Cluster algebra structures on module categories over quantum affine algebras

TL;DR

This paper explores the connection between cluster algebra structures and module categories over quantum affine algebras, revealing new monoidal categorifications and identifying modules corresponding to cluster monomials as real simple modules.

Contribution

It introduces new monoidal categorifications of certain subcategories of quantum affine algebra modules using quiver Hecke algebras, especially for types A and B.

Findings

Cluster algebra structures coincide with those from quiver Hecke algebra modules.

Modules corresponding to cluster monomials are identified as real simple modules.

The results unify and extend previous categorification frameworks for quantum affine algebras.

Abstract

We study monoidal categorifications of certain monoidal subcategories $\mathcal{C}_J$ of finite-dimensional modules over quantum affine algebras, whose cluster algebra structures coincide and arise from the category of finite-dimensional modules over quiver Hecke algebra of type A${}_\infty$. In particular, when the quantum affine algebra is of type A or B, the subcategory coincides with the monoidal category $\mathcal{C}_{\mathfrak{g}}^0$ introduced by Hernandez-Leclerc. As a consequence, the modules corresponding to cluster monomials are real simple modules over quantum affine algebras.

Tables

Table 1. Table 1. Dynkin diagrams and fundamental weights

Type	Dynkin diagram	${}^{}$ Fundamental weights
$A_1^{\left(1\right)}$		${\varpi }_1=\mathrm{cl}\left({\Lambda }_1-{\Lambda }_0\right)$
$A_n^{\left(1\right)}$ ($n\ge 2$)		${\varpi }_i=\mathrm{cl}\left({\Lambda }_i-{\Lambda }_0\right)$ $\left(1\le i\le n\right)$
$A_2^{\left(2\right)}$		${\varpi }_1=\mathrm{cl}\left(2{\Lambda }_1-{\Lambda }_0\right)$
$A_3^{\left(2\right)}$		${\varpi }_1=\mathrm{cl}\left({\Lambda }_1-{\Lambda }_0\right)$, ${\varpi }_2=\mathrm{cl}\left({\Lambda }_2-2{\Lambda }_0\right)$
$A_{2n-1}^{\left(2\right)}$ ($n\ge 3$)		${\varpi }_i=\mathrm{cl}\left({\Lambda }_i-{\Lambda }_0\right)$ $\left(i=1,n\right)$, ${\varpi }_i=\mathrm{cl}\left({\Lambda }_i-2{\Lambda }_0\right)$ $\left(2\le i\le n-1\right)$
$A_{2n}^{\left(2\right)}$ ($n\ge 2$)		${\varpi }_i=\mathrm{cl}\left({\Lambda }_i-{\Lambda }_0\right)$ $\left(i=1,\dots ,n-1\right)$, ${\varpi }_n=\mathrm{cl}\left(2{\Lambda }_n-{\Lambda }_0\right)$
$B_n^{\left(1\right)}$ ($n\ge 3$)		${\varpi }_1=\mathrm{cl}\left({\Lambda }_1-{\Lambda }_0\right)$, ${\varpi }_i=\mathrm{cl}\left({\Lambda }_i-2{\Lambda }_0\right)$ $\left(2\le i\le N-1\right)$, ${\varpi }_n=\mathrm{cl}\left({\Lambda }_n-{\Lambda }_0\right)$
$B_2^{\left(1\right)}$		${\varpi }_1=\mathrm{cl}\left({\Lambda }_1-{\Lambda }_0\right)$, ${\varpi }_2=\mathrm{cl}\left({\Lambda }_2-{\Lambda }_0\right)$

