1.1 Quantum groups and unipotent quantum coordinate rings

Let $I$ be an index set (for simple roots) and let $(A,\mathsf{P},\unicode[STIX]{x1D6F1},\mathsf{P}^{\vee },\unicode[STIX]{x1D6F1}^{\vee })$ be a symmetric Cartan datum. It consists of a symmetric generalized Cartan matrix $A=(a_{i,j})_{i,j\in I}$ , a free abelian group $\mathsf{P}$ called the weight lattice, a set $\unicode[STIX]{x1D6F1}=\{\unicode[STIX]{x1D6FC}_{i}\in \mathsf{P}\mid i\in I\}$ of linearly independent elements called simple roots, the group $\mathsf{P}^{\vee }:=\operatorname{Hom}_{\mathbb{Z}}(\mathsf{P},\mathbb{Z})$ called the coweight lattice, and a set $\unicode[STIX]{x1D6F1}^{\vee }=\{h_{i}\mid i\in I\}\subset \mathsf{P}^{\vee }$ of simple coroots. We assume that $\langle h_{i},\unicode[STIX]{x1D6FC}_{j}\rangle =a_{ij}$ for all $i,j\in I$ , and for each $i\in I$ there exists $\unicode[STIX]{x1D71B}_{i}\in \mathsf{P}$ such that $\langle h_{j},\unicode[STIX]{x1D71B}_{i}\rangle =\unicode[STIX]{x1D6FF}_{ij}$ for all $j\in I$ . The elements $\unicode[STIX]{x1D71B}_{i}$ are called the fundamental weights. The free abelian group $\mathsf{Q}:=\bigoplus _{i\in I}\mathbb{Z}\unicode[STIX]{x1D6FC}_{i}$ is called the root lattice. Set $\mathsf{Q}^{+}=\sum _{i\in I}\mathbb{Z}_{{\geqslant}0}\unicode[STIX]{x1D6FC}_{i}\subset \mathsf{Q}$ and $\mathsf{Q}^{-}=\sum _{i\in I}\mathbb{Z}_{{\leqslant}0}\unicode[STIX]{x1D6FC}_{i}\subset \mathsf{Q}$ . For $\unicode[STIX]{x1D6FD}=\sum _{i\in I}m_{i}\unicode[STIX]{x1D6FC}_{i}\in \mathsf{Q}$ , we set $|\unicode[STIX]{x1D6FD}|=\sum _{i\in I}|m_{i}|$ . For $\unicode[STIX]{x1D6FD}\in \mathsf{Q}^{+}$ with $n=|\unicode[STIX]{x1D6FD}|$ , we set $I^{\unicode[STIX]{x1D6FD}}:=\{\unicode[STIX]{x1D708}=(\unicode[STIX]{x1D708}_{1},\ldots ,\unicode[STIX]{x1D708}_{n})\in I^{n}\mid \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D708}_{1}}+\cdots +\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D708}_{n}}=\unicode[STIX]{x1D6FD}\}.$ Note that the symmetric group $\mathfrak{S}_{n}=\langle s_{1},\ldots ,s_{n-1}\rangle$ acts on $I^{\unicode[STIX]{x1D6FD}}$ by place permutations. Set $\mathfrak{h}=\mathbb{Q}\otimes _{\mathbb{Z}}\mathsf{P}^{\vee }$ . Then there exists a non-degenerate symmetric bilinear form $(\,,\,)$ on $\mathfrak{h}^{\ast }$ satisfying $\langle h_{i},\unicode[STIX]{x1D706}\rangle =(\unicode[STIX]{x1D6FC}_{i},\unicode[STIX]{x1D706})$ for any $\unicode[STIX]{x1D706}\in \mathfrak{h}^{\ast }$ and $i\in I$ . The Weyl group $W$ of $\mathfrak{g}$ is the subgroup $\langle r_{i};~i\in I\rangle \leqslant \operatorname{GL}(\mathfrak{h}^{\ast })$ , where $r_{i}(\unicode[STIX]{x1D706}):=\unicode[STIX]{x1D706}-\langle h_{i},\unicode[STIX]{x1D706}\rangle \unicode[STIX]{x1D6FC}_{i}$ for $\unicode[STIX]{x1D706}\in \mathfrak{h}^{\ast },i\in I$ . An element $\unicode[STIX]{x1D707}\in \mathsf{P}$ is called a dominant integral weight if $\langle h_{i},\unicode[STIX]{x1D707}\rangle \geqslant 0$ for all $i\in I$ . Let us denote the set of dominant integral weights by $\mathsf{P}^{+}$ .

Let $q$ be an indeterminate. The quantum group associated with $(A,\mathsf{P},\unicode[STIX]{x1D6F1},\mathsf{P}^{\vee },\unicode[STIX]{x1D6F1}^{\vee })$ is the associative algebra $U_{q}(\mathfrak{g})$ over $\mathbb{Q}(q)$ generated by $e_{i},f_{i}$ $(i\in I)$ and $q^{h}$ $(h\in \mathsf{P}^{\vee })$ with the defining relations

where $[k]=(q^{k}-q^{-k})/(q-q^{-1})$ for $k\in \mathbb{Z}_{{\geqslant}0}$ , $[k]!=[1]\cdots [k]$ and $\big[\!\begin{smallmatrix}n\\ m\\ \end{smallmatrix}\!\big]=[n]!/([m]![n-m]!)$ .

Note that $U_{q}(\mathfrak{g})=\bigoplus _{\unicode[STIX]{x1D6FD}\in \mathsf{Q}}U_{q}(\mathfrak{g})_{\unicode[STIX]{x1D6FD}}$ , where

Let $U_{q}^{+}(\mathfrak{g})$ be the $\mathbb{Q}(q)$ -subalgebra of $U_{q}(\mathfrak{g})$ generated by $\{e_{i}\mid i\in I\}$ . Let $U_{q}^{+}(\mathfrak{g})_{\mathbb{Z}[q^{\pm 1}]}$ be the $\mathbb{Z}[q^{\pm 1}]$ -subalgebra of $U_{q}^{+}(\mathfrak{g})$ generated by $e_{i}^{(n)}:=e_{i}^{n}/[n]!$ ( $i\in I$ , $n\in \mathbb{Z}_{{>}0}$ ).

Let $\unicode[STIX]{x1D6E5}_{\mathfrak{n}}$ be the algebra homomorphism $U_{q}^{+}(\mathfrak{g})\rightarrow U_{q}^{+}(\mathfrak{g})\otimes U_{q}^{+}(\mathfrak{g})$ given by $\unicode[STIX]{x1D6E5}_{\mathfrak{n}}(e_{i})=e_{i}\otimes 1+1\otimes e_{i}$ , where the algebra structure on $U_{q}^{+}(\mathfrak{g})\otimes U_{q}^{+}(\mathfrak{g})$ is defined by $(x_{1}\otimes x_{2})\cdot (y_{1}\otimes y_{2})=q^{-(\operatorname{wt}(x_{2}),\operatorname{wt}(y_{1}))}(x_{1}y_{1}\otimes x_{2}y_{2}).$ Set

Then $A_{q}(\mathfrak{n})$ is an algebra with multiplication given by $(\unicode[STIX]{x1D713}\cdot \unicode[STIX]{x1D703})(x)=\unicode[STIX]{x1D703}(x_{(1)})\unicode[STIX]{x1D713}(x_{(2)})$ , when $\unicode[STIX]{x1D6E5}_{\mathfrak{n}}(x)=x_{(1)}\otimes x_{(2)}$ in Sweedler’s notation.

Let us denote by $A_{q}(\mathfrak{n})_{\mathbb{Z}[q^{\pm 1}]}$ the $\mathbb{Z}[q^{\pm 1}]$ -submodule of $A_{q}(\mathfrak{n})$ consisting of $\unicode[STIX]{x1D713}\in A_{q}(\mathfrak{n})$ such that $\unicode[STIX]{x1D713}(U_{q}^{+}(\mathfrak{g})_{\mathbb{Z}[q^{\pm 1}]})\subset \mathbb{Z}[q^{\pm 1}]$ . Then it is a $\mathbb{Z}[q^{\pm 1}]$ -subalgebra of $A_{q}(\mathfrak{n})$ . Moreover, the upper global basis $\mathbf{B}^{\operatorname{up}}$ is a $\mathbb{Z}[q^{\pm 1}]$ -module basis of $A_{q}(\mathfrak{n})_{\mathbb{Z}[q^{\pm 1}]}$ . For details of the upper global basis, see [Reference Kang, Kashiwara, Kim and OhKKKO18, § 1.3]. For $w\in W$ , let $A_{q}(\mathfrak{n}(w))_{\mathbb{Z}[q^{\pm 1}]}$ be the $\mathbb{Z}[q^{\pm 1}]$ -submodule of $A_{q}(\mathfrak{n})_{\mathbb{Z}[q^{\pm 1}]}$ consisting of the elements $\unicode[STIX]{x1D713}\in A_{q}(\mathfrak{n})_{\mathbb{Z}[q^{\pm 1}]}$ such that $e_{i_{1}}\cdots e_{i_{n}}\unicode[STIX]{x1D713}=0$ for any $\unicode[STIX]{x1D6FD}\in (\mathsf{Q}^{+}\cap w\mathsf{Q}^{+})\setminus \{0\}$ and any sequence $(i_{1},\ldots ,i_{n})\in I^{\unicode[STIX]{x1D6FD}}$ . Then $A_{q}(\mathfrak{n}(w))_{\mathbb{Z}[q^{\pm 1}]}$ is a $\mathbb{Z}[q^{\pm 1}]$ -subalgebra of $A_{q}(\mathfrak{n})_{\mathbb{Z}[q^{\pm 1}]}$  [Reference Kashiwara, Kim, Oh and ParkKKOP18, Theorem 2.20]. We call this algebra the quantum unipotent coordinate ring associated with $w$ . The set $\mathbf{B}^{\operatorname{up}}(w):=\mathbf{B}^{\operatorname{up}}\cap A_{q}(\mathfrak{n}(w))$ forms a $\mathbb{Z}[q^{\pm 1}]$ -basis of $A_{q}(\mathfrak{n}(w))_{\mathbb{Z}[q^{\pm 1}]}$  [Reference KimuraKim12].

