{\scriptstyle OM}}_{R({\mspace{1mu}\beta})}(M,N)_a \]

with $\mathrm {H{\scriptstyle OM}}_{R({\mspace {1mu}\beta })}(M,N)_a\mathbin {:=}\operatorname {Hom}_{R({\mspace {1mu}\beta })\text {-}\mathrm {gmod}}(q^aM,N)_a$. We write $\deg (f) \mathbin {:=} a$ for an $f \in \mathrm {H{\scriptstyle OM}}_{R({\mspace {1mu}\beta })} (M,N)_a$.

For an $R({\mspace {1mu}\beta })$-module $M$ and an $R(\gamma )$-module $N$, we can obtain $R({\mspace {1mu}\beta }+\gamma )$-module

\[ M \circ N \mathbin{:=} R({\mspace{1mu}\beta}+\gamma) e({\mspace{1mu}\beta},\gamma) \mathop\otimes_{R({\mspace{1mu}\beta}) \otimes R(\gamma)} (M \otimes N), \]

where $e({\mspace {1mu}\beta },\gamma ) \mathbin {:=} \sum _{\nu \in I^{\mspace {1mu}\beta }, \nu ' \in I^\gamma } e(\nu * \nu ') \in R({\mspace {1mu}\beta }+\gamma )$. Here $\nu *\nu '$ denotes the concatenation of $\nu$ and $\nu '$, and $\circ$ is called the convolution product. We say that two simple $R$-modules $M$ and $N$ strongly commute if $M \circ N$ is simple. If a simple module $M$ strongly commutes with itself, then $M$ is called real. A simple $R$-module $M$ is said to be prime if there are no non-trivial simple $R$-modules $N_1$ and $N_2$ such that $M \simeq N_1 \circ N_2$.

For an $R({\mspace {1mu}\beta })$-module $M$, the dual space $M^* \mathbin {:=} \operatorname {Hom}_\mathbf {k}(M,\mathbf {k})$ admits an $R({\mspace {1mu}\beta })$-module structure via

\[ (r \cdot f)(u)= f(\psi(r)u) \quad (r \in R({\mspace{1mu}\beta}), \ u \in M, \ f \in M^*). \]

Here $\psi$ denotes the $\mathbf {k}$-algebra anti-involution $R({\mspace {1mu}\beta })$ which fixes the generators of $R({\mspace {1mu}\beta })$. A simple $R({\mspace {1mu}\beta })$-module $M$ is called self-dual if $M^* \simeq M$.

Set $R\text {-}\mathrm {gmod} \mathbin {:=} \oplus_{{\mspace {1mu}\beta } \in \mathsf {Q}^+}R({\mspace {1mu}\beta })\text {-}\mathrm {gmod}$. Then the category $R\text {-}\mathrm {gmod}$ is a monoidal category with the tensor product $\circ$ and the unit object $\mathbf {1} \mathbin {:=} \mathbf {k} \in R(0)\text {-}\mathrm {gmod}$. Hence, the Grothendieck group $K(R\text {-}\mathrm {gmod} )$ has the $\mathbb {Z}\mspace {1mu}[q^{\pm 1}]$-algebra structure derived from $\circ$ and the degree shift functors $q^{\pm 1}$.

For a monoidal abelian subcategory $\mathcal {C}$ of $R\text {-}\mathrm {gmod}$ stable by grading shifts, we set

\[ \mathcal{K}_\mathbb{A}(\mathcal{C}) \mathbin{:=} \mathbb{A} \mathop\otimes_{\mathbb{Z}\mspace{1mu}[q^{\pm1}]} K(\mathcal{C} ), \]

where $K(\mathcal {C} )$ denotes the Grothendieck ring of $\mathcal {C}$. For a subcategory $\mathcal {C}$ of $R\text {-}\mathrm {gmod}$, we denote by ${\rm Irr}(\mathcal {C})$ the set of the isomorphism classes of self-dual modules in $\mathcal {C}$. Note that ${\rm Irr}(R\text {-}\mathrm {gmod})$ forms an $\mathbb {A}$-basis of $\mathcal {K}_\mathbb {A}(R\text {-}\mathrm {gmod})$.

It is proved in [Reference Khovanov and LaudaKL09, Reference Khovanov and LaudaKL11, Reference RouquierRou08] that there exists an $\mathbb {A}$-algebra isomorphism

(2.3)\begin{equation} \Omega \colon \mathcal{K}_{\mathbb{A}}(R\text{-}\mathrm{gmod}) {\mathop\longrightarrow\limits^{\sim}} \mathcal{A}_{\mathbb{A}}(\mathfrak{n}). \end{equation}

Proposition 2.2 [Reference Kashiwara, Kim, Oh and ParkKKOP18, Proposition 4.1]

For $\varpi \in \mathsf {P}^+$ and $\mu,\zeta \in \mathsf {W} \varpi$ with $\mu \preccurlyeq \zeta$, there exists a self-dual real simple $R( \zeta -\mu )$-module $M(\mu,\zeta )$ such that

\[ \Omega ([M(\mu,\zeta)]) = D(\mu,\zeta). \]

Here, $[M(\mu,\zeta )]$ denotes the isomorphism class of $M(\mu,\zeta )$ which is called the determinantal module associated with $D(\mu,\zeta )$.

For an $R({\mspace {1mu}\beta })$-module $M$, we define

\begin{align*} W(M) & \mathbin{:=} \{ \gamma \in \mathsf{Q}^+ \cap ({\mspace{1mu}\beta}-\mathsf{Q}^+) \mid e(\gamma,{\mspace{1mu}\beta}-\gamma)M \ne 0 \}, \\ W^*(M) & \mathbin{:=} \{ \gamma \in \mathsf{Q}^+ \cap ({\mspace{1mu}\beta}-\mathsf{Q}^+) \mid e({\mspace{1mu}\beta}-\gamma,\gamma)M \ne 0 \}. \end{align*}

An ordered pair $(M, N)$ of $R$-modules is called unmixed [Reference Tingley and WebsterTW16, Definition 2.5] if

\[ W^*(M) \cap W(N) \subset \{0\}. \]

For $w \in \mathsf {W}$, we denote by $\mathscr {C}_w$ the full subcategory of $R\text {-}\mathrm {gmod}$ whose objects $M$ satisfy $W(M) \subset \mathsf {Q}^+ \cap w\mathsf {Q}^-$. Then the category $\mathscr {C}_w$ is the smallest monoidal abelian category of $R\text {-}\mathrm {gmod}$ which (i) is stable under taking subquotients, extensions, grading shifts and (ii) contains $\big \{ S^{\boldsymbol {i}}_k \mathbin {:=} M(w^{\boldsymbol {i}}_{\leqslant k}\varpi _{i_k},w^{\boldsymbol {i}}_{< k}\varpi _{i_k}) \mid 1 \leqslant k \leqslant r \big \}$ for any reduced sequence ${\boldsymbol {i}}$ of $w$. We call $S^{\boldsymbol {i}}_k$ the $k$th cuspidal module associated with ${\boldsymbol {i}}$. Defining ${\mspace {1mu}\beta }_k^{\boldsymbol {i}} \mathbin {:=} w^{\boldsymbol {i}}_{< k} {\mspace {1mu}\alpha }_{i_k}$ for $1 \leqslant k \leqslant r$, one can see that $\{{\mspace {1mu}\beta }_k^{\boldsymbol {i}} \mid 1\leqslant k\leqslant r\}=\Delta ^+ \cap w \Delta ^-$, and $-{\rm wt}(S^{\boldsymbol {i}}_k) = {\mspace {1mu}\beta }^{\boldsymbol {i}}_k$. Then we have [Reference KimuraKim12, § 4]

\[ \Omega( \mathcal{K}_\mathbb{A}(\mathscr{C}_w ) ) \simeq \mathcal{A}_\mathbb{A}(\mathfrak{n}(w)). \]

2.4 $R$-matrices and affreal simple modules

For ${\mspace {1mu}\beta } \in \mathsf {Q}^+$ and $i \in I$, let

\[ \mathfrak{p}_{i,{\mspace{1mu}\beta}} = \sum_{ \eta\, \in I^{\mspace{1mu}\beta}} \bigg( \prod_{a \in [1,|{\mspace{1mu}\beta}|]; \ \eta_a = i} x_a \bigg) e(\eta) \in \mathcal{Z}(R({\mspace{1mu}\beta})), \]

where $\mathcal {Z}(R({\mspace {1mu}\beta }))$ denotes the center of $R({\mspace {1mu}\beta })$.

It is known that any $M \in R({\mspace {1mu}\beta })\text {-}\mathrm {gmod}$ admits an affinization if $R({\mspace {1mu}\beta })$ is symmetric. However, when $R({\mspace {1mu}\beta })$ is not symmetric, it is widely open whether an $R({\mspace {1mu}\beta })$-module $M$ admits an affinization or not.

Proposition 2.5 [Reference Kang, Kashiwara, Kim and OhKKKO18, Reference Kashiwara, Kim, Oh and ParkKKOP21]

Let $M$ and $N$ be simple modules such that one of them is affreal. Then there exists a unique $R$-module homomorphism $\mathbf {r}_{M,N} \in \mathrm {H{\scriptstyle OM}}_R(M,N)$ satisfying

\[ \mathrm{H{\scriptstyle OM}}_R(M \circ N,N \circ M)=\mathbf{k}\, \mathbf{r}_{M,N}. \]

We call the homomorphism $\mathbf {r}_{M,N}$ the $R$-matrix.