Equations

$${\mathrm{M}}_{\tilde{\text{w}}}(p,0{)}^{\prime }\simeq \{\begin{array}{ll}{\mathrm{W}}_{m,{\mathfrak{j}}_p+1}^{(\ell )}& \text{ if }\ell +m\equiv 0\text{mspace}3.0mu\mathrm{mod}\text{mspace}1.0mu2,\\ {\mathrm{W}}_{m,{\mathfrak{j}}_p-1}^{(\ell )}& \text{ if }\ell +m\equiv 1\text{mspace}3.0mu\mathrm{mod}\text{mspace}1.0mu2.\end{array}$$

$${\mu }_{{\Sigma }^{+}}({\mathrm{M}}_{\tilde{\text{w}}}(p,0))\simeq {\mathrm{W}}_{m,{\mathfrak{j}}_p+1}^{(\ell )}$$

$$\mathcal{A}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$P$$

$$({\alpha }_i,{\alpha }_j)={\mathsf{d}}_i{a}_{i,j}(i,j\in I)\text{ and }⟨h_i,\lambda ⟩=\frac{2({\alpha }_i,\lambda )}{({\alpha }_i,{\alpha }_i)}\text{for any λ∈h∗ and i∈I}.$$

$$\phi (e_i)=f_i,\phi (f_i)=e_i,\phi (q^h)=q^h.$$

$${\mathsf{P}}^{+}:=\{\mu \in \mathsf{P}|⟨h_i,\mu ⟩\ge 0\text{for all}i\in I\}.$$

$$(x_1\otimes x_2)\cdot (y_1\otimes y_2)=q^{-(\mathrm{wt}(x_2),\mathrm{wt}(y_1))}(x_1{y}_1\otimes x_2{y}_2),$$

$${\Delta }_{\mathfrak{n}}(e_i)=e_i\otimes 1+1\otimes e_i.$$

$$A_{q}({\mathfrak{n}})=\mathop{\mbox{\normalsize$\bigoplus$}}\limits_{\beta\in\mathsf{Q}^{-}}A_{q}({\mathfrak{n}})_{\beta}\qquad\text{ where $A_{q}({\mathfrak{n}})_{\beta}\mathbin{:=}\operatorname{Hom}_{\mathbb{Q}(q)}(U_{q}^{+}({\mathfrak{g}})_{-\beta},\mathbb{Q}(q))$}.$$

$$(\psi \cdot \theta )(x):=\theta (x_{(1)})\psi (x_{(2)})\text{ where }{\Delta }_{\mathfrak{n}}(x)=x_{(1)}\otimes x_{(2)}.$$

$${\mathbf{B}}^{\mathrm{up}}(A_q(\mathfrak{n})):=\{G^{\mathrm{up}}(b)|b\in B(A_q(\mathfrak{n}))\},$$

$$D(\mu ,\zeta )(x)=(x{u}_{\mu },u_{\zeta }{)}_{\Lambda }$$

$$\mathrm{D}(u\lambda ,v\lambda )\mathrm{D}(u\mu ,v\mu )=q^{-(v\lambda ,v\mu -u\mu )}\mathrm{D}(u(\lambda +\mu ),v(\lambda +\mu )).$$

$$\mathrm{D}(s^{\prime }s\lambda ,t^{\prime }\lambda )\mathrm{D}(s^{\prime }\mu ,t^{\prime }t\mu )=q^{(s^{\prime }s\lambda +t^{\prime }\lambda ,s^{\prime }\mu -t^{\prime }t\mu )}\mathrm{D}(s^{\prime }\mu ,t^{\prime }t\mu )\mathrm{D}(s^{\prime }s\lambda ,t^{\prime }\lambda ).$$

$$\text{w}=s_{i_1}{s}_{i_2}\dots s_{i_{\ell }}.$$

$$p_{+}$$

$$p_{-}$$

$$p^{+}(j)$$

$$p^{-}(j)$$

$${\mathrm{D}}_{\tilde{w}}(s,t):=\{\begin{array}{ll}\mathrm{D}({\lambda }_s,{\lambda }_t)& \text{ if }t>0\text{ and }{i}_s=i_t,\\ \mathrm{D}({\lambda }_s,{\Lambda }_{i_s})& \text{ if }0=t<s\le \ell ,\\ 1& \text{ otherwise}.\end{array}$$