For a dominant weight $\unicode[STIX]{x1D706}\in \mathsf{P}^{+}$ , let $V(\unicode[STIX]{x1D706})$ be the irreducible highest weight $U_{q}(\mathfrak{g})$ -module with highest weight vector $u_{\unicode[STIX]{x1D706}}$ of weight $\unicode[STIX]{x1D706}$ . Let $(~,~)_{\unicode[STIX]{x1D706}}$ be the non-degenerate symmetric bilinear form on $V(\unicode[STIX]{x1D706})$ such that $(u_{\unicode[STIX]{x1D706}},u_{\unicode[STIX]{x1D706}})_{\unicode[STIX]{x1D706}}=1$ and $(xu,v)_{\unicode[STIX]{x1D706}}=(u,\unicode[STIX]{x1D711}(x)v)_{\unicode[STIX]{x1D706}}$ for $u,v\in V(\unicode[STIX]{x1D706})$ and $x\in U_{q}(\mathfrak{g})$ , where $\unicode[STIX]{x1D711}$ is the algebra antiautomorphism on $U_{q}(\mathfrak{g})$ defined by $\unicode[STIX]{x1D711}(e_{i})=f_{i}$ , $\unicode[STIX]{x1D711}(f_{i})=e_{i}$ and $\unicode[STIX]{x1D711}(q^{h})=q^{h}$ . For each $\unicode[STIX]{x1D707},\unicode[STIX]{x1D701}\in W\unicode[STIX]{x1D706}$ , the unipotent quantum minor $D(\unicode[STIX]{x1D707},\unicode[STIX]{x1D701})$ is an element in $A_{q}(\mathfrak{n})$ given by $D(\unicode[STIX]{x1D707},\unicode[STIX]{x1D701})(x)=(xu_{\unicode[STIX]{x1D707}},u_{\unicode[STIX]{x1D701}})_{\unicode[STIX]{x1D706}}$ for $x\in U_{q}^{+}(\mathfrak{g})$ , where $u_{\unicode[STIX]{x1D707}}$ and $u_{\unicode[STIX]{x1D701}}$ are unique members of the upper global basis $\mathbf{B}^{\operatorname{up}}(\unicode[STIX]{x1D706})$ of $V(\unicode[STIX]{x1D706})$ of weight $\unicode[STIX]{x1D707}$ and $\unicode[STIX]{x1D701}$ , respectively (see [Reference Kang, Kashiwara, Kim and OhKKKO18, § 1.3] for the definition of $\mathbf{B}^{\operatorname{up}}(\unicode[STIX]{x1D706})$ ). For each reduced expression $\text{}\underline{w}=s_{i_{1}}s_{i_{2}}\cdots s_{i_{l}}$ of $w\in W$ , define $w_{{\leqslant}k}:=s_{i_{1}}\cdots s_{i_{k}}$ and $w_{{<}k}:=s_{i_{1}}\cdots s_{i_{k-1}}$ for $1\leqslant k\leqslant l$ . Then the set $\{D(w_{{\leqslant}k}\unicode[STIX]{x1D71B}_{i_{k}},w_{{<}k}\unicode[STIX]{x1D71B}_{i_{k}})\mid 1\leqslant k\leqslant l\}$ of unipotent quantum minors generates $A_{q}(\mathfrak{n}(w))_{\mathbb{Z}[q^{\pm }1]}$ as a $\mathbb{Z}[q^{\pm 1}]$ -algebra.

1.2 Quantum cluster algebras

Fix finite index sets $J_{\text{ex}}$ and $J_{\text{fr}}$ and we call the elements in $J_{\text{ex}}$ and $J_{\text{fr}}$ exchangeable indices and frozen indices, respectively. Set $J:=J_{\text{ex}}\sqcup J_{\text{fr}}$ . For a given skew-symmetric integer-valued $J\times J$ matrix $L=(\unicode[STIX]{x1D706}_{ij})_{i,j\in J}$ , we define $\mathscr{P}(L)$ as the $\mathbb{Z}[q^{\pm 1/2}]$ -algebra generated by a family of elements $\{{X_{i}\}}_{i\in J}$ with the defining relations $X_{i}X_{j}=q^{\unicode[STIX]{x1D706}_{ij}}X_{j}X_{i}\;(i,j\in J)$ . For $\mathbf{a}=(a_{i})_{i\in J}\in \mathbb{Z}^{J}$ , set

where $\mathscr{F}(L)$ is the skew field of fractions of $\mathscr{P}(L)$ , and  with $i_{1}<\cdots <i_{r}$ in a fixed total order on $J$ . Note that $X^{\mathbf{a}}$ does not depend on the choice of a total order. Then we have

and $\{X^{\mathbf{a}}\}_{\mathbf{a}\in \mathbb{Z}_{{\geqslant}0}^{J}}$ is a basis of $\mathscr{P}(L)$ as a $\mathbb{Z}[q^{\pm 1/2}]$ -module. The subalgebra $\mathbb{Z}[q^{\pm 1/2}][X_{i}^{\pm 1}\mid i\in J]$ of $\mathscr{F}(L)$ is called a quantum torus.

Let $\widetilde{B}=(b_{ij})_{(i,j)\in J\times J_{\text{ex}}}$ be an integer-valued $J\times J_{\text{ex}}$ matrix whose principal part $B:=(b_{ij})_{i,j\in J_{\text{ex}}}$ is skew-symmetric. The matrix $\widetilde{B}$ is called an exchange matrix. A pair $(L,\widetilde{B})$ of skew-symmetric integer-valued $J\times J$ matrix $L$ and an exchange matrix $\widetilde{B}$ is called compatible if there exists a positive integer $d$ such that $\sum _{k\in J}\unicode[STIX]{x1D706}_{ik}b_{kj}=\unicode[STIX]{x1D6FF}_{ij}d$ for all $i\in J$ and $j\in J_{\text{ex}}$  [Reference Berenstein and ZelevinskyBZ05, § 3]. In this paper we treat only the case $d=2$ . Let $(L,\widetilde{B})$ be a compatible pair and $A$ a $\mathbb{Z}[q^{\pm 1/2}]$ -algebra.

We say that ${\mathcal{S}}=(\{{x_{i}\}}_{i\in J},L,\widetilde{B})$ is a quantum seed in $A$ if $\{{x_{i}\}}_{i\in J}$ satisfies $x_{i}x_{j}=q^{\unicode[STIX]{x1D706}_{ij}}x_{j}x_{i}$ for any $i,j\in J$ and the algebra map $\mathscr{P}(L)\rightarrow A$ given by $X_{i}\mapsto x_{i}$ is injective. In that case, we call the set $\{{x_{i}\}}_{i\in J}$ the cluster of ${\mathcal{S}}$ and its elements the cluster variables. The elements $x^{\mathbf{a}}$ ( $\mathbf{a}\in \mathbb{Z}_{{\geqslant}0}^{J}$ ) are called the quantum cluster monomials.

Let ${\mathcal{S}}=(\{{x_{i}\}}_{i\in J},L,\widetilde{B})$ be a quantum seed in a $\mathbb{Z}[q^{\pm 1/2}]$ -algebra $A$ which is contained in a skew field $K$ . For each $k\in J_{\text{ex}}$ , we define

and

where $\mathbf{a}^{\prime }=(a_{i}^{\prime })_{i\in J}$ and $\mathbf{a}^{\prime \prime }=(a_{i}^{\prime \prime })_{i\in J}$ are given by

Then the triple

becomes a new quantum seed in $K$ and we call it the mutation of ${\mathcal{S}}$ at $k$ .

The quantum cluster algebra $\mathscr{A}_{q^{1/2}}({\mathcal{S}})$ associated to the quantum seed ${\mathcal{S}}$ is the $\mathbb{Z}[q^{\pm 1/2}]$ -subalgebra of the skew field $K$ generated by all the cluster variables in the quantum seeds obtained from ${\mathcal{S}}$ by any sequence of mutations.