Definition 2.6 For simple $R$-modules $M$ and $N$ such that one of them is affreal, we define

\begin{align*} \Lambda(M,N) &\mathbin{:=} \deg (\mathbf{r}_{M,N}) , \\ \widetilde{\Lambda}(M,N) &\mathbin{:=} \tfrac{1}{2} \big( \Lambda(M,N) + \big({\rm wt}(M), {\rm wt}(N)\big) \big) , \\ \mathfrak{d}(M,N) &\mathbin{:=} \tfrac{1}{2} \big( \Lambda(M,N) + \Lambda(N,M)\big). \end{align*}

It is proved in [Reference Kang, Kashiwara, Kim and OhKKKO18, Reference Kashiwara, Kim, Oh and ParkKKOP21] that the invariants $\widetilde {\Lambda }(M,N)$ and $\mathfrak {d}(M,N)$ in Definition 2.6 belong to $\mathbb {Z}\mspace {1mu}_{\geqslant 0}$.

For simple modules $M$ and $N$, $M \triangledown N$ and $M \triangle N$ denote the head and the socle of $M \circ N$, respectively.

An ordered sequence of simple modules $\underline {L}=(L_1,\ldots,L_r)$ is called almost affreal if all $L_i$ $(1 \leqslant i \leqslant r)$ are affreal except for at most one.

Definition 2.9 An almost affreal sequence $\underline {L}$ of simple modules is called a normal sequence if the composition of $R$-matrices

\begin{align*} {\mathbf r}{\underline{L}} & \mathbin{:=} \prod_{1 \leqslant i < k \leqslant r} {\mathbf r}{L_i,L_k } = ({\mathbf r}{L_{r-1},L_r}) \circ \cdots ({\mathbf r}{L_2,L_r}\circ \cdots \circ {\mathbf r}{L_2,L_3}) \circ ({\mathbf r}{L_1,L_r} \circ \cdots \circ {\mathbf r}{L_1,L_2}) \\ & \colon \; q^{\sum_{1 \leqslant i < k \leqslant r} \Lambda(L_i,L_k)} L_1 \circ \cdots \circ L_r \rightarrow L_r \circ \cdots \circ L_1 \quad \text{does not vanish.} \end{align*}

For a given almost affreal sequence $\underline {L}$ of $R$-modules, the sufficient conditions for $\underline {L}$ being normal are studied in [Reference Kashiwara and KimKK19, Reference Kashiwara, Kim, Oh and ParkKKOP23]. In this paper, we will use the conditions frequently.

2.5 Commuting families

Let $J$ be an index set. We say that a family of affreal simple modules $\mathcal {M}= \{ M_j \}_{j \in J}$ in $R\text {-}\mathrm {gmod}$ is a commuting family if

\[ M_i \circ M_j \simeq M_j \circ M_i \quad \text{up to a grading shift for any $i,j \in J$.} \]

For a commuting family $\mathcal {M}= \{ M_j \ | \ j \in J\}$ in $R\text {-}\mathrm {gmod}$, let us take ${\mathbf \lambda} \colon \mathbb {Z}\mspace {1mu}^{\oplus J} \times \mathbb {Z}\mspace {1mu}^{\oplus J} \to \mathbb {Z}\mspace {1mu}$ such that

(2.4)\begin{equation} {\mathbf \lambda}(\mathbf{e}_i,\mathbf{e}_j)-{\mathbf \lambda}(\mathbf{e}_j,\mathbf{e}_i) = \Lambda(M_i,M_j) \quad \text{for any $ i,j \in J$.}\end{equation}

Here $\{ \mathbf {e}_j \mid j \in J\}$ is the standard basis of $\mathbb {Z}\mspace {1mu}^{\oplus J}$. Then there exists a family $\{ \mathcal {M}_{\mathbf \lambda} (\mathbf {a}) \mid \mathbf {a} \in \mathbb {Z}\mspace {1mu}_{\geqslant 0}^{\oplus J}\}$ of simple modules in $\mathscr {C}_w$ such that

(2.5)\begin{equation} \begin{array}{ll} \mathcal{M}_{\mathbf \lambda}(0) = \mathbf{1}, \quad \mathcal{M}_{\mathbf \lambda}(\mathbf{e}_j) =M_j & \text{for any }j\in J, \\ \mathcal{M}_{\mathbf \lambda}(\mathbf{a}) \circ \mathcal{M}_{\mathbf \lambda}(\mathbf{b}) \simeq q^{ -{\mathbf \lambda}(a,b) } \mathcal{M}_{\mathbf \lambda}(\mathbf{a}+\mathbf{b}) & \text{for any }\mathbf{a},\mathbf{b} \in \mathbb{Z}\mspace{1mu}_{\geqslant 0}^{J}. \end{array} \end{equation}

We sometimes omit $_{\mathbf \lambda}$ for notational simplicity.

The following lemma is obvious.

2.6 Localization of $\mathscr {C}_w$

Throughout this subsection, we fix $w \in \mathsf {W}$ and set

\[ I_w\mathbin{:=}\{i\in I\mid w\varpi_i\not=\varpi_i\}. \]

For notational simplicity, let us write

\[ C_i \mathbin{:=} M(w \varpi_i,\varpi_i) \in R\text{-}\mathrm{gmod} \quad \text{for $i \in I$.} \]

Then $\{ \Omega ([C_i]) \mid i \in I_w\}$ forms the set of frozen variables of $\mathcal {A}_\mathbb {A}(\mathfrak {n}(w))$. For each $\mu = \sum _{i \in I} \mu _i\varpi _i \in \mathsf {P}^+$, we set $C_\mu = M(w\mu,\mu )$, which is a self-dual convolution product $q^c \circ _{i \in I} C_i^{\circ \mu _i}$ for some $c\in \mathbb {Z}\mspace {1mu}$.

It is proved in [Reference Kashiwara, Kim, Oh and ParkKKOP21, Reference Kashiwara, Kim, Oh and ParkKKOP23, Reference Kashiwara, Kim, Oh and ParkKKOP24a] that there exists a monoidal abelian category $\widetilde {\mathscr {C}}_w = \mathscr {C}_w[C_i^{\circ -1} \mid i \in I]$ with a tensor product $\circ$, a degree shift functor $q$ and a monoidal exact fully faithful functor $\Phi _w\colon \mathscr {C}_w \to \widetilde {\mathscr {C}}_w$ satisfying the following properties.

(2.6)

(For the precise properties, see [Reference Kashiwara, Kim, Oh and ParkKKOP21, Reference Kashiwara, Kim, Oh and ParkKKOP23, Reference Kashiwara, Kim, Oh and ParkKKOP24a].)

Theorem 2.15 ([Reference Kashiwara, Kim, Oh and ParkKKOP21] (see also [Reference Kashiwara, Kim, Oh and ParkKKOP23, Remark 3.6]))

There exists an $\mathbb {A}$-algebra isomorphism

\[ \widetilde{\Omega} \colon \mathcal{K}_\mathbb{A}(\widetilde{\mathscr{C}}_w) \mathbin{:=} \mathbb{A} \mathop\otimes_{\mathbb{Z}\mspace{1mu}[q^{\pm1}]} K(\widetilde{\mathscr{C}}_w) {\mathop\longrightarrow\limits^{\sim}} \mathcal{A}_\mathbb{A}(\mathfrak{n}^w) \quad \text{such that} \quad \widetilde{\Omega}|_{\mathcal{K}_\mathbb{A}(\mathscr{C}_w)} = \Omega. \]

Here $K(\widetilde {\mathscr {C}}_w)$ denotes the Grothendieck ring of $\widetilde {\mathscr {C}}_w$.

A pair $(\varepsilon \colon X \otimes Y \to \mathbf {1}, \; \eta \colon \mathbf {1} \to Y \otimes X)$ of morphisms in a monoidal category with a unit object $\mathbf {1}$ is called an adjunction if the following two conditions hold.

1. (a) The composition $X \simeq X \otimes \mathbf {1} \mathop\longrightarrow\limits^{[X \otimes \eta]} X \otimes Y \otimes X \mathop\longrightarrow\limits^{[\varepsilon \otimes X]} \mathbf {1} \otimes X \simeq X$ is the identity.

2. (b) The composition $Y \simeq \mathbf {1} \otimes Y \mathop\longrightarrow\limits^{[\eta \otimes Y]} Y \otimes X \otimes Y \mathop\longrightarrow\limits^{[Y \otimes \varepsilon]} Y \otimes \mathbf {1} \simeq Y$ is the identity.

In the case when $(\varepsilon,\eta )$ is an adjunction, we say that $X$ is a left dual to $Y$, $Y$ is a right dual to $X$ and $(X,Y)$ is a dual pair.

2.7 Determinantal modules and monoidal clusters

In this subsection, we denote by $\mathcal {C}$ the category of $\mathscr {C}_w$ or $\widetilde {\mathscr {C}}_w$. Recall that

\begin{align*} \mathscr{A}&=\mathcal{K}_\mathbb{A}(\mathcal{C})\ \text{has a quantum cluster algebra structure via an isomorphism}\\ \Omega &=\Omega\ \text{or}\ \widetilde{\Omega}. \end{align*}

Let ${\boldsymbol {i}}=(i_1,\ldots,i_r)$ be a reduced sequence of $w \in \mathsf {W}$. For $k$ such that $1\leqslant k\leqslant r$, set

\[ M_k^{\boldsymbol{i}}=M(w^{\boldsymbol{i}}_{\leqslant k}\varpi_{i_k},\varpi_{i_k}) \]

(see Proposition 2.2 for the notation).