$$(\text{updelta},\lambda )=⟨\mathsf{c},\lambda ⟩\text{for any }\lambda \in \mathsf{P},$$

$$\Delta (e_i)=e_i\otimes t_i^{-1}+1\otimes e_i,\Delta (f_i)=f_i\otimes 1+t_i\otimes f_i,\Delta (q^h)=q^h\otimes q^h.$$

$$e_i\mapsto e_i,f_i\mapsto f_i,t_i\mapsto t_i^{-1},q^{1/\gamma }\to q^{-1/\gamma }.$$

$$e_i(u_z)=z^{{\delta }_{i,0}}(e_{i}u{)}_z,f_i(u_z)=z^{-{\delta }_{i,0}}(f_{i}u{)}_z,t_i(u_z)=(t_{i}u{)}_z.$$

$${\varpi }_i:=\gcd ({\mathsf{c}}_0,{\mathsf{c}}_i{)}^{-1}\mathrm{cl}({\mathsf{c}}_0{\Lambda }_i-{\mathsf{c}}_i{\Lambda }_0)\in {\mathsf{P}}_{\mathrm{cl}}.$$

$$\overline{M_a}\simeq (\overline{M}{)}_{\overline{a}},\overline{M\otimes N}\simeq \overline{N}\otimes \overline{M}.$$

$${\mathrm{Hom}}_{U_q^{\prime }(\mathfrak{g})}(M\otimes X,Y)\simeq {\mathrm{Hom}}_{U_q^{\prime }(\mathfrak{g})}(X,{}^{\ast }M\otimes Y),{\mathrm{Hom}}_{U_q^{\prime }(\mathfrak{g})}(X\otimes {}^{\ast }M,Y)\simeq {\mathrm{Hom}}_{U_q^{\prime }(\mathfrak{g})}(X,Y\otimes M),$$

$${\mathrm{Hom}}_{U_q^{\prime }(\mathfrak{g})}(M^{\ast }\otimes X,Y)\simeq {\mathrm{Hom}}_{U_q^{\prime }(\mathfrak{g})}(X,M\otimes Y),{\mathrm{Hom}}_{U_q^{\prime }(\mathfrak{g})}(X\otimes M,Y)\simeq {\mathrm{Hom}}_{U_q^{\prime }(\mathfrak{g})}(X,Y\otimes M^{\ast }),$$

$$\bigl{(}V(\varpi_{i})_{x}\bigr{)}^{*}\simeq V(\varpi_{i^{*}})_{x(p^{*})^{-1}},\qquad{}^{*}\bigl{(}V(\varpi_{i})_{x}\bigr{)}\simeq V(\varpi_{i^{*}})_{xp^{*}}$$

$$M\simeq V_c$$

$$R_{M_{z_M},N_{z_N}}^{\mathrm{univ}}:\mathbf{k}[[z_N/z_M]]{\otimes }_{\mathbf{k}[z_N/z_M]}(M_{z_M}\otimes N_{z_N})\to \mathbf{k}[[z_N/z_M]]{\otimes }_{\mathbf{k}[z_N/z_M]}(N_{z_N}\otimes M_{z_M}).$$

Full text

The research of M. Kashiwara was supported by Grant-in-Aid for Scientific Research (B) 15H03608, Japan Society for the Promotion of Science.

The research of M. Kim was supported by the National Research Foundation of Korea(NRF) Grant funded by the Korea government(MSIP) (NRF-2017R1C1B2007824).

The research of S.-j. Oh was supported by the National Research Foundation of Korea(NRF) Grant funded by the Korea government(MSIP) (NRF-2016R1C1B2013135).

The research of E. P. was supported by the National Research Foundation of Korea(NRF) Grant funded by the Korea Government(MSIP)(NRF-2017R1A1A1A05001058).