A quantum cluster algebra structure on a $\mathbb{Z}[q^{\pm 1/2}]$ -algebra $A$ is a family $\mathscr{F}$ of quantum seeds in $A$ such that:

1. (a) for any quantum seed ${\mathcal{S}}$ in $\mathscr{F}$ , $\mathscr{A}_{q^{1/2}}({\mathcal{S}})$ is isomorphic to $A$ ;

2. (b) for any pair ${\mathcal{S}},{\mathcal{S}}^{\prime }$ of quantum seeds in $\mathscr{F}$ , ${\mathcal{S}}^{\prime }$ is obtained from ${\mathcal{S}}$ by a sequence of mutations;

3. (c) any mutation of a quantum seed in $\mathscr{F}$ is in $\mathscr{F}$ .

We call a quantum seed in $\mathscr{F}$ a quantum seed of the quantum cluster algebra $A$ .

The quantum Laurent phenomenon [Reference Berenstein and ZelevinskyBZ05, Corollary 5.2] asserts that for any quantum seed ${\mathcal{S}}$ , every element $x\in \mathscr{A}_{q^{1/2}}({\mathcal{S}})$ can be expressed as

in $K$ , where $X=(x_{i})_{i\in J}$ is the cluster in ${\mathcal{S}}$ , and $c_{\mathbf{a}}\in \mathbb{Z}[q^{\pm 1/2}]$ .

Remark that in [Reference Berenstein and ZelevinskyBZ05] the quantum cluster algebra associated with ${\mathcal{S}}$ is defined as the subalgebra of $K$ generated by all the cluster variables together with the inverses of the frozen variables. Then it is the (right) ring of quotients of $\mathscr{A}_{q^{1/2}}({\mathcal{S}})$ with respect to the multiplicative subset consisting of monomials in the frozen variables.

Geiß, Leclerc and Schröer showed that the algebra $\mathbb{Z}[q^{\pm 1/2}]\otimes _{\mathbb{Z}[q^{\pm 1}]}A_{q}(\mathfrak{n}(w))_{\mathbb{Z}[q^{\pm 1}]}$ has a quantum cluster algebra structure [Reference Geiß, Leclerc and SchröerGLS13b]. In particular, for each choice of reduced expression $\text{}\underline{w}=s_{i_{1}}s_{i_{2}}\cdots s_{i_{l}}$ , they gave a distinguished quantum seed whose cluster consists of $D(w_{{\leqslant}k}\unicode[STIX]{x1D71B}_{i_{k}},\unicode[STIX]{x1D71B}_{i_{k}})$ for $1\leqslant k\leqslant l$ up to a power of $q^{1/2}$ . For a detailed description, see § 2.3 below.

1.3 Quiver Hecke algebras

Let $\mathbf{k}$ be a base field. Take a family of polynomials $(Q_{ij})_{i,j\in I}$ in $\mathbf{k}[u,v]$ which are of the form

such that $Q_{i,j}(u,v)=Q_{j,i}(v,u)$ .

The (symmetric) quiver Hecke algebra (also called the Khovanov–Lauda–Rouquier algebra) $R(\unicode[STIX]{x1D6FD})$ at $\unicode[STIX]{x1D6FD}\in \mathsf{Q}^{+}$ associated with $(A,\mathsf{P},\unicode[STIX]{x1D6F1},\mathsf{P}^{\vee },\unicode[STIX]{x1D6F1}^{\vee })$ and $(Q_{ij})_{i,j\in I}$ is the algebra over $\mathbf{k}$ generated by the elements $\{e(\unicode[STIX]{x1D708})\}_{\unicode[STIX]{x1D708}\in I^{\unicode[STIX]{x1D6FD}}}$ , $\{{x_{k}\}}_{1\leqslant k\leqslant n}$ , $\{{\unicode[STIX]{x1D70F}_{m}\}}_{1\leqslant m\leqslant n-1}$ satisfying the following defining relations:

The algebra $R(\unicode[STIX]{x1D6FD})$ is $\mathbb{Z}$ -graded with $\deg e(\unicode[STIX]{x1D708})=0$ , $\deg x_{k}e(\unicode[STIX]{x1D708})=2$ , and $\deg \unicode[STIX]{x1D70F}_{m}e(\unicode[STIX]{x1D708})=-(\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D708}_{m}},\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D708}_{m+1}})$ .

Note that the choice of $\pm$ in (1.1) is irrelevant. That is to say, the isomorphism class of the quiver Hecke algebra $R(\unicode[STIX]{x1D6FD})$ does not depend on the choice of $\pm$ .

Let us denote by $R(\unicode[STIX]{x1D6FD})\text{-}\text{gmod}$ the category of finite-dimensional graded $R(\unicode[STIX]{x1D6FD})$ -modules with homomorphisms of degree $0$ . For an $R(\unicode[STIX]{x1D6FD})$ -module $M$ , we set $\operatorname{wt}(M):=-\unicode[STIX]{x1D6FD}\in \mathsf{Q}^{-}$ . For the sake of simplicity, we say that $M$ is an $R$ -module instead of saying that $M$ is a graded $R(\unicode[STIX]{x1D6FD})$ -module for $\unicode[STIX]{x1D6FD}\in \mathsf{Q}^{+}$ . Let us denote by $q$ the grading shift functor: for $M\in R\text{-}\text{gmod}$ , we have $(qM)_{n}=M_{n-1}$ by definition.

We also sometimes ignore grading shifts if there is no danger of confusion. Hence, for $R$ -modules $M$ and $N$ , we sometimes say that $f:M\rightarrow N$ is a homomorphism if $f:q^{a}M\rightarrow N$ is a morphism in $R\text{-}\text{gmod}$ for some $a\in \mathbb{Z}$ . If we want to emphasize that $f:q^{a}M\rightarrow N$ is a morphism in $R\text{-}\text{gmod}$ , we say so. We set

For $M\in R(\unicode[STIX]{x1D6FD})\text{-}\text{gmod}$ , the space $M^{\ast }:=\operatorname{Hom}_{\mathbf{k}}(M,\mathbf{k})$ admits an $R(\unicode[STIX]{x1D6FD})$ -module structure induced by the $\mathbf{k}$ -algebra antiautomorphism $\unicode[STIX]{x1D713}$ on $R(\unicode[STIX]{x1D6FD})$ which fixes the generators $e(\unicode[STIX]{x1D708})$ , $x_{k}$ , and $\unicode[STIX]{x1D70F}_{m}$ . Hence we have a contravariant auto-equivalence $^{\ast }$ on $R\text{-}\text{gmod}$ . We say that a simple $R(\unicode[STIX]{x1D6FD})$ -module $M$ is self-dual if $M\simeq M^{\ast }$ in $R\text{-}\text{gmod}$ .

For $\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D6FE}\in \mathsf{Q}^{+}$ , set $e(\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D6FE}):=\sum _{\unicode[STIX]{x1D708}\in I^{\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D708}^{\prime }\in I^{\unicode[STIX]{x1D6FE}}}e(\unicode[STIX]{x1D708}\ast \unicode[STIX]{x1D708}^{\prime })\in R(\unicode[STIX]{x1D6FD}+\unicode[STIX]{x1D6FE})$ , where $\unicode[STIX]{x1D708}\ast \unicode[STIX]{x1D708}^{\prime }$ denotes the concatenation of $\unicode[STIX]{x1D708}$ and $\unicode[STIX]{x1D708}^{\prime }$ . Then there is a $\mathbf{k}$ -algebra homomorphism $R(\unicode[STIX]{x1D6FD})\otimes R(\unicode[STIX]{x1D6FE})\rightarrow e(\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D6FE})R(\unicode[STIX]{x1D6FD}+\unicode[STIX]{x1D6FE})e(\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D6FE})$ . The convolution product

is a bifunctor given by

Set $R\text{-}\text{gmod}:=\bigoplus _{\unicode[STIX]{x1D6FD}\in \mathsf{Q}_{+}}R(\unicode[STIX]{x1D6FD})\text{-}\text{gmod}$ . Then the category $R\text{-}\text{gmod}$ is a monoidal category with the tensor product $\circ$ and the unit object $\mathbf{1}:=\mathbf{k}\in R(0)\text{-}\text{gmod}$ .

For $M\in R(\unicode[STIX]{x1D6FD})\text{-}\text{gmod}$ and $N\in R(\unicode[STIX]{x1D6FE})\text{-}\text{gmod}$ , we have $(M\circ N)^{\ast }\simeq q^{(\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D6FE})}(N^{\ast }\circ M^{\ast })$ .

1.4 $R$ -matrices and real simple modules

For $1\leqslant a<|\unicode[STIX]{x1D6FD}|$ , we define the intertwiner $\unicode[STIX]{x1D711}_{a}\in R(\unicode[STIX]{x1D6FD})$ by

For $m,n\in \mathbb{Z}_{{\geqslant}0}$ , let us denote by $w[m,n]$ the element of $\mathfrak{S}_{m+n}$ defined by

and set $\unicode[STIX]{x1D711}_{[m,n]}:=\unicode[STIX]{x1D711}_{i_{1}}\cdots \unicode[STIX]{x1D711}_{i_{\ell }}$ , where $s_{i_{1}}\cdots s_{i_{t}}$ is a reduced expression of $w[m,n]$ . It does not depend on the choice of a reduced expression, since $\{\unicode[STIX]{x1D711}_{a}\}$ satisfies the braid relation [Reference Kang, Kashiwara and KimKKK18, § 1.3.2].