Proposition 2.17 [Reference Kashiwara, Kim, Oh and ParkKKOP18, Theorem 4.12]

Let ${\boldsymbol {i}}=(i_1,\ldots,i_r)$ be a reduced sequence of $w \in \mathsf {W}$. For $s < t \in \mathsf {K}$, we have

\[ -\Lambda(M_s^{\boldsymbol{i}},M_t^{\boldsymbol{i}}) = (\varpi_{i_s}-w^{\boldsymbol{i}}_{\leqslant s}\varpi_{i_s}, \varpi_{i_t}+w^{\boldsymbol{i}}_{\leqslant t}\varpi_{i_t} ) = l^{\boldsymbol{i}}_{st} = (L^{\boldsymbol{i}})_{st}. \]

We say that a commuting family $\mathcal {M}=\{M_i\}_{i\in \mathsf {K}}$ in $\mathcal {C}$ is a monoidal cluster if there exists a quantum seed $(\{ X_i \}_{i \in \mathsf {K}}, L=(l_{i,j})_{i,j \in \mathsf {K}}, \widetilde {B}=(b_{i,j})_{i \in \mathsf {K},j\in {\mathsf {K}_\mathrm {ex}}} )$ of $\mathscr {A}$ such that

\[ X_i = \Omega( q^{{1/4}({\rm wt}(M_i),{\rm wt}(M_i))}[M_i]) \quad \text{and} \quad l_{i,j} = -\Lambda(M_i,M_j). \]

Note that every monoidal seed is independent since the quantum cluster monomials in a cluster are linearly independent over $\mathbb {A}$ by the definition.

With Proposition 2.2 and (2.2), Proposition 2.17 says that

(2.7)\begin{equation} \mathcal{M}^{\boldsymbol{i}} \mathbin{:=} \{ M^{\boldsymbol{i}}_k \}_{1 \leqslant k \leqslant r} \text{ is a monoidal cluster in $\mathscr{C}_w$,} \end{equation}

for any reduced sequence ${\boldsymbol {i}}=(i_1,\ldots,i_r)$ of $w$. We call $\mathcal {M}^{\boldsymbol {i}}$ the GLS cluster (associated with ${\boldsymbol {i}}$).

3.1 Definition

Let $J$ be a finite index set. Let $\mathscr {C}$ be a full monoidal subcategory of $R\text {-}\mathrm {gmod}$ stable by taking subquotients, extensions and grading shifts.

Proof. Since the proof are similar, we prove only part (i). We have

\begin{align*} \mathcal{M}(\mathbf{b}+\mathbf{a}') \simeq \mathcal{M}(\mathbf{b}) \triangledown \mathcal{M}(\mathbf{a}') &\simeq( X \triangledown \mathcal{M}(\mathbf{a}) ) \triangledown \mathcal{M}(\mathbf{a}') \simeq {\operatorname{hd}}(X \circ \mathcal{M}(\mathbf{a}') \circ \mathcal{M}(\mathbf{a}) )\\ &\simeq( X \triangledown \mathcal{M}(\mathbf{a}') ) \triangledown \mathcal{M}(\mathbf{a}) \simeq \mathcal{M}(\mathbf{b}'+\mathbf{a}), \end{align*}

and, hence, we have $\mathbf {a}'+\mathbf {b}=\mathbf {a}+\mathbf {b}'$ since $\mathcal {M}$ is independent.

Proof. Since the proofs are similar, we shall only prove the first statement. Let us take $\mathbf {a}^{(1)} \in \mathbb {Z}\mspace {1mu}^J_{\geqslant 0}$ such that $\mathbf {a}^{(1)}_j \gg 0$ for all $j \in J$. Then Proposition 2.8 says that the simple module $Y \mathbin {:=} X \triangledown \mathcal {M}(\mathbf {a}^{(1)})$ commutes with all $M_j$. Since $\mathcal {M}$ is a quasi-Laurent family, there exists $\mathbf {a}^{(2)},\mathbf {b} \in \mathbb {Z}\mspace {1mu}_{\ge 0}^J$ such that $Y \circ \mathcal {M}(\mathbf {a}^{(2)}) \simeq \mathcal {M}(\mathbf {b})$. Hence, by taking $\mathbf {a} = \mathbf {a}^{(1)}+\mathbf {a}^{(2)}$, we have

\[ X \triangledown \mathcal{M}(\mathbf{a}) \simeq \mathcal{M}(\mathbf{b}), \]

as desired.

By applying the similar argument as in the lemma above to composition factors of $X \circ \mathcal {M}(\mathbf {a})$, we have the following corollary.

Corollary 3.4 Let $\mathcal {M}$ be a quasi-Laurent family in $\mathscr {C}$. Then, for any $X\in \mathscr {C}$, there exist $\mathbf {a} \in \mathbb {Z}\mspace {1mu}_{\geqslant 0}$, a finite index set $\mathsf {S}$, $c(s) \in \mathbb {Z}\mspace {1mu}$ and $\mathbf {b}(s) \in \mathbb {Z}\mspace {1mu}_{\geqslant 0}^J$ $(s \in \mathsf {S})$ such that

(3.1)\begin{equation} [X \circ \mathcal{M}(\mathbf{a})] = \sum_{s \in \mathsf{S}} q^{c(s)}[\mathcal{M}(\mathbf{b}(s))]. \end{equation}

Note that if $K(\mathscr {C})\vert _{q=1}$ is factorial, then every $[M_k]$ is prime in $K(\mathscr {C})\vert _{q=1}$ (see [Reference Geiss, Leclerc and SchröerGLS13b]).

Proof. Let $X$ be a simple module in $\mathscr {C}$ commuting with all $M_k$ $(k\in \mathsf {K})$. The quantum Laurent phenomenon states that there exists $\mathbf {a} \in \mathbb {Z}\mspace {1mu}_{\geqslant 0}^\mathsf {K}$ such that

(3.2)\begin{equation} [X] = \dfrac{\sum_{s \in \mathsf{S}} c(s)[\mathcal{M}(\mathbf{b}^{(s)})]}{[\mathcal{M}(\mathbf{a})]} \iff [X \circ \mathcal{M}(\mathbf{a})] = \sum_{s \in \mathsf{S}} c(s)[\mathcal{M}(\mathbf{b}^{(s)})] \end{equation}

for some $\ell \in \mathbb {Z}\mspace {1mu}_{\geqslant 1}$, $\mathbf {b}^{(s)} \in \mathbb {Z}\mspace {1mu}_{\geqslant 0}^\mathsf {K}$ and $c(s) \in \mathbb {Z}\mspace {1mu}[q^{\pm 1/2}]$. Since $X \circ \mathcal {M}(\mathbf {a})$ is simple, the right-hand side of (3.2) must coincide with $q^c\mathcal {M}(\mathbf {c})$ for some $\mathbf {c} \in \mathbb {Z}\mspace {1mu}_{\geqslant 0}^J$ and $c \in \mathbb {Z}\mspace {1mu}$. Hence, $\mathcal{M}$ is a quasi-Laurent family.

Let us show that $\mathcal {M}$ is a Laurent family. If $X\circ \mathcal {M}(\mathbf {a})\simeq \mathcal {M}(\mathbf {b})$, then we have $\mathbf {a}_k\leqslant \mathbf {b}_k$ for all $k$, since each $[M_k]$ is a prime element of $K(\mathscr {C})\vert _{q=1}$. Hence, setting $\mathbf {c}=\mathbf {b}-\mathbf {a}\in \mathbb {Z}\mspace {1mu}_{\ge 0}^\mathsf {K}$, we have $X\circ \mathcal {M}(\mathbf {a})\simeq \mathcal {M}(\mathbf {c})\circ \mathcal {M}(\mathbf {a})$, which implies that $X\simeq \mathcal {M}(\mathbf {c})$. Hence, $\mathcal {M}$ is a Laurent family.

Definition 3.7 For a simple module $X \in \mathscr {C}$ and a quasi-Laurent family $\mathcal {M}=\{ M_j \mid j \in J \}$, take $\mathbf {a},\mathbf {b},\mathbf {a}',\mathbf {b}' \in \mathbb {Z}\mspace {1mu}_{\geqslant 0}^J$ such that $X \triangledown \mathcal {M}(\mathbf {a}) \simeq \mathcal {M}(\mathbf {b})$ and $X \triangle \mathcal {M}(\mathbf {a}') \simeq \mathcal {M}(\mathbf {b}')$ up to grading shifts. Then we define

\[ \mathbf{g}_\mathcal{M}^R(X) \mathbin{:=} \mathbf{b} -\mathbf{a} \quad \text{and} \quad \mathbf{g}_\mathcal{M}^L(X) \mathbin{:=} \mathbf{b}' -\mathbf{a}' \in \mathbb{Z}\mspace{1mu}^J. \]

The following lemma can be proved by the same arguments in [Reference Kashiwara and KimKK19].

4.1 PBW decomposition vector

Let us take $w \in \mathsf {W}$ and its reduced sequence ${\boldsymbol {i}}=(i_1,\ldots,i_r)$. Recall the following.