1. 1 Quantum groups, quantum coordinate rings and quantum affine algebras
2. 2 Quantum cluster algebras and monoidal categorification
3. 3 Quiver Hecke algebras
4. 4 Cluster structure on R^{A_{\infty}}\mbox{-\mathrm{gmod}}
5. 5 The category $\mathcal{T}_{N}$ and its cluster structure
6. 6 Main result

Introduction

The quiver Hecke algebras (or Khovanov-Lauda-Rouquier algebras), introduced independently by Khovanov-Lauda ([37, 38]) and Rouquier ([47]), provide the categorification of a half of quantum group $U_{q}({\mathfrak{g}})$. Since then, quiver Hecke algebras have been studied actively and various new features were discovered in the viewpoint of categorification. Studying quantum groups via the quiver Hecke algebras has become one of the main research themes on quantum groups in aspect of categorification. In particular, the studies on representations of a quantum affine algebra $U_{q}^{\prime}({\mathfrak{g}})$ via the generalized quantum Schur-Weyl duality functors ([25]) and the quantum cluster algebra structures of quantum unipotent coordinate algebras $A_{q}({\mathfrak{n}}(w))$ via the monoidal categorifications ([29]) drew big attention of researchers on various areas.

The generalized quantum Schur-Weyl duality functor was developed in [25], which is a vast generalization of quantum affine Schur-Weyl duality. Let $U_{q}^{\prime}({\mathfrak{g}})$ be a quantum affine algebra of arbitrary type over a base field $\mathbf{k}$, and let $\{V_{j}\}_{j\in J}$ be a family of quasi-good $U_{q}^{\prime}({\mathfrak{g}})$-modules. The generalized quantum Schur-Weyl duality provides a procedure to make a symmetric quiver Hecke algebra $R^{J}$ from the R-matrices among $\{V_{j}\}_{j\in J}$ and to construct a monoidal functor \mathcal{F}\colon R^{J}\mbox{-\mathrm{gmod}}\to\mathscr{C}_{\mathfrak{g}} enjoying good properties. Here R^{J}\mbox{-\mathrm{gmod}} (resp. $\mathscr{C}_{\mathfrak{g}}$) denotes the monoidal category of graded $R^{J}$-modules (resp. integrable $U_{q}^{\prime}({\mathfrak{g}})$-modules) which are finite-dimensional over $\mathbf{k}$.

Let us recall briefly the results of [25]. Let $N\in\mathbb{\mspace{1.0mu}Z}_{>1}$ and $J=\mathbb{\mspace{1.0mu}Z}$, and consider the family $\{V(\varpi_{1})_{q^{2k}}\}_{k\in\mathbb{\mspace{1.0mu}Z}}$ of the quantum affine algebra $U_{q}^{\prime}({\mathfrak{g}})$ of type $A_{N-1}^{(1)}$. Then the corresponding quiver Hecke algebra $R^{J}$ is of type $A_{\infty}$ and the generalized quantum Schur-Weyl duality associated with $\{V(\varpi_{1})_{q^{2k}}\}_{k\in\mathbb{\mspace{1.0mu}Z}}$ gives a monoidal functor \mathcal{F}\colon R^{J}\mbox{-\mathrm{gmod}}\to\mathscr{C}_{J}\subset\mathscr{C}^{0}_{\mathfrak{g}}. Here $\mathscr{C}_{{\mathfrak{g}}}^{0}$ is the smallest Serre subcategory of $\mathscr{C}_{{\mathfrak{g}}}$ which is stable by taking tensor products and contains a sufficiently large family of fundamental representations, and $\mathscr{C}_{J}$ is a certain subcategory of $\mathscr{C}^{0}_{\mathfrak{g}}$ determined by the functor $\mathcal{F}$ (see Section 1.7 and 6.1). Let $\mathcal{S}_{N}$ be the smallest Serre subcategory of \mathcal{A}\mathbin{:=}R^{J}\mbox{-\mathrm{gmod}} such that

1. (i)

$\mathcal{S}_{N}$ contains $L[a,a+N]$ for any $a\in J$, 2. (ii)