Let $M$ be an $R(\unicode[STIX]{x1D6FD})$ -module and $N$ an $R(\unicode[STIX]{x1D6FE})$ -module. Then the map $M\otimes N\rightarrow N\circ M$ given by $u\otimes v\longmapsto \unicode[STIX]{x1D711}_{[|\unicode[STIX]{x1D6FE}|,|\unicode[STIX]{x1D6FD}|]}(v\otimes u)$ induces an $R(\unicode[STIX]{x1D6FD}+\unicode[STIX]{x1D6FE})$ -module homomorphism

Let $z$ be an indeterminate which is homogeneous of degree $2$ , and let $\unicode[STIX]{x1D713}_{z}$ be the graded algebra homomorphism

given by

For an $R(\unicode[STIX]{x1D6FD})$ -module $M$ , we denote by $M_{z}$ the $(\mathbf{k}[z]\otimes R(\unicode[STIX]{x1D6FD}))$ -module $\mathbf{k}[z]\otimes M$ with the action of $R(\unicode[STIX]{x1D6FD})$ twisted by $\unicode[STIX]{x1D713}_{z}$ .

For a non-zero $R(\unicode[STIX]{x1D6FD})$ -module $M$ and a non-zero $R(\unicode[STIX]{x1D6FE})$ -module $N$ , let $s$ be the largest non-negative integer such that the image of $R_{M_{z},N}$ is contained in $z^{s}(N\circ M_{z})$ . We denote by $R_{M_{z},N}^{\text{ren}}$ the $R(\unicode[STIX]{x1D6FD})$ -module homomorphism $z^{-s}R_{M_{z},N}$ and call it the renormalized $R$ -matrix. We denote $\mathbf{r}_{M,N}:=R_{M_{z},N}^{\text{ren}}|_{z=0}:M\circ N\rightarrow N\circ M$ and call $\mathbf{r}_{M,N}$ the $R$ -matrix. By the definition, $\mathbf{r}_{M,N}$ never vanishes. We denote by $\unicode[STIX]{x1D6EC}(M,N)$ the homogeneous degree of $\mathbf{r}_{M,N}$ and set

Then we have

by [Reference Kang, Kashiwara and KimKKK18, Proposition 1.10(iii)] and [Reference Kang, Kashiwara, Kim and OhKKKO18, Lemma 3.2.1] for any simple modules $M$ and $N$ . For more properties of $\widetilde{\unicode[STIX]{x1D6EC}}(M,N)$ and $\mathfrak{d}(M,N)$ , see [Reference Kang, Kashiwara, Kim and OhKKKO18, § 3.2].

In such a case, $\mathbf{r}_{M,N}:q^{\unicode[STIX]{x1D6EC}(M,N)}M\circ N\rightarrow N\circ M$ is an isomorphism. If a simple module $M$ strongly commutes with itself, then $M$ is called real. Assume that $M$ and $N$ are simple modules and one of them is real. Then $M$ and $N$ strongly commute if and only if $M\circ N$ is isomorphic to $N\circ M$ up to a grading shift, which is equivalent to the condition $\mathfrak{d}(M,N)=0$ (see [Reference Kang, Kashiwara, Kim and OhKKKO18, Lemma 3.2.3]).

For simple $R$ -modules $M$ and $N$ , we denote by $M\,\unicode[STIX]{x1D6FB}\,N$ the head of $M\circ N$ and by $M\unicode[STIX]{x1D6E5}N$ the socle of $M\circ N$ .

Assume that $M$ and $N$ are self-dual simple modules and one of them is real. Then $q^{\widetilde{\unicode[STIX]{x1D6EC}}(M,N)}M\,\unicode[STIX]{x1D6FB}\,N$ is a self-dual simple module.

Let $\{{M_{i}\}}_{i\in J}$ be a mutually strongly commuting family of self-dual real simple modules. For an element $\mathbf{a}=(\mathbf{a}_{i})_{i\in J}\in \mathbb{Z}_{{\geqslant}0}^{J}$ , there exists a unique self-dual simple module which is a convolution product of modules $M_{i}$ with multiplicities $\mathbf{a}_{i}$ . We will denote it by $M(\mathbf{a})$ for short. For any sequence $(k_{1},\ldots ,k_{m})$ in which each $i\in J$ appears $\mathbf{a}_{i}$ times, we have

in $R\text{-}\text{gmod}$ , since the latter is a self-dual simple module.

Note that we have

2.1 Monoidal categorification of $A_{q}(\mathfrak{n}(w))$

The Grothendieck group $K(R\text{-}\text{gmod})$ has a $\mathbb{Z}[q^{\pm 1}]$ -algebra structure. The multiplication structure on $K(R\text{-}\text{gmod})$ is induced by the convolution product and the action of $q$ is induced by the grading shift functor $q$ . Moreover, the isomorphism classes of self-dual simple modules form a $\mathbb{Z}[q^{\pm 1}]$ -module basis of $K(R\text{-}\text{gmod})$ .

There is a $\mathbb{Z}[q^{\pm 1}]$ -algebra isomorphism between the Grothendieck ring $K(R\text{-}\text{gmod})$ and $A_{q}(\mathfrak{n})_{\mathbb{Z}[q^{\pm 1}]}$  [Reference Khovanov and LaudaKL09].

If we assume further that $\mathbf{k}$ is of characteristic $0$ , then the basis consisting of the classes of self-dual simple modules in $R\text{-}\text{gmod}$ corresponds to the upper global basis $\mathbf{B}^{\operatorname{up}}$ under the isomorphism [Reference Varagnolo and VasserotVV11, Reference RouquierRou12]. We have $[M^{\ast }]=\overline{[M]}$ for $M\in R\text{-}\text{gmod}$ , where ‘ $-$ ’ denotes the bar involution which is the $\mathbb{Z}$ -linear involution on $K(R\text{-}\text{gmod})$ sending $q\mapsto q^{-1}$ and fixing the classes of self-dual simple modules.

In particular, the quantum unipotent minors $D(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707})$ belong to $\mathbf{B}^{\operatorname{up}}$ and we denote the corresponding self-dual simple modules $M(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707})$ and call them determinantial modules. Even if $\operatorname{char}(\mathbf{k})\neq 0$ , there exists a unique simple module $M(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707})$ such that $[M(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707})]=D(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707})$  [Reference Kashiwara, Kim, Oh and ParkKKOP18, Proposition 4.1].

For an $R(\unicode[STIX]{x1D6FD})$ -module $M$ , define

For $w\in W$ , we denote by ${\mathcal{C}}_{w}$ the full subcategory of $R\text{-}\text{gmod}$ whose objects $M$ satisfy $\text{W}(M)\subset \mathsf{Q}^{+}\cap w\mathsf{Q}^{-}$ . Then the category ${\mathcal{C}}_{w}$ is the smallest monoidal abelian full subcategory of $R\text{-}\text{gmod}$ which is stable under taking subquotients, extensions and grading shifts, and contains the self-dual simple modules $\{M(w_{{\leqslant}k}\unicode[STIX]{x1D71B}_{i_{k}},w_{{<}k}\unicode[STIX]{x1D71B}_{i_{k}})\mid 1\leqslant k\leqslant l\}$ , where $\text{}\underline{w}=s_{i_{1}}s_{i_{2}}\cdots s_{i_{l}}$ is a reduced expression of $w$  [Reference Kashiwara, Kim, Oh and ParkKKOP18, Theorem 2.20]. The Grothendieck ring $K({\mathcal{C}}_{w})$ is isomorphic to $A_{q}(\mathfrak{n}(w))_{\mathbb{Z}[q^{\pm 1}]}$ . Moreover, the set of the classes of self-dual simple modules coincides with the upper global basis, whenever $\mathbf{k}$ is of characteristic $0$ .

We set

These are commutative rings.

In [Reference Kang, Kashiwara, Kim and OhKKKO18], it is shown that the category ${\mathcal{C}}_{w}$ gives a monoidal categorification of the cluster algebra $A_{q}(\mathfrak{n}(w))$ . The following theorem is one of the main results in [Reference Kang, Kashiwara, Kim and OhKKKO18].

If $[\mathscr{S}]$ in (2.1) is a quantum seed and satisfies Theorem 2.1(ii), then we call the triple $\mathscr{S}=(\{{M_{i}\}}_{i\in J},-\unicode[STIX]{x1D6EC},\widetilde{B})$ a monoidal seed in ${\mathcal{C}}_{w}$ , and $\mathscr{C}=\{{M_{i}\}}_{i\in J}$ a monoidal cluster in ${\mathcal{C}}_{w}$ . By Lemma 3.2 below, the monoidal seed $\mathscr{S}$ is completely determined by the monoidal cluster $\mathscr{C}$ .

We call $\unicode[STIX]{x1D707}_{k}(\mathscr{C}):=\{M_{i}^{\prime }\}_{i\in J}$ the mutation of $\mathscr{C}$ at $k$ , where $M_{i}^{\prime }=M_{i}$ $(i\neq k)$ and $M_{k}^{\prime }$ is as in Theorem 2.1(ii).