1. (a) We take an index set $\mathsf {K}=[1,r]$ with a decomposition

   \[ {\mathsf{K}_\mathrm{ex}} \sqcup {\mathsf{K}_\mathrm{fr}} \quad \text{where} \ {\mathsf{K}_\mathrm{ex}}=\{k\in\mathsf{K}\mid k_+\leqslant r\}. \]

2. (b) For each $1 \leqslant k \leqslant r$, we set ${\mspace {1mu}\beta }^{\boldsymbol {i}}_k \in \Delta ^+ \cap w\Delta ^-$ and define simple modules $S^{\boldsymbol {i}}_k = M(w^{\boldsymbol {i}}_{\leqslant k}\varpi _{i_k},w^{\boldsymbol {i}}_{< k}\varpi _{i_k})$ and $M^{\boldsymbol {i}}_k = M(w^{\boldsymbol {i}}_{\leqslant k}\varpi _{i_k},\varpi _{i_k})$. Note that

   (4.1)\begin{equation} M^{\boldsymbol{i}}_k \simeq S^{\boldsymbol{i}}_k \triangledown M^{\boldsymbol{i}}_{k_-} \quad \text{and} \quad \mathcal{M}^{\boldsymbol{i}} \mathbin{:=} \{M_k^{\boldsymbol{i}}\mid k\in \mathsf{K}\} \text{ forms a commuting family.} \end{equation}

For any $\mathbf {a} = (\mathbf {a}_k)_{1 \leqslant k \leqslant r} \in \mathbb {Z}\mspace {1mu}_{\geqslant 0}^\mathsf {K}$, the convolution product

\[ P_{\boldsymbol{i}}(\mathbf{a}) \mathbin{:=} q^{{1/2} \sum_{k=1}^r \mathbf{a}_k(\mathbf{a}_k-1) \mathsf{d}_{i_k} } {S^{\boldsymbol{i}}_r}^{\;\circ \mathbf{a}_r} \circ \cdots \circ {S^{\boldsymbol{i}}_1}^{\;\circ \mathbf{a}_1} \]

has a self-dual simple head. Conversely, every self-dual simple module in $\mathscr {C}_w$ is isomorphic to ${\operatorname {hd}}(P_{\boldsymbol {i}}(\mathbf {a}))$ for some $\mathbf {a} \in \mathbb {Z}\mspace {1mu}_{\ge 0}^\mathsf {K}$ in a unique way (see [Reference McNamaraMcN15, Theorem 3.1] and [Reference Tingley and WebsterTW16, Theorem 2.19]). We call $\{ P_{\boldsymbol {i}}(\mathbf {a}) \mid \mathbf {a} \in \mathbb {Z}\mspace {1mu}_{\ge 0}^\mathsf {K} \}$ the PBW basis of $\mathscr {C}_w$ associated with ${\boldsymbol {i}}$.

For a simple module $X$ such that $X\simeq {\operatorname {hd}}(P_{\boldsymbol {i}}(\mathbf {a}))$, we set

\[ {\rm PBW}_{\boldsymbol{i}}(X) \mathbin{:=} \mathbf{a} = (\mathbf{a}_1,\ldots,\mathbf{a}_r). \]

The following lemma says that the operation ${\rm PBW}_{\boldsymbol {i}}( - \triangledown \mathcal {M}^{\boldsymbol {i}}(\mathbf {a}))$ on the set of simple modules behaves very nicely, where $\mathcal {M}^{\boldsymbol {i}}(\mathbf {a})$ is defined in (2.5).

Lemma 4.1 (cf. [Reference Kashiwara and KimKK19, Lemma 3.11 and Proposition 3.14])

For $M=\mathcal {M}^{\boldsymbol {i}}(\mathbf {a})$ with $\mathbf {a} \in \mathbb {Z}\mspace {1mu}_{\ge 0}^{\mathsf {K}}$ and a simple module $X$, we have

\[ {\rm PBW}_{\boldsymbol{i}}(X \triangledown M) ={\rm PBW}_{\boldsymbol{i}}(X)+{\rm PBW}_{\boldsymbol{i}}(M). \]

In particular, $\mathbf {c}\mathbin {:=}{\rm PBW}_{\boldsymbol {i}}\big (\mathcal {M}^{\boldsymbol {i}}(\mathbf {a})\big )$ is given by $\mathbf {c}_k=\sum _{j}\mathbf {a}_j$ where $j$ ranges over $j\in [1,r]$ such that $j\geqslant k$ and ${\boldsymbol {i}}_j={\boldsymbol {i}}_k$.

Proof. It is enough to show it when $M=M^{\boldsymbol {i}}_k$. Note that

\[ X \simeq {\operatorname{hd}}\bigg(\circ_{1 \leqslant k \leqslant r}^{\to} {S^{\boldsymbol{i}}_k}^{\circ \mathbf{n}_k} \bigg) = {S^{\boldsymbol{i}}_{r}}^{\circ \mathbf{n}_r} \triangledown Y, \]

where $\mathbf {n}=(\mathbf {n}_1,\ldots,\mathbf {n}_r) = {\rm PBW}_{\boldsymbol {i}}(X)$, $Y \simeq {\operatorname {hd}}\Big (\displaystyle \circ _{1 \leqslant k \leqslant r-1}^{\to } {S^{\boldsymbol {i}}_k}^{\circ \mathbf {n}_k } \Big )$ and $\displaystyle \circ _{p \leqslant k \leqslant q}^{\to } X_k$ denotes the ordered convolution product $X_q \circ X_{q-1} \circ \cdots X_{p+1} \circ X_p$ for $X_k$ in $R\text {-}\mathrm {gmod}$.

If $r > k$, then $({S^{\boldsymbol {i}}_r}^{\circ \mathbf {n}_r},M^{\boldsymbol {i}}_k)$ is unmixed and, hence, ${S^{\boldsymbol {i}}_r}^{\circ \mathbf {n}_r} \circ Y \circ M^{\boldsymbol {i}}_k$ has a simple head. We have

\begin{align*} X \triangledown M^{\boldsymbol{i}}_k & \simeq {\operatorname{hd}}({S^{\boldsymbol{i}}_r}^{\circ \mathbf{n}_r} \circ Y \circ M^{\boldsymbol{i}}_k) \\ & \simeq {S^{\boldsymbol{i}}_r}^{\circ \mathbf{n}_r} \triangledown (Y \triangledown M^{\boldsymbol{i}}_k) \simeq {S^{\boldsymbol{i}}_r}^{\circ \mathbf{n}_r} \triangledown {\operatorname{hd}}\bigg( \displaystyle\circ_{1 \leqslant k \leqslant r-1}^{\to} {S^{\boldsymbol{i}}_k}^{\circ \mathbf{c}_k} \bigg), \end{align*}

where $\mathbf {c} ={\rm PBW}_{\boldsymbol {i}}(Y)+{\rm PBW}_{\boldsymbol {i}}(M^{\boldsymbol {i}}_k)$ by induction on $r$. Thus, our assertion follows in this case.

If $r = k$, then $M^{\boldsymbol {i}}_r$ commutes with all the objects of $\mathscr {C}_w$ and, hence, we have

\begin{align*} X \triangledown M^{\boldsymbol{i}}_r & \simeq {\operatorname{hd}}({S^{\boldsymbol{i}}_r}^{\circ \mathbf{n}_r} \circ M^{\boldsymbol{i}}_r \circ {S^{\boldsymbol{i}}_{r-1}}^{\circ \mathbf{n}_{r-1}} \circ \cdots {S^{\boldsymbol{i}}_{1}}^{\circ \mathbf{n}_{1}} ) \\ & \simeq {\operatorname{hd}}({S^{\boldsymbol{i}}_r}^{\circ \mathbf{n}_r} \circ M^{\boldsymbol{i}}_r) \triangledown Y \simeq {\operatorname{hd}}({S^{\boldsymbol{i}}_r}^{\circ \mathbf{n}_r} \circ (S^{\boldsymbol{i}}_r \triangledown M^{\boldsymbol{i}}_{r_-})) \triangledown Y \\ & \simeq {\operatorname{hd}}({S^{\boldsymbol{i}}_r}^{\circ \mathbf{n}_r+1} \triangledown M^{\boldsymbol{i}}_{r_-} ) \triangledown Y \simeq {S^{\boldsymbol{i}}_r}^{\circ \mathbf{n}_r+1} \triangledown (M^{\boldsymbol{i}}_{r_-} \circ Y ), \end{align*}

where the last isomorphism follows from the commutativity of $M^{\boldsymbol {i}}_{r_-}$ and $Y$. Then our assertion follows from the induction hypothesis.

The lemma above gives a direct proof of the following corollary although it follows immediately from Proposition 3.6.

The following proposition is proved in [Reference Kashiwara and KimKK19, Proposition 2.11] for symmetric quiver Hecke algebra and the same proof also works for the general case.

Proposition 4.3 For any $\mathbf {a} = (\mathbf {a}_1,\ldots,\mathbf {a}_r)$, $\mathbf {b} = (\mathbf {b}_1,\ldots,\mathbf {b}_r) \in \mathbb {Z}\mspace {1mu}_{\geqslant 0}^\mathsf {K}$, the ordered sequence

\[ ({S^{\boldsymbol{i}}_r}^{\circ \mathbf{a}_r},(S_{r-1}^{\boldsymbol{i}})^{\circ \mathbf{a}_{r-1}},\ldots, {S^{\boldsymbol{i}}_1}^{\circ \mathbf{a}_1},{M^{\boldsymbol{i}}_1}^{\circ\mathbf{b}_1},{M^{\boldsymbol{i}}_2}^{\circ\mathbf{b}_2},\ldots,{M^{\boldsymbol{i}}_r}^{\circ\mathbf{b}_r}) \]

is a normal sequence.