X\mathop{\mathbin{\mbox{\large\circ}}}Y,\ Y\mathop{\mathbin{\mbox{\large\circ}}}X\in\mathcal{S}_{N} for all $X\in\mathcal{A}$ and $Y\in\mathcal{S}_{N}$,

where $L[a,a+N]$ is the 1-dimensional $R^{J}(\epsilon_{a}-\epsilon_{a+N+1})$-module (see Section 4.2 for the definition of $L[a,b]$). Then they constructed the monoidal category $\mathcal{T}_{N}$ with the modified convolution product $\star$ by localizing the quotient category $\mathcal{A}/\mathcal{S}_{N}$ at the commuting family coming from the objects $L_{a}=L[a,a+N-1]$’s. Moreover they showed that $\mathcal{T}_{N}$ is rigid, the functor $\mathcal{F}\colon\mathcal{A}\to\mathscr{C}_{J}$ factors through the canonical functor $\Upomega_{N}\colon\mathcal{A}\to\mathcal{T}_{N}$, and the resulting functor $\mathcal{T}_{N}\to\mathscr{C}_{J}$ induces an isomorphism between the Grothendieck rings of $\mathscr{C}_{J}$ and $\mathcal{T}_{N}|_{q=1}$. The category $\mathcal{T}_{N}$ also plays the same role in the studies on generalized quantum Schur-Weyl duality for $\mathscr{C}_{A_{N-1}^{(2)}}^{0}$ ([28]) and $\mathscr{C}_{B_{N/2}^{(1)}}^{0}$ (for $N\in 2\mathbb{\mspace{1.0mu}Z}$) ([32]).

The monoidal categorification of a quantum unipotent coordinate algebra $A_{q}({\mathfrak{n}}(w))$ using a certain monoidal subcategory $\mathcal{C}_{w}$ (see Definition 3.19) of R\mbox{-\mathrm{gmod}} for symmetric quiver Hecke algebras was provided in [29], which gives a monoidal categorical explanation on the quantum cluster algebra structure of $A_{q}({\mathfrak{n}}(w))$ given in Geiß–Leclerc–Schröer [13]. For each reduced expression ${\widetilde{w}}$ of a Weyl group element $w$, the initial quantum monoidal seed in $\mathcal{C}_{w}$ is given by the determinantial modules $\mathrm{M}_{\widetilde{w}}(s,0)$ which correspond to certain unipotent quantum minors of $A_{q}({\mathfrak{n}}(w))$. It was shown in [29] that every cluster monomials is a member of the upper global basis of $A_{q}({\mathfrak{n}}(w))$ by using the monomial categorification of $A_{q}({\mathfrak{n}}(w))$ arising from $\mathcal{C}_{w}$.

The cluster algebra structures also appear in certain monoidal subcategories of the category $\mathscr{C}_{\mathfrak{g}}$ of finite-dimensional integrable representations of a quantum affine algebra $U_{q}^{\prime}({\mathfrak{g}})$. The Grothendieck ring of the subcategory $\mathscr{C}_{\ell}$ ($\ell\in\mathbb{\mspace{1.0mu}Z}_{>0}$) of $\mathscr{C}_{\mathfrak{g}}$ introduced in Hernandez–Leclerc [16, Section 3.8] was studied by using cluster algebra structures ([16]), and an algorithm for calculating $q$-character of Kirillov-Reshetikhin modules for any untwisted quantum affine algebras was described in [17] by studying the cluster algebra $\mathscr{A}$ which is isomorphic to the Grothendieck ring of the subcategory $\mathscr{C}^{-}_{\mathfrak{g}}$ of $\mathscr{C}_{\mathfrak{g}}$ (see Section 2.3). It was conjectured in [17] that all cluster monomials of $\mathscr{A}$ correspond to the classes of certain simple objects of $\mathscr{C}^{-}_{\mathfrak{g}}$ (see Conjecture 2.7).