2.2 Laurent phenomenon

Let $\mathscr{C}=\{{M_{i}\}}_{i\in J}$ be a monoidal cluster in ${\mathcal{C}}_{w}$ . For $\mathbf{a}\in \mathbb{Z}_{{\geqslant}0}^{J}$ , we denote by $M_{\mathscr{C}}(\mathbf{a})$ the module $M(\mathbf{a})$ defined in (1.3) starting from $\mathscr{C}=\{{M_{i}\}}_{i\in J}$ . If there is no risk of confusion, we simply write $M(\mathbf{a})$ instead of $M_{\mathscr{C}}(\mathbf{a})$ .

The subalgebra $\mathbb{Z}[q^{\pm 1}][[M_{i}];i\in J]$ of $A_{q}(\mathfrak{n}(w))$ is isomorphic to the polynomial ring with the $[M_{i}]$ as variables. The quantum Laurent phenomenon claims that $K({\mathcal{C}}_{w})$ is contained in the quantum Laurent polynomial ring $\mathbb{Z}[q^{\pm 1}][[M_{i}];i\in J_{\text{fr}}][[M_{i}]^{\pm 1};i\in J_{\text{ex}}]$ in the skew field of fractions of $A_{q}(\mathfrak{n}(w))$ .

Hence, for a module $X$ in ${\mathcal{C}}_{w}$ , there exists $\mathbf{a}\in \mathbb{Z}_{{\geqslant}0}^{J}$ with $a_{j}=0$ for $j\in J_{\text{fr}}$ such that

for some $r\geqslant 0$ , $\mathbf{b}(s)\in \mathbb{Z}_{{\geqslant}0}^{J}$ and $c(\mathbf{b}(s))\in \mathbb{Z}[q^{\pm 1/2}]$ . Since $[X\circ M(\mathbf{a})]$ is the class of a module in ${\mathcal{C}}_{w}$ and each $[M(\mathbf{b}(s))]$ is the class of a real simple module, the coefficient $c(\mathbf{b}(s))$ lies in $\mathbb{Z}_{{\geqslant}0}[q^{\pm 1}]$ for all $1\leqslant s\leqslant r$ .

Summing up, the following lemma is an immediate consequence of the quantum Laurent phenomenon.

In the sequel, we write

for a monoidal cluster $\mathscr{C}=\{{M_{i}\}}_{i\in J}$ .

The following theorem is an immediate consequence of the lemma above.

2.3 Initial seeds and dual PBW basis

Let $\text{}\underline{w}=s_{i_{1}}\cdots s_{i_{l}}$ be a reduced expression of $w\in W$ , and set $\mathbf{i}=(i_{1},\ldots ,i_{l})$ . Recalling that $w_{{\leqslant}k}:=s_{i_{1}}\cdots s_{i_{k}}$ , $w_{{<}k}:=s_{i_{1}}\cdots s_{i_{k-1}}$ , we set $\unicode[STIX]{x1D6FD}_{k}:=w_{{<}k}(\unicode[STIX]{x1D6FC}_{i_{k}})$ for $1\leqslant k\leqslant l$ . Then $\unicode[STIX]{x1D6E5}_{+}\cap w\unicode[STIX]{x1D6E5}_{-}=\{\unicode[STIX]{x1D6FD}_{k}\mid 1\leqslant k\leqslant l\}$ , where $\unicode[STIX]{x1D6E5}_{+}$ and $\unicode[STIX]{x1D6E5}_{-}$ denote the set of positive roots and the set of negative roots, respectively. Let $\preccurlyeq$ be a convex order on the set $\unicode[STIX]{x1D6E5}_{+}$ satisfying

See [Reference Tingley and WebsterTW16, Reference Kashiwara, Kim, Oh and ParkKKOP18] for details of convex orders on $\unicode[STIX]{x1D6E5}_{+}$ .

Define

Then $S_{k}$ is a self-dual $R(\unicode[STIX]{x1D6FD}_{k})$ -module such that $\text{W}(S_{k})\subset \operatorname{Span}_{\mathbb{R}_{{\geqslant}0}}\{\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E5}_{+}\mid \unicode[STIX]{x1D6FE}\preccurlyeq \unicode[STIX]{x1D6FD}_{k}\}$ (i.e.  $S_{k}$ is a $\preccurlyeq$ -cuspidal module). For any sequence $\mathbf{a}=(\mathbf{a}_{i})_{1\leqslant i\leqslant l}\in \mathbb{Z}_{{\geqslant}0}^{l}$ , the convolution product

has a self-dual simple head. Moreover, every self-dual simple module in ${\mathcal{C}}_{w}$ is isomorphic to $\operatorname{hd}(P_{\mathbf{i}}(\mathbf{a}))$ for some $\mathbf{a}\in \mathbb{Z}_{{\geqslant}0}^{l}$ . Moreover, such an $\mathbf{a}$ is unique.

Hence, $\{[P_{\mathbf{i}}(\mathbf{a})]\}_{\mathbf{a}\in \mathbb{Z}_{{\geqslant}0}^{l}}$ is a $\mathbb{Z}[q^{\pm 1}]$ -basis of $K({\mathcal{C}}_{w})$ . It is called the dual Poincaré–Birkhoff–Witt (PBW) basis.

It is shown in [Reference Geiß, Leclerc and SchröerGLS13b] that the triple $(\{q^{-(d_{k},d_{k})/4}[M_{k}]\},-\unicode[STIX]{x1D6EC},\widetilde{B})$ is an initial quantum seed for the quantum cluster algebra $\mathbb{Z}[q^{\pm 1/2}]\otimes _{\mathbb{Z}[q^{\pm 1}]}A_{q}(\mathfrak{n}(w))_{\mathbb{Z}[q^{\pm 1}]}$ , where $d_{k}:=\operatorname{wt}(M_{k})=w_{{\leqslant}k}\unicode[STIX]{x1D71B}_{i_{k}}-\unicode[STIX]{x1D71B}_{i_{k}}$ , $\unicode[STIX]{x1D6EC}=(\unicode[STIX]{x1D6EC}(M_{i},M_{j}))_{1\leqslant i,j\leqslant l}$ , and $\widetilde{B}$ is the $l\times (l-n)$ matrix given in [Reference Kang, Kashiwara, Kim and OhKKKO18, Definition 11.1.1]. In this case, the index set of cluster variables are $J=\{1,2,\ldots ,l\}$ , $J_{\text{fr}}=\{k\in J\mid k_{+}=l+1\}$ , where $k_{+}:=\min (\{s\mid k<s\leqslant l,i_{k}=i_{s}\}\sqcup \{l+1\})$ , and $J_{\text{ex}}=J\setminus J_{\text{fr}}$ , $n=\sharp \{i_{1},\ldots ,i_{\ell }\}=\sharp J_{\text{fr}}$ . On the other hand, the family $\{[S_{k}]\mid k\in J\}$ forms a generating set of the algebra $A_{q}(\mathfrak{n}(w))_{\mathbb{Z}[q^{\pm 1}]}$ , and hence it is called the set of dual PBW generators.

We introduce the following notion in order to describe a property of the modules $S_{k}$ and $M_{k}$ .

By applying [Reference Kang, Kashiwara, Kim and OhKKKO18, Proposition 3.2.8] inductively, we obtain the following lemma.

In a similar way, we have the following lemma.

The following proposition is a consequence of Corollary 2.9.

The following proposition is a direct consequence of Proposition 2.11 and Lemma 2.8.

3.1 Localization and duals of simple objects

Let $w$ be an element of the Weyl group $W$ . In this subsection we recall the localization of the monoidal category ${\mathcal{C}}_{w}$ via a real commuting family of central objects following [Reference Kashiwara, Kim, Oh and ParkKKOP19]. Set $C_{i}:=M(w\unicode[STIX]{x1D71B}_{i},\unicode[STIX]{x1D71B}_{i})$ . Then $\{[C_{i}]\mid i\in I,C_{i}\not \simeq \mathbf{1}\}$ is the set of frozen variables of $A_{q}(\mathfrak{n}(w))$ up to a power of $q$ . For each $\unicode[STIX]{x1D707}=\sum \unicode[STIX]{x1D707}_{i}\unicode[STIX]{x1D71B}_{i}\in \mathsf{P}^{+}$ , we denote by $C_{\unicode[STIX]{x1D707}}$ the self-dual convolution product $q^{c}\,\circ _{i\in I}\,C_{i}^{\circ \unicode[STIX]{x1D707}_{i}}$ for some $c\in \mathbb{Z}$ . Then we have $C_{\unicode[STIX]{x1D707}}\simeq M(w\unicode[STIX]{x1D707},\unicode[STIX]{x1D707})$ . Note that

holds for any $\unicode[STIX]{x1D707},\unicode[STIX]{x1D707}^{\prime }\in \mathsf{P}^{+}$ . Here $\unicode[STIX]{x1D706}$ is the bilinear form on $\mathsf{P}$ defined by $\unicode[STIX]{x1D706}(\unicode[STIX]{x1D707},\unicode[STIX]{x1D707}^{\prime })=(\unicode[STIX]{x1D707},w\unicode[STIX]{x1D707}^{\prime }-\unicode[STIX]{x1D707}^{\prime })$ .