The statement and proof of following proposition are the same as [Reference Kashiwara and KimKK19, Proposition 3.14] even though [Reference Kashiwara and KimKK19] dealt only with symmetric quiver Hecke algebras. Here we repeat it in order to show relations between explicit $\mathbb {Z}\mspace {1mu}_{\geqslant 0}$-vectors associated with a simple module $X$ in $\mathscr {C}_w$ for the readers’ convenience.

Proposition 4.4 For a simple module $X$ in $\mathscr {C}_w$ or $\widetilde {\mathscr {C}}_w$, there exist $\mathbf {a}, \mathbf {b} \in \mathbb {Z}\mspace {1mu}_{\geqslant 0}^\mathsf {K}$ such that

\[ X \triangledown \mathcal{M}^{\boldsymbol{i}}(\mathbf{a}) \simeq \mathcal{M}^{\boldsymbol{i}}(\mathbf{b}) \quad \text{up to a grading shift.} \]

Proof. In this proof, we sometimes drop $^{\boldsymbol {i}}$ for notational simplicity. It is enough to consider when $X \in \mathscr {C}_w$ by part (D) in (2.6). Note that there exists a unique $\mathbf {c} = (\mathbf {c}_1,\ldots,\mathbf {c}_r) \in \mathbb {Z}\mspace {1mu}^\mathsf {K}_{\geqslant 0}$ such that

\[ X \simeq {\operatorname{hd}}(P_{\boldsymbol{i}}(\mathbf{c})) \simeq {\operatorname{hd}}({S^{\boldsymbol{i}}_r}^{\circ \mathbf{c}_r} \circ \cdots \circ {S^{\boldsymbol{i}}_1}^{\circ \mathbf{c}_1}) . \]

Set $\mathbf {c}_+ \mathbin {:=} \sum _{k=1}^r \mathbf {c}_{k_+} \mathbf {e}_k= \sum _{j \in \mathsf {K}} \mathbf {c}_k \mathbf {e}_{k_-}$, where $\{ \mathbf {e}_j \mid j \in \mathsf {K}\}$ is the standard basis of $\mathbb {Z}\mspace {1mu}^\mathsf {K}$ such that $\mathbf {c} = \sum _{j \in \mathsf {K}} \mathbf {c}_j \mathbf {e}_j$. Then we have $\mathcal {M}^{\boldsymbol {i}}(\mathbf {c}_+) \simeq M_{1_-}^{\circ \mathbf {c}_1} \circ \cdots \circ M_{r_-}^{\circ \mathbf {c}_r}$. By (4.1) and Proposition 4.3, we have

\begin{align*} X \triangledown \mathcal{M}^{\boldsymbol{i}}(\mathbf{c}_+) &\simeq {\operatorname{hd}}(S_r^{\circ \mathbf{c}_r} \circ S_{r-1}^{\circ \mathbf{c}_{r-1}} \circ \cdots \circ S_1^{\circ \mathbf{c}_{1}} \circ \mathcal{M}^{\boldsymbol{i}}(\mathbf{c}_+) ) \\ &\simeq {\operatorname{hd}}(S_r^{\circ \mathbf{c}_r} \circ S_{r-1}^{\circ \mathbf{c}_{r-1}} \circ \cdots \circ S_2^{\circ \mathbf{c}_{2}} \circ S_1^{\circ \mathbf{c}_{1}} \circ M_{1_-}^{\circ \mathbf{c}_1} \circ M_{2_-}^{\circ \mathbf{c}_2} \circ \cdots \circ M_{r_-}^{\circ \mathbf{c}_r} ) \\ &\simeq {\operatorname{hd}}((S_r^{\circ \mathbf{c}_r} \circ S_{r-1}^{\circ \mathbf{c}_{r-1}} \circ \cdots \circ S_2^{\circ \mathbf{c}_{2}}) \circ (S_1^{\circ \mathbf{c}_{1}} \triangledown M_{1_-}^{\circ \mathbf{c}_1}) \circ (M_{2_-}^{\circ \mathbf{c}_2} \circ \cdots \circ M_{r_-}^{\circ \mathbf{c}_r} )) \\ &\simeq {\operatorname{hd}}((S_r^{\circ \mathbf{c}_r} \circ S_{r-1}^{\circ \mathbf{c}_{r-1}} \circ \cdots \circ S_1^{\circ \mathbf{c}_{2}}) \circ M_{1}^{\circ \mathbf{c}_1} \circ (M_{2_-}^{\circ \mathbf{c}_2} \circ \cdots \circ M_{r_-}^{\circ \mathbf{c}_r} )) \\ &\simeq {\operatorname{hd}}((S_r^{\circ \mathbf{c}_r} \circ S_{r-1}^{\circ \mathbf{c}_{r-1}} \circ \cdots \circ S_2^{\circ \mathbf{c}_{2}}) \circ (M_{2_-}^{\circ \mathbf{c}_2} \circ \cdots \circ M_{r_-}^{\circ \mathbf{c}_r} )\circ M_{1}^{\circ \mathbf{c}_1} ) \\ &\simeq \cdots \simeq {\operatorname{hd}}(M_r^{\circ \mathbf{c}_r} \circ \cdots \circ M_1^{\circ \mathbf{c}_1}) \simeq \mathcal{M}^{\boldsymbol{i}}(\mathbf{c}), \end{align*}

which implies our assertion.

As seen by the proof of the above proposition and Proposition 3.6, we have the following.

Proposition 4.5 The commuting family $\mathcal {M}^{\boldsymbol {i}}$ is a Laurent family. Moreover, for a simple module $M$, two vectors $\mathbf {a}={\rm PBW}_{\boldsymbol {i}}(M)$ and $\mathbf {g}\mathbin {:=}\mathbf {g}_{\boldsymbol {i}}^R(M)$ are related by

\[ \mathbf{g}_k=\mathbf{a}_k-\mathbf{a}_{k_+},\quad \mathbf{a}_k=\sum_{j; j\geqslant k,\ {\boldsymbol{i}}_j={\boldsymbol{i}}_k}\mathbf{g}_j, \]

where $\mathbf {a}_{r+1}=0$.

The following corollary can be proved by the same arguments in [Reference Kashiwara and KimKK19].

5.1 Skew-symmetric pairing associated with a quasi-Laurent family

Let $\mathscr {C}$ be a full monoidal subcategory of $R\text {-}\mathrm {gmod}$ stable by taking subquotients, extensions and grading shifts, and let $\mathcal {M}=\{ M_j \mid j \in J\}$ be a quasi-Laurent family in $\mathscr {C}$ labeled by a finite index set $J$.

For $X,Y \in {\rm Irr}(\mathscr {C})$, let us define

(5.1) \begin{align} {\mathrm G}^R_\mathcal{M}(X,Y) &\mathbin{:=} \sum\limits_{ a,b \in J} (\mathbf{g}^R_\mathcal{M}(X))_a (\mathbf{g}^R_\mathcal{M}(Y))_b \, \Lambda(M_a,M_b) \quad \text{and}\nonumber\\ {\mathrm G}^L_\mathcal{M}(X,Y) &\mathbin{:=} \sum\limits_{ a,b \in J} (\mathbf{g}^L_\mathcal{M}(X))_a (\mathbf{g}^L_\mathcal{M}(Y))_b \, \Lambda(M_a,M_b). \end{align}

The following lemma immediately follows from Lemma 3.9.

Lemma 5.1 For $M=\mathcal {M}(\mathbf {a})$ with $\mathbf {a} \in \mathbb {Z}\mspace {1mu}_{\geqslant 0}^{J}$ and $X,Y \in {\rm Irr}(\mathscr {C})$, we have

\begin{align*} &{\mathrm G}^R_\mathcal{M}(X \triangledown M,Y) = {\mathrm G}^R_\mathcal{M}(X,Y)+ {\mathrm G}^R_\mathcal{M}(M,Y), \quad {\mathrm G}^R_\mathcal{M}(X,Y) = -{\mathrm G}^R_\mathcal{M}(Y,X), \\ &{\mathrm G}^L_\mathcal{M}(X \triangle M,Y) = {\mathrm G}^L_\mathcal{M}(X,Y)+ {\mathrm G}^L_\mathcal{M}(M,Y), \quad {\mathrm G}^L_\mathcal{M}(X,Y) = -{\mathrm G}^L_\mathcal{M}(Y,X). \end{align*}

Proposition 5.2 Let $X$ be a simple module in $\mathscr {C}$. Then for any $\mathbf {c}\in \mathbb {Z}\mspace {1mu}_{\ge 0}^J$, we have

\[ {\rm{(i)}} \ \Lambda(X,\mathcal{M}(\mathbf{c})) ={\mathrm G}^R_\mathcal{M}(X,\mathcal{M}(\mathbf{c})) \quad \text{and} \quad {\rm{(ii)}}\ \Lambda(\mathcal{M}(\mathbf{c}),X) ={\mathrm G}^L_\mathcal{M}(\mathcal{M}(\mathbf{c}),X) . \]

Proposition 5.3 For any simple modules $X,Y$ in $\mathscr {C}$ such that one of them is affreal, we have

\[ {\mathrm G}^R_\mathcal{M}(X,Y), {\mathrm G}^L_\mathcal{M}(X,Y) \leqslant \Lambda(X,Y). \]

Proof. Since the proofs are similar, we will consider the case of ${\mathrm G}^R_\mathcal {M}$. Take $\mathbf {a},\mathbf {b}\in \mathbb {Z}\mspace {1mu}_{\ge 0}^J$ such that $Y \triangledown \mathcal {M}(\mathbf {a}) \simeq \mathcal {M}(\mathbf {b})$. Then we have

\begin{align*} {\mathrm G}^R_\mathcal{M}(X,Y) + {\mathrm G}^R_\mathcal{M}(X,\mathcal{M}(\mathbf{a}))& = {\mathrm G}^R_\mathcal{M}(X,Y\triangledown\mathcal{M}(\mathbf{a})) \\ & = \Lambda(X,Y \triangledown \mathcal{M}(\mathbf{a})) \leqslant \Lambda(X,Y)+ \Lambda(X,\mathcal{M}(\mathbf{a})) \\ & = \Lambda(X,Y)+ {\mathrm G}^R_\mathcal{M}(X,\mathcal{M}(\mathbf{a})) , \end{align*}

which yields our assertion. Here, the inequality follows from [Reference Kang, Kashiwara, Kim and OhKKKO18, Proposition 3.2.10].