In a viewpoint of the categorification using quiver Hecke algebras, it is natural to study a cluster algebra structure on monoidal subcategories of quantum affine algebras using the monoidal categorification of quantum unipotent coordinate algebras via the generalized quantum Schur-Weyl duality. In fact, it is what we perform in this paper. We study a quantum cluster algebra structure of the category \mathcal{A}\mathbin{:=}R^{J}\mbox{-\mathrm{gmod}} of type $A_{\infty}$ and show that this quantum cluster algebra structure is compatible with the functor $\mathcal{A}\to\mathcal{T}_{N}$ for any $N\in\mathbb{\mspace{1.0mu}Z}_{>1}$ (see Theorem 5.24).

We further provide the condition $\eqref{eq: general assumption}$ on quasi-good modules of a quantum affine algebra $U_{q}^{\prime}({\mathfrak{g}})$ to make the generalized Schur-Weyl duality functor $\mathcal{F}\colon\mathcal{A}\to\mathscr{C}_{J}$ factor through the canonical functor $\Upomega_{N}\colon\mathcal{A}\to\mathcal{T}_{N}$ so that we have the exact monoidal functor $\widetilde{\mathcal{F}}\colon\mathcal{T}_{N}\to\mathscr{C}_{J}$ by using the framework given in [32] (see Theorem 6.7). In the cases of types $A_{N-1}^{(t)}$ ($t=1,2$), $B_{N/2}^{(1)}$ (for $N\in 2\mathbb{\mspace{1.0mu}Z}$), $C_{N-1}^{(1)}$ (for $N>3$), $D_{N}^{(t)}$ (for ($N\geq 4$ and $t=1,2$) or ($N=4$ and $t=3$)), we give explicit quasi-good modules satisfying the condition, which provide the cluster algebra structure on the Grothendieck ring $K(\mathscr{C}_{J})$ induced from $\mathcal{T}_{N}$. Moreover, all cluster monomials of $K(\mathscr{C}_{J})$ correspond to the classes of real simple modules in $\mathscr{C}_{J}$ (Theorem 6.10). Note that the families for the types $A$ and $B$ appeared in [25, 28], and [32], but the ones for types $C$ and $D$ are new. Since $\mathscr{C}_{J}=\mathscr{C}_{\mathfrak{g}}^{0}$ in types $A$ and $B$, there exists a cluster algebra structure on $K(\mathscr{C}_{\mathfrak{g}}^{0})$ coming from the quantum cluster algebra structure of $\mathcal{T}_{N}$. In particular, we prove that the conjecture given in [17] (see Theorem 6.15) is true when ${\mathfrak{g}}$ is of type $A_{N-1}^{(1)}$: all cluster monomials of $\mathscr{A}$ correspond to the classes of certain real simple objects of $\mathscr{C}_{A_{N-1}^{(1)}}^{-}$. Remark that for the case of the categories $\mathscr{C}_{\ell}$, the corresponding property was proved in [46].