In [Reference Kashiwara, Kim, Oh and ParkKKOP19] it is proved that there exist a monoidal category $\widetilde{{\mathcal{C}}}_{w}={\mathcal{C}}_{w}[C_{i}^{\circ -1};i\in I]$ and a monoidal functor $\unicode[STIX]{x1D6F7}:{\mathcal{C}}_{w}\rightarrow \widetilde{{\mathcal{C}}}_{w}$ satisfying the following conditions.

1. (1) The objects $\unicode[STIX]{x1D6F7}(C_{i})$ are invertible in $\widetilde{{\mathcal{C}}}_{w}$ ; that is, there exists an object $\unicode[STIX]{x1D6F7}(C_{i})^{-1}$ in $\widetilde{{\mathcal{C}}}_{w}$ such that $\unicode[STIX]{x1D6F7}(C_{i})\circ \unicode[STIX]{x1D6F7}(C_{i})^{-1}\simeq \mathbf{1}$ and $\unicode[STIX]{x1D6F7}(C_{i})^{-1}\circ \unicode[STIX]{x1D6F7}(C_{i})\simeq \mathbf{1}$ .

2. (2) For any monoidal functor $\unicode[STIX]{x1D6F9}:{\mathcal{C}}_{w}\rightarrow {\mathcal{T}}$ to another monoidal category ${\mathcal{T}}$ in which $\unicode[STIX]{x1D6F9}(C_{i})$ is invertible for every $i$ , there exists a monoidal functor $\unicode[STIX]{x1D6F9}^{\prime }:\widetilde{{\mathcal{C}}}_{w}\rightarrow {\mathcal{T}}$ such that $\unicode[STIX]{x1D6F9}\simeq \unicode[STIX]{x1D6F9}^{\prime }\circ \unicode[STIX]{x1D6F7}$ . Moreover, $\unicode[STIX]{x1D6F9}^{\prime }$ is unique up to an isomorphism.

Then there exists a real commuting family $\{{\widetilde{C}_{\unicode[STIX]{x1D707}}\}}_{\unicode[STIX]{x1D707}\in \mathsf{P}}$ in $\widetilde{{\mathcal{C}}}_{w}$ such that $\widetilde{C}_{\unicode[STIX]{x1D707}}=\unicode[STIX]{x1D6F7}(C_{\unicode[STIX]{x1D707}})$ for every $\unicode[STIX]{x1D707}\in \mathsf{P}^{+}$ and $\widetilde{C}_{\unicode[STIX]{x1D707}}\circ \widetilde{C}_{\unicode[STIX]{x1D707}^{\prime }}\simeq q^{\unicode[STIX]{x1D706}(\unicode[STIX]{x1D707},\unicode[STIX]{x1D707}^{\prime })}\widetilde{C}_{\unicode[STIX]{x1D707}+\unicode[STIX]{x1D707}^{\prime }}$ for every $\unicode[STIX]{x1D707},\unicode[STIX]{x1D707}^{\prime }\in \mathsf{P}$ .

The localization $\widetilde{{\mathcal{C}}}_{w}$ also satisfies the following properties.

(3)

Every simple object of $\widetilde{{\mathcal{C}}}_{w}$ is isomorphic to $\unicode[STIX]{x1D6F7}(S)\circ \widetilde{C}_{\unicode[STIX]{x1D707}}$ for some simple object $S$ of ${\mathcal{C}}_{w}$ and $\unicode[STIX]{x1D707}\in \mathsf{P}$ .

(4)

For two simple objects $S$ and $S^{\prime }$ in ${\mathcal{C}}_{w}$ , $\unicode[STIX]{x1D6F7}(S)\circ \widetilde{C}_{\unicode[STIX]{x1D707}}\simeq \unicode[STIX]{x1D6F7}(S^{\prime })\circ \widetilde{C}_{\unicode[STIX]{x1D707}^{\prime }}$ in $\widetilde{{\mathcal{C}}}_{w}$ if and only if $S\circ C_{\unicode[STIX]{x1D707}+\unicode[STIX]{x1D706}}\simeq S^{\prime }\circ C_{\unicode[STIX]{x1D707}^{\prime }+\unicode[STIX]{x1D706}}$ in ${\mathcal{C}}_{w}$ for some $\unicode[STIX]{x1D706}\in \mathsf{P}$ such that $\unicode[STIX]{x1D707}+\unicode[STIX]{x1D706}$ , $\unicode[STIX]{x1D707}^{\prime }+\unicode[STIX]{x1D706}\in \mathsf{P}^{+}$ . Here we ignore grading shifts.

Note that the category $\widetilde{{\mathcal{C}}}_{w}$ is abelian and every object has finite length. Moreover, $\unicode[STIX]{x1D6F7}:{\mathcal{C}}_{w}\rightarrow \widetilde{{\mathcal{C}}}_{w}$ is exact. The grading shift functor $q$ and the contravariant functor $M\mapsto M^{\ast }$ on ${\mathcal{C}}_{w}$ are extended to $\widetilde{{\mathcal{C}}}_{w}$ . Moreover, they satisfy

for $M,N\in \widetilde{{\mathcal{C}}}_{w}$ . Here, for $M\simeq \unicode[STIX]{x1D6F7}(L)\circ \widetilde{C}_{\unicode[STIX]{x1D706}}\in \widetilde{{\mathcal{C}}}_{w}$ with $L\in {\mathcal{C}}_{w}$ and $\unicode[STIX]{x1D706}\in \mathsf{P}$ , we set $\operatorname{wt}(M)=\operatorname{wt}(L)+(w\unicode[STIX]{x1D706}-\unicode[STIX]{x1D706})$ .

For any simple object $M\in \widetilde{{\mathcal{C}}}_{w}$ , there exists a unique $n\in \mathbb{Z}$ such that $(q^{n}M)^{\ast }\simeq q^{n}M$ . We say that $q^{n}M$ is self-dual in this case.

We denote by ${\mathcal{I}}rr_{\text{sd}}(\widetilde{{\mathcal{C}}}_{w})$ the set of the isomorphism classes of self-dual simple objects of $\widetilde{{\mathcal{C}}}_{w}$ .

The Grothendieck ring $K(\widetilde{{\mathcal{C}}}_{w})$ of $\widetilde{{\mathcal{C}}}_{w}$ is a $\mathbb{Z}[q^{\pm 1}]$ -algebra with a basis consisting of the class of self-dual simple objects. Moreover, it is isomorphic to the localization of the ring $K({\mathcal{C}}_{w})$ by the set

We refer to [Reference Berenstein and ZelevinskyBZ05, Appendix A] for the definition of the localization of a ring $R$ by a multiplicative subset $S$ under the condition that $aR\cap Rb\neq \{0\}$ for any non-zero $a,b\in R$ and $sR=Rs$ for all $s\in S$ .

Hence the ring $\mathbb{Z}[q^{\pm 1/2}]\otimes _{\mathbb{Z}[q^{\pm 1}]}K(\widetilde{{\mathcal{C}}}_{w})$ , which is a localization of $\mathbb{Z}[q^{\pm 1/2}]\otimes _{\mathbb{Z}[q^{\pm 1}]}K({\mathcal{C}}_{w})$ , is the $\mathbb{Z}[q^{\pm 1/2}]$ -subalgebra of the skew field of fractions $K$ of an initial quantum torus $\mathbb{Z}[q^{\pm 1/2}][X_{i}^{\pm 1}\mid i\in J]$ generated by all the exchangeable cluster variables, the frozen variables, and the inverses of the frozen variables. Thus it is the quantum cluster algebra in the sense of [Reference Berenstein and ZelevinskyBZ05].

Recall that a pair $(\unicode[STIX]{x1D700}:X\otimes Y\rightarrow \mathbf{1},\unicode[STIX]{x1D702}:\mathbf{1}\rightarrow Y\otimes X)$ of morphisms in a monoidal category with a unit object $\mathbf{1}$ is called an adjunction if:

1. (a) is the identity of $X$ ; and

2. (b) is the identity of $Y$ .

In this case, the pair $(X,Y)$ is called a dual pair. When $(\unicode[STIX]{x1D700},\unicode[STIX]{x1D702})$ is an adjunction, we say that $X$ is a left dual of $Y$ and $Y$ is a right dual of $X$ . Note that a left dual (respectively, right dual) of an object is unique up to a unique isomorphism if it exists.

The following theorem plays an important role in this paper.

Note that any object of ${\mathcal{C}}_{w}$ with a non-zero weight cannot have a dual in ${\mathcal{C}}_{w}$ , because the sum of the weights of a dual pair is equal to zero.

We denote by $\mathscr{D}(M)$ the right dual of a simple object $M$ in $\widetilde{{\mathcal{C}}}_{w}$ and by $\mathscr{D}^{-1}(M)$ the left dual of $M$ .

3.2 Cluster variables and exchange matrices

Set

where $k_{+}:=\min (\{j\mid k<j\leqslant l,\;i_{j}=i_{k}\}\cup \{l+1\})$ . Let $\{M_{j}\mid j\in J\}$ be a monoidal cluster in ${\mathcal{C}}_{w}$ . Recall that $l=\sharp J$ and $n=\sharp J_{\text{fr}}=\sharp I^{\prime }$ . Taking a total order on $J$ such that $j<j^{\prime }$ for all $j\in J_{\text{ex}}$ and $j^{\prime }\in J_{\text{fr}}$ , we shall identify $J$ , $J_{\text{ex}}$ and $J_{\text{fr}}$ with $\{1,\ldots ,l\}$ , $\{1,\ldots ,l-n\}$ and $\{l-n+1,\ldots ,l\}$ , respectively.