Proposition 5.4 If simple modules $X$ and $Y$ in $\mathscr {C}$ commute and one of them is affreal, then we have

\[ \Lambda(X,Y) = {\mathrm G}^R_\mathcal{M}(X,Y) = {\mathrm G}^L_\mathcal{M}(X,Y). \]

Proof. Since the proofs are similar, we will only give the proof for ${\mathrm G}^R_\mathcal {M}$. By the preceding proposition, we have

\begin{align*} 0&=\big(\Lambda(X,Y)+\Lambda(Y,X)\big)-\big({\mathrm G}^R_\mathcal{M}(X,Y)+{\mathrm G}^R_\mathcal{M}(Y,X)\big)\\ &=\big(\Lambda(X,Y)-{\mathrm G}^R_\mathcal{M}(X,Y)\big)+\big( \Lambda(Y,X)-{\mathrm G}^R_\mathcal{M}(Y,X)\big) \ge0, \end{align*}

which implies $\Lambda (X,Y)-{\mathrm G}^R_\mathcal {M}(X,Y)=0$.

Remark 5.5 The two invariants ${{\mathrm G}}^R_{{\mathcal {M}}}(X,Y)$ and ${{\mathrm G}}^L_{{\mathcal {M}}}(X,Y)$ are different in general and depend on the choice of $\mathcal {M}$.

Let $w_0$ be the longest element of finite type $A_2$. For a reduced sequence ${\boldsymbol {i}}=(1,2,1)$ of $w_0$, we have

(5.2)\begin{equation} \{\!S_1^{{\boldsymbol{i}}}=\langle 1\rangle , S_2^{{\boldsymbol{i}}}=\langle 12\rangle , S_3^{{\boldsymbol{i}}}=\langle 2\rangle \}\quad \text{and}\quad \mathcal{M}^{\boldsymbol{i}} =\{ M_1^{{\boldsymbol{i}}}=\langle 1\rangle , M_2^{{\boldsymbol{i}}}=\langle 12\rangle , M_3^{{\boldsymbol{i}}}=\langle 21\rangle\!\}, \end{equation}

while

(5.3)\begin{equation} \{\!S_1^{\boldsymbol{j}}=\langle 2\rangle , S_2^{\boldsymbol{j}}=\langle 21\rangle , S_3^{\boldsymbol{j}}=\langle 1\rangle \}\quad \text{and}\quad \mathcal{M}^{\boldsymbol{j}} =\{ M_1^{\boldsymbol{j}}=\langle 2\rangle , M_2^{\boldsymbol{j}}=\langle 21\rangle , M_3^{\boldsymbol{j}}=\langle 12\rangle\!\} \end{equation}

for the other reduced sequence ${\boldsymbol {j}}=(2,1,2)$ of $w_0$. Here $\langle k\rangle$ $(k=1,2)$ is a one-dimensional $R({\mspace {1mu}\alpha }_k)$-module, and $\langle 12 \rangle$ and $\langle 21 \rangle$ are one-dimensional $R({\mspace {1mu}\alpha }_1+{\mspace {1mu}\alpha }_2)$-modules (see [Reference Kang, Kashiwara and KimKKK18] for more details on these modules).

Since $A_2$ is symmetric, $\mathcal {M}' \mathbin {:=} \mu _1(\mathcal {M}^{\boldsymbol {i}})$ is also a Laurent family given as follows:

\[ \mathcal{M}' = \{ M'_1 =\mu_1(M^{\boldsymbol{i}}_1) \simeq \langle 2\rangle, M'_2 = M^{\boldsymbol{i}}_2 \simeq \langle 12 \rangle, M'_3 = M^{\boldsymbol{i}}_3 \simeq \langle 21 \rangle\}. \]

Note that $\mathcal {M}'=\mathcal {M}^{\boldsymbol {j}}$ (up to an index permutation). We have

\begin{align*} & \mathbf{g}^R_{\mathcal{M}^{\boldsymbol{i}}}(\langle 1 \rangle) = \mathbf{g}^L_{\mathcal{M}^{\boldsymbol{i}}}(\langle 1 \rangle) = (1,0,0), \quad \mathbf{g}^R_{\mathcal{M}^{\boldsymbol{i}}}(\langle 2 \rangle ) = (-1,0,1), \quad \mathbf{g}^L_{\mathcal{M}^{\boldsymbol{i}}}(\langle 2 \rangle ) = (-1,1,0), \\ & \mathbf{g}^R_{\mathcal{M}'}(\langle 1 \rangle) = (-1,1,0), \quad \mathbf{g}^L_{\mathcal{M}'}(\langle 1 \rangle) = (-1,0,1), \quad \mathbf{g}^R_{\mathcal{M}'}(\langle 2 \rangle) = \mathbf{g}^L_{\mathcal{M}'}(\langle 2 \rangle) = (1,0,0), \end{align*}

and

\begin{align*} & \Lambda(\langle 1 \rangle,\langle 2\rangle) = \Lambda(\langle 2\rangle,\langle 1\rangle)=1, \quad \Lambda(\langle 1\rangle,\langle 12\rangle)=1, \\ & \Lambda(\langle 1\rangle,\langle 21\rangle) =-1, \quad \Lambda(\langle 12\rangle,\langle 2\rangle) = 1, \quad \Lambda(\langle 21\rangle,\langle 2\rangle) = -1. \end{align*}

Note that $\Lambda (X,X)=0$ for an affreal simple module $X$. Thus, we have

\begin{align*} \begin{array}{ll} {\mathrm G}^R_{\mathcal{M}^{\boldsymbol{i}}}(\langle 1\rangle,\langle 2\rangle) = 1 \times \Lambda(\langle 1\rangle,\langle 21\rangle) =-1, \quad & {\mathrm G}^R_{\mathcal{M}'}(\langle 1\rangle,\langle 2\rangle) =1 \times \Lambda(\langle 12 \rangle,\langle 2 \rangle) =1, \\ {\mathrm G}^L_{\mathcal{M}^{\boldsymbol{i}}}(\langle 1\rangle,\langle 2\rangle) = 1 \times \Lambda(\langle 1\rangle,\langle 12\rangle) =1, \quad & {\mathrm G}^L_{\mathcal{M}'}(\langle 1\rangle,\langle 2\rangle) =1 \times \Lambda(\langle 21 \rangle,\langle 2 \rangle) =-1. \end{array} \end{align*}

Thus, for a non-commuting pair of simple modules $(X,Y)$ in $\mathscr {C}$, the $\mathbb {Z}\mspace {1mu}$-values ${\mathrm G}^R_\mathcal {M}(X,Y)$ and ${\mathrm G}^L_\mathcal {M}(X,Y)$ do depend on the choice of a quasi-Laurent commuting family $\mathcal {M}$.

5.2 Skew-symmetric pairing associated with the GLS cluster

Let $w \in \mathsf {W}$ and ${\boldsymbol {i}}=(i_1,\ldots,i_r)$ a reduced sequence of $w$. Let $\mathcal {M}^{\boldsymbol {i}}$ be the associated GLS cluster. For such a Laurent family, we can define ${\mathrm G}_{\mathcal {M}^{\boldsymbol {i}}}^R$ in terms of PBW decompositions.

We define a skew-symmetric $\mathbb {Z}\mspace {1mu}$-valued map $\lambda ^{\boldsymbol {i}}\colon [1,r]\times [1,r]\to \mathbb {Z}\mspace {1mu}$ by

(5.4)\begin{equation} \lambda^{\boldsymbol{i}} _{a,b} \mathbin{:=} (-1)^{\delta(a>b)} \delta(a \ne b) ({\mspace{1mu}\beta}^{\boldsymbol{i}}_a,{\mspace{1mu}\beta}^{\boldsymbol{i}}_b) \end{equation}

for $1 \leqslant a,b\leqslant r$.

Let us recall the notion of ${\boldsymbol {i}}$-box and an affreal simple module $M^{\boldsymbol {i}}[a,b]$ in $\mathscr {C}_w$ for an ${\boldsymbol {i}}$-box $[a,b]$, which are introduced in [Reference Kashiwara, Kim, Oh and ParkKKOP24b].

1. (a) For $1 \leqslant a \leqslant b \leqslant r$ such that $i_a=i_b$, we call an interval $[a,b]$ an ${\boldsymbol {i}}$-box.

2. (b) For an ${\boldsymbol {i}}$-box $[a,b]$, we set $[a,b]_{\boldsymbol {i}} \mathbin {:=} \{ u \mid a \leqslant u \leqslant b, \; i_a=i_u\}$.