Let us explain our results more precisely. Let $J=\mathbb{\mspace{1.0mu}Z}$ and let $A^{J}$ be the Cartan matrix of type $A_{\infty}$. Let $R^{J}$ be the symmetric quiver Hecke algebra of type $A_{\infty}$. We first choose a special infinite sequence ${\widetilde{\textbf{{w}}}}$ of simple reflections $s_{j}$ $(j\in J)$ of the Weyl group $W_{J}$ of type $A_{\infty}$ (see $\eqref{eq: def tw}$). It is shown in Proposition 4.1 that every $p$-prefix ${\widetilde{\textbf{{w}}}}_{\leq p}$ of ${\widetilde{\textbf{{w}}}}$ is a reduced expression in $W_{J}$ for each $p\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$ and, if $p=t(2t+1)$ for $t\in\frac{1}{2}\mathbb{\mspace{1.0mu}Z}_{\geq 1}$, then ${\widetilde{\textbf{{w}}}}_{\leq p}$ can be viewed as a reduced expression of the longest element of the parabolic subgroup of $W_{J}$ generated by $s_{j}$ for $j\in[-\lfloor t-\frac{1}{2}\rfloor,\lfloor t\rfloor]$ (see Remark 4.2). In this sense, the category R^{J}\mbox{-\mathrm{gmod}} can be understood as a limit of the subcategories $\mathcal{C}_{{\widetilde{\textbf{{w}}}}_{\leq p}}$. We then consider the determinantial modules $\mathrm{M}_{\widetilde{\textbf{{w}}}}(p,0)$ and the cuspidal modules $\mathrm{M}_{\widetilde{\textbf{{w}}}}(p^{+},p)$ associated with ${\widetilde{\textbf{{w}}}}$. Let $c\colon\mathbb{\mspace{1.0mu}Z}_{\geq 1}\rightarrow\mathbb{\mspace{1.0mu}Z}_{\geq 1}\times\mathbb{\mspace{1.0mu}Z}_{\geq 1}$ be the bijection given in Definition 4.3 and let $L[a,b]$ be the 1-dimensional $R$-module for $a,b\in\mathbb{\mspace{1.0mu}Z}$ defined in Section 4.2. For $p\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$ with $c(p)=(\ell,m)$, we prove that $\mathrm{M}_{\widetilde{\textbf{{w}}}}(p,0)$ is isomorphic to the head $\mathrm{W}^{(\ell)}_{m,\mathfrak{j}_{p}}$ of a certain convolution product of $m$-many $R$-modules $L[a,b]$ with the length $\ell=|b-a|+1$ (Proposition 4.10), and describe also the cuspidal module $\mathrm{M}_{\widetilde{\textbf{{w}}}}(p^{+},p)$ in terms of $L[a,b]$ (Corollary 4.13). We next describe the quiver Q associated with the initial seed given by ${\widetilde{\textbf{{w}}}}$, which can be viewed as the square product of the bipartite Dynkin quiver $Q_{A_{\infty}}$ (see $\eqref{eq: Q with W}$). Theorem 4.18 tells that the category R^{J}\mbox{-\mathrm{gmod}} is a monoidal categorification of the quantum cluster algebras $\mathscr{A}_{q^{1/2}}(\infty)$ with the quantum seed $[\mathscr{S}_{\infty}]$ arising from the determinantial modules $\mathrm{M}_{\widetilde{\textbf{{w}}}}(p,0)$ and the quiver Q.

We next consider the monoidal categorification structure of R^{J}\mbox{-\mathrm{gmod}} which we constructed. In Proposition 5.7, we describe the mutation $\mathrm{M}_{\widetilde{\textbf{{w}}}}(p,0)^{\prime}$ of $\mathrm{M}_{\widetilde{\textbf{{w}}}}(p,0)$ as follows:

[TABLE]

We then find sequences of mutations $\mu_{\Sigma^{+}}$ and $\mu_{\Sigma^{-}}$ such that $\mu_{\Sigma^{+}}(\textbf{{Q}})$ and $\mu_{\Sigma^{-}}(\textbf{{Q}})$ are the same as Q as quivers and

[TABLE]

for $p\in{K}$ with $c(p)=(\ell,m)$ (Proposition 5.9). In this viewpoint, since $\mathrm{M}_{\widetilde{\textbf{{w}}}}(p,0)$ is isomorphic to $\mathrm{W}^{(\ell)}_{m,\mathfrak{j}_{p}}$, applying $\mu_{\Sigma^{+}}$ (resp.  $\mu_{\Sigma^{-}}$) to Q is understood as shifting the indices $\mathfrak{j}_{p}$ of $\mathrm{W}^{(\ell)}_{m,\mathfrak{j}_{p}}$ at all vertices of Q by $1$ (resp. $-1$). For each $N\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$, we define the subquiver $\textbf{{Q}}_{N}$ of Q as in $\eqref{eq: the procedure}$ and investigate compatibility with $\mu_{\Sigma^{+}}$, $\mu_{\Sigma^{-}}$ and the procedure from $\mathcal{A}$ to $\mathcal{T}_{N}$. Then we can conclude that the category $\mathcal{T}_{N}$ is a monoidal categorification of the quantum cluster algebras $\mathscr{A}_{q^{1/2}}(N)$ with the quantum seed $[\mathscr{S}_{N}]$ arising from $\{\Upomega_{N}(\mathrm{M}_{{\widetilde{\textbf{{w}}}}}(p,0))\}_{c(p)\in{{K}}^{\mathrm{ex}}_{N}}$ and the quiver $\textbf{{Q}}_{N}$ (Theorem 5.24).