Then the matrices $\unicode[STIX]{x1D6EC}=(\unicode[STIX]{x1D6EC}(M_{i},M_{j}))_{1\leqslant i,j\leqslant l}$ and the exchange matrix $\widetilde{B}$ satisfy the following relation (see [Reference Berenstein and ZelevinskyBZ05]):

Let $D=(d_{t,j})_{t\in I^{\prime },j\in J}$ be an $n\times l$ matrix defined by

so that $\sum _{t\in I^{\prime }}(D\mathbf{b})_{t}\unicode[STIX]{x1D6FC}_{t}=\operatorname{wt}(M(\mathbf{b}))$ for $\mathbf{b}\in \mathbb{Z}_{{\geqslant}0}^{J}$ . Then we have

Now let $\unicode[STIX]{x1D6EC}^{\prime }$ be the $(l-n)\times l$ matrix obtained by taking the first $l-n$ rows of $\unicode[STIX]{x1D6EC}$ . Then we have

3.3 Dominance order

Let $\mathscr{C}=\{M_{j}\mid j\in J\}$ be a monoidal cluster in ${\mathcal{C}}_{w}$ and $\mathscr{S}$ the monoidal seed determined by $\mathscr{C}$ . We define the order $\succcurlyeq _{\mathscr{S}}$ on $\mathbb{Z}_{{\geqslant}0}^{J}$ by

if and only if

1. (1) $\operatorname{wt}(M(\mathbf{b}))=\operatorname{wt}(M(\mathbf{b}^{\prime }))$ ,

2. (2) $\unicode[STIX]{x1D6EC}(M(\mathbf{b}),M_{k})\geqslant \unicode[STIX]{x1D6EC}(M(\mathbf{b}^{\prime }),M_{k})$ for all $k\in J_{\text{ex}}$ .

Note that condition (1) implies that $\unicode[STIX]{x1D6EC}(M(\mathbf{b}),M_{k})=\unicode[STIX]{x1D6EC}(M(\mathbf{b}^{\prime }),M_{k})$ for all $k\in J_{\text{fr}}$ , because $\unicode[STIX]{x1D6EC}(M(\mathbf{b}),M_{k})=(w\unicode[STIX]{x1D71B}_{i_{k}}+\unicode[STIX]{x1D71B}_{i_{k}},\operatorname{wt}(M(\mathbf{b})))$ depends only on $\operatorname{wt}(M(\mathbf{b}))$ (see [Reference Kashiwara, Kim, Oh and ParkKKOP18, Theorem 4.12]).

Note that $\mathbf{b}+\mathbf{a}\succcurlyeq _{\mathscr{S}}\mathbf{b}^{\prime }+\mathbf{a}$ for any $\mathbf{b},\mathbf{b}^{\prime },\mathbf{a}\in \mathbb{Z}_{{\geqslant}0}^{J}$ with $\mathbf{b}\succcurlyeq _{\mathscr{S}}\mathbf{b}^{\prime }$ . Hence we can extend the relation to the one on $\mathbb{Z}^{J}$ by

Recall that $\widetilde{\unicode[STIX]{x1D6EC}}(X,Y)=\frac{1}{2}(\unicode[STIX]{x1D6EC}(X,Y)+(\operatorname{wt}(X),\operatorname{wt}(Y)))\in \mathbb{Z}$ for $X,Y\in R\text{-}\text{gmod}$ . Hence $\mathbf{b}\succcurlyeq _{\mathscr{S}}\mathbf{b}^{\prime }$ implies that $\unicode[STIX]{x1D6EC}(M(\mathbf{b}),M_{k})-\unicode[STIX]{x1D6EC}(M(\mathbf{b}^{\prime }),M_{k})\in 2\mathbb{Z}_{{\geqslant}0}$ for all $k\in J_{\text{ex}}$ .

Thus we have

where $D$ and $\unicode[STIX]{x1D6EC}^{\prime }$ are the matrices given in § 3.2.

3.4 Heads and socles

Let $\mathscr{C}=\{M_{j}\mid j\in J\}$ be a monoidal cluster in ${\mathcal{C}}_{w}$ , $\mathscr{S}$ the monoidal seed determined by $\mathscr{C}$ . Then any simple module $X\in {\mathcal{C}}_{w}$ satisfies the following relation in the Grothendieck ring $K({\mathcal{C}}_{w})$ by Lemma 2.3:

for a family $\{\mathbf{b}(k)\}_{k\in K}$ in $\mathbb{Z}_{{\geqslant}0}^{J}$ , a family $\{{c_{k}\}}_{k\in K}$ of integers and $\mathbf{a}\in \mathbb{Z}_{{\geqslant}0}^{J}$ such that $\mathbf{a}_{j}=0$ for $j\in J_{\text{fr}}$ .

Recall that $X\circ M(\mathbf{a})$ has a simple head $X\,\unicode[STIX]{x1D6FB}\,M(\mathbf{a})$ and a simple socle $X\unicode[STIX]{x1D6E5}M(\mathbf{a})$ , since $M(\mathbf{a})$ is a real simple module.

Note that $\mathbf{g}_{\mathscr{S}}^{R}$ and $\mathbf{g}_{\mathscr{S}}^{L}$ are well defined by Lemma 3.7.

The following lemma is obvious.

These maps are closely related with the duality.

3.5 Tropical transformations

Let us fix a monoidal cluster $\mathscr{C}=\{M_{j}\mid j\in J\}$ in ${\mathcal{C}}_{w}$ and let $\mathscr{S}$ be the associated monoidal seed. For $i\in J_{\text{ex}}$ and $j\in J_{\text{fr}}$ , we set $b_{ij}:=-b_{ji}$ . Let us denote by $\{{\mathbf{e}_{j}\}}_{j\in J}$ the natural basis of $\mathbb{Z}^{J}$ .

Fix $k\in J_{\text{ex}}$ and denote by $\unicode[STIX]{x1D707}_{k}(\mathscr{S})=\{M_{j}^{\prime }\mid j\in J\}$ the mutation of $\mathscr{S}$ in the direction $k$ . Recall that $M_{j}^{\prime }=M_{j}$ for $j\not =k$ and there exists an exact sequence (ignoring the gradings)

where

Note that $U$ and $V$ are the kernel and the image of the $R$ -matrix $\mathbf{r}_{M_{k},M_{k}^{\prime }}$ , respectively.

Set $[a]_{+}:=\max \{a,0\}$ for $a\in \mathbb{Z}$ .

For $\mathbf{g}\in \mathbb{Z}^{J}$ , define

where

and

where

Setting $\unicode[STIX]{x1D707}_{k}(\mathscr{S}):=\mathscr{S}^{\prime }$ , the maps $\unicode[STIX]{x1D719}_{\mathscr{S}^{\prime },\mathscr{S}}^{R}$ and $\unicode[STIX]{x1D719}_{\mathscr{S},\mathscr{S}^{\prime }}^{R}$ (respectively, $\unicode[STIX]{x1D719}_{\mathscr{S}^{\prime },\mathscr{S}}^{L}$ and $\unicode[STIX]{x1D719}_{\mathscr{S},\mathscr{S}^{\prime }}^{L}$ ) are inverse to each other.

Now the following corollary is another consequence of Lemma 3.13.

3.6 Injective reachable seeds and common triangular basis

It is known (see, for example, [Reference QinQin17]) that, for every monoidal cluster $\mathscr{C}=\{M_{j}\mid j\in J\}$ of ${\mathcal{C}}_{w}$ , there exist a sequence of mutations $\unicode[STIX]{x1D6F4}$ and a permutation $\unicode[STIX]{x1D70E}$ on $J_{\text{ex}}$ such that

where $\mathscr{S}$ is the monoidal seed associated with $\mathscr{C}$ , $\{\mathbf{e}_{j}\in \mathbb{Z}^{J}\mid j\in J\}$ denotes the natural basis of $\mathbb{Z}^{J}$ , $\widetilde{B}=(b_{i,j})_{i\in J,j\in J_{\text{ex}}}$ and $\unicode[STIX]{x1D6F4}(\widetilde{B})=((\unicode[STIX]{x1D6F4}b)_{i,j})_{i\in J,j\in J_{\text{ex}}}$ are the exchange matrices of the monoidal seeds $\mathscr{S}$ and $\unicode[STIX]{x1D6F4}(\mathscr{S})$ , respectively, and $M_{\unicode[STIX]{x1D6F4}(\mathscr{C})}(\mathbf{a})$ denotes the self-dual monomials in the monoidal cluster $\unicode[STIX]{x1D6F4}(\mathscr{C})$ . See [Reference Geiß, Leclerc and SchröerGLS11, Proposition 13.4] for a precise description of $\unicode[STIX]{x1D6F4}$ . A quantum seed $[\mathscr{S}]$ satisfying the above condition is called injective-reachable in [Reference QinQin17]. By Corollary 3.18, for any $k\in J_{\text{ex}}$ , there exists $\mathbf{c}_{k}\in \mathbb{Z}^{J_{\text{fr}}}$ such that

Note that $M(\mathbf{c}_{k})$ is a central invertible object of $\widetilde{{\mathcal{C}}}_{w}$ .