3. (c) For an ${\boldsymbol {i}}$-box $[a,b]$, we set

   \begin{align*} M^{\boldsymbol{i}}[a,b] \mathbin{:=} M(w^{\boldsymbol{i}}_{\leqslant b}\varpi_{i_a},w^{\boldsymbol{i}}_{< a}\varpi_{i_a})& \simeq {\operatorname{hd}}\Big( \circ_{u \in [a,b]_{\boldsymbol{i}}}^{\to} S^{\boldsymbol{i}}_u \Big) \\ &\simeq S^{\boldsymbol{i}}_b \triangledown M^{\boldsymbol{i}}[a,b_-] \simeq M^{\boldsymbol{i}}[a_+,b] \triangledown S^{\boldsymbol{i}}_a, \end{align*}
   up to grading shifts. In particular, $M^{\boldsymbol {i}}_k = M^{\boldsymbol {i}}[k_{\min },k]$ and $S^{\boldsymbol {i}}_k = M^{\boldsymbol {i}}[k,k]$.

Note that $M^{\boldsymbol {i}}[a,b]$ is an affreal simple module in $\mathscr {C}_w$.

Proposition 5.7 For ${\boldsymbol {i}}$-boxes $[x,y]$ and $[x',y']$ in an interval $[1,r]$, assume that

(5.5)\begin{equation} {\rm{(a)}} \ \ x > x'_- \quad \text{or} \quad {\rm{(b)}} \ \ y_+ > y'. \end{equation}

Then we have

(5.6)\begin{equation} \Lambda( M^{\boldsymbol{i}}[x,y], M^{\boldsymbol{i}}[x',y']) = \sum_{ u \in [x,y]_{\boldsymbol{i}} , \ v \in [x',y']_{\boldsymbol{i}} } \lambda_{u,v}. \end{equation}

Proof. Since the proof is similar, we shall give only the proof of case (a). Let us divide into sub-cases as below.

(i) [$x=y > x'_-$]  If $x>x'$, we have

\begin{align*} \Lambda(S^{\boldsymbol{i}}_x, M^{\boldsymbol{i}}[x',y']) & = \Lambda(S^{\boldsymbol{i}}_x,M^{\boldsymbol{i}}[x'_+,y'] \triangledown S^{\boldsymbol{i}}_{x'}) \\ & \underset{(1)}= \Lambda(S^{\boldsymbol{i}}_x,M^{\boldsymbol{i}}[ x'_+,y'])+ \Lambda(S^{\boldsymbol{i}}_x, S^{\boldsymbol{i}}_{x'})= \Lambda(S^{\boldsymbol{i}}_x,M^{\boldsymbol{i}}[x'_+,y'])+ \lambda^{\boldsymbol{i}}_{x,x'}. \end{align*}

Here $\underset {(1)}=$ holds by [Reference Kashiwara, Kim, Oh and ParkKKOP23, Proposition 2.12] and the fact that $(S^{\boldsymbol {i}}_x,S^{\boldsymbol {i}}_{x'})$ is an unmixed pair. Then by the induction hypothesis on $|[x',y']_{\boldsymbol {i}}|$, we have

\[ \Lambda(S^{\boldsymbol{i}}_x, M^{\boldsymbol{i}}[x',y']) = \lambda^{\boldsymbol{i}}_{x,x'} + \sum_{v \in [x'_+,y']_{\boldsymbol{i}}} \lambda^{\boldsymbol{i}}_{x,v} = \sum_{v \in [x', y' ]_{\boldsymbol{i}}} \lambda^{\boldsymbol{i}}_{x,v}, \]

as we desired.

Now, the remainder of case (i) can be described as follows:

\[ x'_-< x =y \leqslant x' \leqslant y'. \]

Since $S^{\boldsymbol {i}}_x$ commutes with $M^{\boldsymbol {i}}[x',y']$ and $M^{\boldsymbol {i}}[x',y'_-]$ by [Reference Kashiwara, Kim, Oh and ParkKKOP21, Proposition 3.27],

\begin{align*} \Lambda(S^{\boldsymbol{i}}_x,M^{\boldsymbol{i}}[x',y']) & = - \Lambda(M^{\boldsymbol{i}}[x',y'],S^{\boldsymbol{i}}_x) = -\Lambda( S^{\boldsymbol{i}}_{y'} \triangledown M^{\boldsymbol{i}}[x',y'_-],S^{\boldsymbol{i}}_x) \\ & = -\Lambda( S^{\boldsymbol{i}}_{y'} ,S^{\boldsymbol{i}}_x) -\Lambda( M^{\boldsymbol{i}}[x',y'_-],S^{\boldsymbol{i}}_x) \\ & = ({\mspace{1mu}\beta}_{y'},{\mspace{1mu}\beta}_x) + \Lambda( S^{\boldsymbol{i}}_x, M^{\boldsymbol{i}}[x',y'_-]) = \lambda^{\boldsymbol{i}}_{x,y'} + \Lambda( S^{\boldsymbol{i}}_x, M^{\boldsymbol{i}}[x',y'_-]), \end{align*}

then our assertion follows from the induction hypothesis on $|\,[x',y']_{\boldsymbol {i}}|$.

(ii) [$x< y$] Assume first that $y>y'$. Then we have

\begin{align*} \Lambda(M^{\boldsymbol{i}}[x,y],M^{\boldsymbol{i}}[x',y']) & = \Lambda(S^{\boldsymbol{i}}_y \triangledown M^{\boldsymbol{i}}[x,y_-],M^{\boldsymbol{i}}[x',y']) \\ & = \Lambda(S^{\boldsymbol{i}}_y,M^{\boldsymbol{i}}[x',y']) + \Lambda(M^{\boldsymbol{i}}[x,y_-],M^{\boldsymbol{i}}[x',y']). \end{align*}

Note that $(S^{\boldsymbol {i}}_y,M^{\boldsymbol {i}}[x',y'])$ is an unmixed pair. Then, by induction on $|[x,y]_{\boldsymbol {i}}|$, we have

\begin{align*} \Lambda(M^{\boldsymbol{i}}[x,y],M^{\boldsymbol{i}}[x',y'])& = \Lambda(S^{\boldsymbol{i}}_y \triangledown M^{\boldsymbol{i}}[x,y_-],M^{\boldsymbol{i}}[x',y'])\\ & = \sum_{v \in [x',y']_{\boldsymbol{i}}} \lambda^{\boldsymbol{i}}_{y,v} + \sum_{ u \in [x,y_-]_{\boldsymbol{i}}; \ v \in [x',y']_{\boldsymbol{i}} } \lambda^{\boldsymbol{i}}_{u,v}, \end{align*}

which yields our assertion for this case.

Now let us assume that $y \leqslant y'$. Then we have

\[ x'_- < x < y \leqslant y'. \]

Then for any $u \in [x,y]_{\boldsymbol {i}}$, $S^{\boldsymbol {i}}_u$ commutes with $M^{\boldsymbol {i}}[x',y']$ by [Reference Kashiwara, Kim, Oh and ParkKKOP21, Proposition 3.27]. By [Reference Kang, Kashiwara, Kim and OhKKKO18, Proposition 3.2.13], we have

\[ \Lambda(M^{\boldsymbol{i}}[x,y],M^{\boldsymbol{i}}[x',y']) = \sum_{u \in [x,y]_{\boldsymbol{i}}} \Lambda(S^{\boldsymbol{i}}_u,M^{\boldsymbol{i}}[x',y']). \]

Then our assertion follows from case (i).

We say that ${\boldsymbol {i}}$-boxes $[a_1,b_1]$ and $[a_2,b_2]$ commute if we have either

\[ (a_1)_- < a_2 \leqslant b_2 < (b_1)_+ \quad \text{or} \quad (a_2)_- < a_1 \leqslant b_1 < (b_2)_+. \]

The following corollary is proved in [Reference Kashiwara, Kim, Oh and ParkKKOP24b, Theorem 4.21] in the quantum affine case.

Proof. By Proposition 5.7, we have

\[ \Lambda(M^{\boldsymbol{i}}[a_1,b_1],M^{\boldsymbol{i}}[a_2,b_2]) = \sum_{\substack{u \in [a_1,b_1]_{\boldsymbol{i}} \\ v \in [a_2,b_2]_{\boldsymbol{i}} }} \lambda^{\boldsymbol{i}}_{u,v} = - \Lambda(M^{\boldsymbol{i}}[a_2,b_2],M^{\boldsymbol{i}}[a_1,b_1]), \]

which implies $\mathfrak {d}(M^{\boldsymbol {i}}[a_1,b_1],M^{\boldsymbol {i}}[a_2,b_2])=0$. Thus, our assertion follows from Proposition 2.7(v).

Proof. If the ${\boldsymbol {i}}$-boxes $[x,y]$ and $[x',y']$ satisfy (5.5), our assertion holds. Thus, it is enough to consider when $x \leqslant x'_-$ and $y_+ \leqslant y'$. Since they commute,

\[ \Lambda(M^{\boldsymbol{i}}[x,y],M^{\boldsymbol{i}}[x',y']) =- \Lambda(M^{\boldsymbol{i}}[x',y'],M^{\boldsymbol{i}}[x,y]). \]

If $x'> x_-$ or $y'_+> y$, Proposition 5.7 says that

\[ \Lambda(M^{\boldsymbol{i}}[x,y],M^{\boldsymbol{i}}[x',y']) = - \sum_{ u \in [x,y]_{\boldsymbol{i}} ; \ v \in [x',y']_{\boldsymbol{i}} } \lambda^{\boldsymbol{i}}_{v,u} = \sum_{ u \in [x,y]_{\boldsymbol{i}} ; \ v \in [x',y']_{\boldsymbol{i}} } \lambda^{\boldsymbol{i}}_{u,v}, \]

which implies the assertion. Thus, we may assume that $x' \leqslant x_-$. However, in this case, we have

\[ x' \leqslant x_- \leqslant x \leqslant x'_-, \]

which yields a contradiction.

Let us define the skew-symmetric pairing ${\mathrm L}_{\boldsymbol {i}}$ on ${\rm Irr}(\mathscr {C}_w)$ as follows:

(5.7)\begin{equation} {\mathrm L}_{\boldsymbol{i}}(X,Y) \mathbin{:=} \sum_{1 \leqslant a,b \leqslant r} ({\rm PBW}_{\boldsymbol{i}}(X))_a ({\rm PBW}_{\boldsymbol{i}}(Y))_b \; \lambda^{\boldsymbol{i}}_{a,b}. \end{equation}

The following lemma follows from Lemma 4.1 and (5.7).

Lemma 5.10 For $M=\mathcal {M}^{\boldsymbol {i}}(\mathbf {a})$ with $\mathbf {a} \in \mathbb {Z}\mspace {1mu}_{\ge 0}^{\mathsf {K}}$, we have

\[ {\mathrm L}_{\boldsymbol{i}}(X \triangledown M,Y) = {\mathrm L}_{\boldsymbol{i}}(X,Y)+ {\mathrm L}_{\boldsymbol{i}}(M,Y) \quad \text{and} \quad {\mathrm L}_{\boldsymbol{i}}(X,Y) = -{\mathrm L}_{\boldsymbol{i}}(Y,X). \]

Proposition 5.11 For any simple $X,Y$ in $\mathscr {C}_w$, we have

\[ {\mathrm L}_{\boldsymbol{i}}(X,Y)={\mathrm G}^R_{\mathcal{M}^{\boldsymbol{i}}}(X,Y). \]

Proof. Let $\mathcal {S}$ be the set of simple modules $Y$ in $\mathscr {C}_w$ such that ${\mathrm L}_{\boldsymbol {i}}(X,Y)={\mathrm G}^R_{\mathcal {M}^{\boldsymbol {i}}}(X,Y)$ for any simple $X\in \mathscr {C}_w$, and let $\mathcal {S}'$ be the set of simple modules $Y$ in $\mathscr {C}_w$ such that ${\mathrm L}_{\boldsymbol {i}}(\mathcal {M}^{\boldsymbol {i}}(\mathbf {a}),Y)={\mathrm G}^R_{\mathcal {M}^{\boldsymbol {i}}}(\mathcal {M}^{\boldsymbol {i}}(\mathbf {a}),Y)$ for any $\mathbf {a}\in \mathbb {Z}\mspace {1mu}_{\ge 0}^\mathsf {K}$. By Proposition 5.9, we have

\[ {\mathrm L}_{\boldsymbol{i}}(M^{\boldsymbol{i}}_s,M^{\boldsymbol{i}}_t)=\Lambda(M^{\boldsymbol{i}}_s,M^{\boldsymbol{i}}_t) \quad \text{for any $s,t \in \mathsf{K}$.} \]

Thus, we have $\mathcal {M}^{\boldsymbol {i}}(\mathbf {a}) \in \mathcal {S}'$ by Lemma 5.10. Now, let us show $\mathcal {S}'\subset \mathcal {S}$. Let $Y\in \mathcal {S}'$. For any simple $X$, there exist $\mathbf {a},\mathbf {b}\in \mathbb {Z}\mspace {1mu}_{\ge 0}^\mathsf {K}$ such that $X\triangledown \mathcal {M}^{\boldsymbol {i}}(\mathbf {a})\simeq \mathcal {M}^{\boldsymbol {i}}(\mathbf {b})$. Hence, we have

\begin{align*} {\mathrm L}_{\boldsymbol{i}}(X,Y)+{\mathrm L}_{\boldsymbol{i}}(\mathcal{M}^{\boldsymbol{i}}(\mathbf{a}),Y)&\underset{(1)}={\mathrm L}_{\boldsymbol{i}}(\mathcal{M}^{\boldsymbol{i}}(\mathbf{b}),Y)= {\mathrm G}^R_{\mathcal{M}^{\boldsymbol{i}}}(\mathcal{M}^{\boldsymbol{i}}(\mathbf{b}),Y)\\ &\underset{(2)}={\mathrm G}^R_{\mathcal{M}^{\boldsymbol{i}}}(X,Y)+{\mathrm G}^R_{\mathcal{M}^{\boldsymbol{i}}}(\mathcal{M}^{\boldsymbol{i}}(\mathbf{a}),Y)= {\mathrm G}^R_{\mathcal{M}^{\boldsymbol{i}}}(X,Y)+{\mathrm L}_{\boldsymbol{i}}(\mathcal{M}^{\boldsymbol{i}}(\mathbf{a}),Y). \end{align*}

Here $\underset {(1)}=$ follows from Lemma 5.10 and $\underset {(2)}=$ follows from Lemma 5.1. Hence, we have ${\mathrm L}_{\boldsymbol {i}}(X,Y)={\mathrm G}^R_{\mathcal {M}^{\boldsymbol {i}}}(X,Y)$. Thus, we have proved $\mathcal {S}'\subset \mathcal {S}$.

Since $\mathcal {M}^{\boldsymbol {i}}(\mathbf {a})\in \mathcal {S}'$, we have $\mathcal {M}^{\boldsymbol {i}}(\mathbf {a})\in \mathcal {S}$, which implies that

\[ {\mathrm L}_{\boldsymbol{i}}(Y,\mathcal{M}^{\boldsymbol{i}}(\mathbf{a}))={\mathrm G}^R_{\mathcal{M}^{\boldsymbol{i}}}(Y,\mathcal{M}^{\boldsymbol{i}}(\mathbf{a})) \]

for any simple $Y$. Hence any simple $Y$ belongs to $\mathcal {S}'$ and hence to $\mathcal {S}$.

5.3 Degree and codegree

In this subsection, we see the relationship between $\mathbf {g}^R_\mathcal {M}(X)$ (respectively, $\mathbf {g}^L_\mathcal {M}(X)$) and the degree (respectively, codegree) in the (quantum) cluster algebra theory. For a commuting family $\mathcal {M}=\{ M_j \mid j \in J \}$ labeled by a finite index set $J$, let us recall the preorder $\preccurlyeq _\mathcal {M}$ on $\mathbb {Z}\mspace {1mu}_{\geqslant 0}^{J}$ given in [Reference Kashiwara and KimKK19, § 3.3] (see also [Reference QinQin17, Definition 3.1.1]):

\begin{align*} \mathbf{b}' \preccurlyeq_\mathcal{M} \mathbf{b} \quad \text{if and only if} \quad & {\rm (1)} \ {\rm wt}(\mathcal{M}(\mathbf{b})) ={\rm wt}(\mathcal{M}(\mathbf{b}')), \\ & {\rm (2)} \ \Lambda(\mathcal{M}(\mathbf{b}'),M_j) \leqslant \Lambda(\mathcal{M}(\mathbf{b}),M_j) \text{ for all } j \in J. \end{align*}

The preorder $\preccurlyeq _\mathcal {M}$ can be extended to the one on $\mathbb {Z}\mspace {1mu}^{J}$ as follows: for $\mathbf {b},\mathbf {b}' \in \mathbb {Z}\mspace {1mu}^{J}$,

\[ \mathbf{b}' \preccurlyeq_\mathcal{M} \mathbf{b} \text{ if } \mathbf{b}'+\mathbf{a} \preccurlyeq_\mathcal{M} \mathbf{b}+\mathbf{a} \text{ for some } \mathbf{a} \in \mathbb{Z}\mspace{1mu}_{\geqslant 0}^{J} \text{ such that } \mathbf{b}+\mathbf{a},\mathbf{b}'+\mathbf{a} \in \mathbb{Z}\mspace{1mu}_{\geqslant 0}^{J}. \]

We write $\mathbf {b}'\prec _\mathcal {M}\mathbf {b}$, if $\mathbf {b}'\preccurlyeq _\mathcal {M}\mathbf {b}$ holds but $\mathbf {b}\preccurlyeq _\mathcal {M}\mathbf {b}'$ does not hold. Hence, $\mathbf {b}'\prec _\mathcal {M}\mathbf {b}$ if and only if $\mathbf {b}'\preccurlyeq _\mathcal {M}\mathbf {b}$ and there exists $j\in J$ such that $\Lambda (\mathcal {M}(\mathbf {b}),M_j)<\Lambda (\mathcal {M}(\mathbf {b}'),M_j)$.

The following proposition is proved for symmetric quiver Hecke algebras and can be extended to general quiver Hecke algebras using almost the same argument:

Proposition 5.13 [Reference Kashiwara and KimKK19, Proposition 3.3]

For a monoidal cluster $\mathcal {M}=\{ M_k \mid k \in \mathsf {K}\}$ associated with a quantum seed $(\{ X_k\}_{k\in \mathsf {K}}, L,\widetilde {B})$,

\[ \mathbf{b}' \preccurlyeq_\mathcal{M} \mathbf{b} \quad \text{if and only if} \quad \mathbf{b}-\mathbf{b}' = \widetilde{B} {\underline{v}} \text{ for some } {\underline{v}} \in \mathbb{Z}\mspace{1mu}_{\ge0}^{\mathsf{K}_\mathrm{ex}}. \]

In particular, the relation $\preccurlyeq _\mathcal {M}$ is an order on $\mathbb {Z}\mspace {1mu}^\mathsf {K}$.