In [Reference QinQin17], Qin provided the notion of common triangular bases for a quantum cluster algebra whose seeds are all injective-reachable. He studied the existence of common triangular basis for some cases including $A_{q}(\mathfrak{n}(w))$ .

Let $\mathscr{C}=\{M_{j}\mid j\in J\}$ be a monoidal cluster in ${\mathcal{C}}_{w}$ . Recall that $K({\mathcal{C}}_{w})[\mathscr{C}^{-1}]$ is the Laurent polynomial ring in $\{[M_{j}]\}_{j\in J}$ (see (2.4)). For $\mathbf{b}\in \mathbb{Z}^{J}$ , we denote by $[M(\mathbf{b})]$ the element in $K({\mathcal{C}}_{w})[\mathscr{C}^{-1}]$ given by

for any $\mathbf{a}\in \mathbb{Z}_{{\geqslant}0}^{J}$ such that $\mathbf{b}+\mathbf{a}\in \mathbb{Z}_{{\geqslant}0}^{J}$ . It does not depend on the choice of $\mathbf{a}$ . Note that we do not define $M(\mathbf{b})$ itself. It coincides with the ordinary $[M(\mathbf{b})]$ when $\mathbf{b}\in \mathbb{Z}_{{\geqslant}0}^{J}$ .

We set

We call $X^{\mathbf{b}}$ a quantum Laurent monomial in $[\mathscr{S}]$ . We have

where $\unicode[STIX]{x1D706}$ is a skew-symmetric bilinear form on $\mathbb{Z}^{J}$ such that $\unicode[STIX]{x1D706}(\mathbf{e}_{i},\mathbf{e}_{j})=-\unicode[STIX]{x1D6EC}(M_{i},M_{j})$ .

A $\mathbb{Z}[q^{\pm 1/2}]$ -basis ${\mathcal{B}}$ of $\mathbb{Z}[q^{\pm 1/2}]\otimes _{\mathbb{Z}[q^{\pm 1}]}K(\widetilde{{\mathcal{C}}}_{w})$ is a common triangular basis if it satisfies the following four conditions for every monoidal seed $\mathscr{S}$ (see [Reference QinQin17, § 6.1]).

1. (1) ${\mathcal{B}}$ contains the quantum cluster monomials in the quantum seed $[\mathscr{S}]$ .

2. (2) (Bar invariance) The expansion of $b\in {\mathcal{B}}$ as a $\mathbb{Z}[q^{\pm 1/2}]$ -linear combination of quantum Laurent monomials in $[\mathscr{S}]$ has bar-invariant coefficients.

3. (3) (Parameterization) The expansion of $b\in {\mathcal{B}}$ as a $\mathbb{Z}[q^{\pm 1/2}]$ -linear combination of the quantum Laurent monomials in $[\mathscr{S}]$ contains a unique non-zero term $c_{\mathbf{a}}X^{\mathbf{a}}$ such that all the other non-zero terms $c_{\mathbf{b}}X^{\mathbf{b}}$ satisfy $\mathbf{a}\succ _{\mathscr{S}}\mathbf{b}$ . Moreover, the map ${\mathcal{B}}\rightarrow \mathbb{Z}^{J}$ , given by $b\mapsto \mathbf{a}$ , is a bijection.

4. (4) (Triangularity) For any quantum cluster variable $X_{i}$ in the seed $[\mathscr{S}]$ and for any element $S$ in ${\mathcal{B}}$ , we can write

   (3.6) $$\begin{eqnarray}X_{i}S=\mathop{\sum }_{k=1}^{t}\unicode[STIX]{x1D700}_{k}q^{\ell _{k}}b_{k}\end{eqnarray}$$
   for some $\unicode[STIX]{x1D700}_{k}=\pm 1$ , $b_{k}\in {\mathcal{B}}$ and $\ell _{k}\in \mathbb{Z}/2$ ( $1\leqslant k\leqslant t$ ) such that $\unicode[STIX]{x1D700}_{1}=1$ , $\mathbf{g}_{\mathscr{S}}^{R}(b_{1})=\mathbf{g}_{\mathscr{S}}^{R}(X_{i})+\mathbf{g}_{\mathscr{S}}^{R}(S)$ , $\mathbf{g}_{\mathscr{S}}^{R}(b_{k})\prec _{\mathscr{S}}\mathbf{g}_{\mathscr{S}}^{R}(b_{1})$ and $\ell _{1}>\ell _{k}$ if $2\leqslant k\leqslant t$ .

The purpose of this subsection is to prove the following proposition. Recall that ${\mathcal{I}}rr_{\text{sd}}(\widetilde{{\mathcal{C}}}_{w})$ is the set of isomorphism classes of self-dual simple objects of $\widetilde{{\mathcal{C}}}_{w}$ .

Let us prove condition (2). For a self-dual simple module $S$ in ${\mathcal{C}}_{w}$ , write

where $\mathbf{a},\mathbf{b}(1),\ldots ,\mathbf{b}(r)\in \mathbb{Z}_{{\geqslant}0}^{J}$ such that $\mathbf{b}(k)\neq \mathbf{b}(k^{\prime })$ for $k\neq k^{\prime }$ , and $p_{k}(q)\in \mathbb{Z}[q^{\pm 1}]$ . Then we have

where $\unicode[STIX]{x1D707}_{\mathbf{c}}=\operatorname{wt}(M(\mathbf{c}))$ for $\mathbf{c}\in \mathbb{Z}_{{\geqslant}0}^{J}$ . It follows that

and hence

Thus condition (2) follows by Lemma 3.21.

The map in condition (3) is $\mathbf{g}_{\mathscr{S}}^{R}$ by Lemma 3.6, and hence the last assertion in condition (3) is nothing more than Corollary 3.17.

It remains to prove condition (4). Taking $-$ to (3.6), it is equivalent to showing that for any self-dual simple module $S$ in ${\mathcal{C}}_{w}$ and a cluster variable $M_{i}$ in the seed $\mathscr{S}$ , we have

where $S_{j}$ is a self-dual simple module in ${\mathcal{C}}_{w}$ and $\unicode[STIX]{x1D700}_{j}=\pm 1$ , $\ell _{j}\in \mathbb{Z}/2$ ( $1\leqslant j\leqslant t$ ) such that $\mathbf{g}_{\mathscr{S}}^{R}(S_{1})=\mathbf{g}_{\mathscr{S}}^{R}(M_{i})+\mathbf{g}_{\mathscr{S}}^{R}(S)$ , $\ell _{1}<\ell _{j}$ and $\mathbf{g}_{\mathscr{S}}^{R}(S_{k})\prec _{\mathscr{S}}\mathbf{g}_{\mathscr{S}}^{R}(S_{1})$ for $2\leqslant k\leqslant t$ .

Since $M_{i}$ is a real simple module, there exist self-dual simple modules $S_{j}$ and $\ell _{j}\in \mathbb{Z}$ such that

$S\,\unicode[STIX]{x1D6FB}\,M_{i}\simeq q^{\ell _{1}}S_{1}$ and $\ell _{1}<\ell _{j}$ , $\unicode[STIX]{x1D6EC}(S_{j},M_{i})<\unicode[STIX]{x1D6EC}(S,M_{i})$ for $2\leqslant j\leqslant t$  [Reference Kang, Kashiwara, Kim and OhKKKO18, Theorems 4.2.1, 4.1.1].

Hence it remains to prove $\mathbf{g}_{\mathscr{S}}^{R}(S_{j})\prec _{\mathscr{S}}\mathbf{g}_{\mathscr{S}}^{R}(S_{1})$ for $2\leqslant j\leqslant t$ .

Assume that

for some $\mathbf{a},\mathbf{b}(k)\in \mathbb{Z}_{{\geqslant}0}^{J}$ with $\mathbf{g}_{\mathscr{S}}^{R}(S)=\mathbf{b}(1)-\mathbf{a}$ (or $S\,\unicode[STIX]{x1D6FB}\,M(\mathbf{a})\simeq M(\mathbf{b}(1))$ ). Hence we have $\mathbf{b}(k)\prec _{\mathscr{S}}\mathbf{b}(1)$ for $2\leqslant k\leqslant r$ .

Now we have

Thus for each $1\leqslant j\leqslant t$ , $S_{j}\,\unicode[STIX]{x1D6FB}\,M(\mathbf{a})$ is isomorphic to $M(\mathbf{b}(k_{j})+\mathbf{e}_{i})$ for some $1\leqslant k_{j}\leqslant r$ up to a grading shift.

Note that $S\circ M_{i}\circ M(\mathbf{a})$ has a simple head $q^{\ell _{1}}S_{1}\,\unicode[STIX]{x1D6FB}\,M(\mathbf{a})$ and hence we have

Hence $k_{1}=1$ . Since $M(\mathbf{b}(1)+\mathbf{e}_{i})$ appears once in (3.8) by Lemma 3.6, we have $k_{j}\not =1$ for $j\not =1$ .

It follows that

Thus we obtain condition (4), and the proof of Proposition 3.19 is complete. ◻