For the construction of generalized quantum Schur-Weyl duality functor $\mathcal{T}_{N}\to\mathscr{C}_{\mathfrak{g}}$, we provide the condition $\eqref{eq: general assumption}$ on quasi-good modules of the quantum affine algebra $U_{q}^{\prime}({\mathfrak{g}})$. Let $\{V_{a}\}_{a\in J}$ be a family of quasi-good modules in $\mathscr{C}_{\mathfrak{g}}$ satisfying the condition $\eqref{eq: general assumption}$. Note that, to define Schur-Weyl duality functor, we have to choose duality coefficients $P_{i,j}(u,v)$ ($i,j\in J$) which are elements in $\mathbf{k}[[u,v]]$ satisfying certain conditions determined by $\{V_{a}\}_{a\in J}$. In Lemma 6.1, we prove that any Schur-Weyl duality functor arising from $\{V_{a}\}_{a\in J}$ sends $L[a,b]$ to [math] for any $a,b\in\mathbb{\mspace{1.0mu}Z}$ with $b-a\geq N$, which implies that it factors through $\mathcal{A}\rightarrow\mathcal{A}/\mathcal{S}_{N}$. We prove that there exists a suitable duality coefficients $\{P_{i,j}(u,v)\}_{i,j\in J}$ such that the corresponding Schur-Weyl duality functor $\mathcal{F}$ factors through the canonical functor $\Upomega_{N}\colon\mathcal{A}\to\mathcal{T}_{N}$ by following the framework given in [32]. Theorem 6.7 gives an exact monoidal functor $\widetilde{\mathcal{F}}\colon\mathcal{T}_{N}\to\mathscr{C}_{J}$ such that the following diagram quasi-commutes$\colon$

[TABLE]

Then, the functor $\widetilde{\mathcal{F}}$ induces an isomorphism K(\mathcal{T}_{N})|_{q=1}\mathop{\xrightarrow[\raisebox{1.29167pt}[0.0pt][1.29167pt]{\scriptstyle{}}]{{\raisebox{-2.58334pt}[0.0pt][-2.58334pt]{\mspace{2.0mu}\sim\mspace{2.0mu}}}}}K(\mathscr{C}_{J}) and the category $\mathscr{C}_{J}$ gives a monoidal categorification of the cluster algebra $\mathscr{A}(N)\mathbin{:=}\mathscr{A}_{q^{1/2}}(N)|_{q=1}$ via the functor $\widetilde{\mathcal{F}}$. Therefore, every cluster monomial in $\mathscr{A}(N)$ corresponds to the isomorphism class of a real simple object in $\mathscr{C}_{J}$ (Theorem 6.10). For types $A_{N-1}^{(t)}$ ($t=1,2$), $B_{N/2}^{(1)}$ (for $N\in 2\mathbb{\mspace{1.0mu}Z}$), $C_{N-1}^{(1)}$ (for $N>3$), $D_{N}^{(t)}$ (for ($N\geq 4$ and $t=1,2$) or ($N=4$ and $t=3$)), we give explicit quasi-good modules satisfying the condition $\eqref{eq: general assumption}$ (see Section 6.2). For types $A$ and $B$, $\mathscr{C}_{J}$ is equal to $\mathscr{C}_{{\mathfrak{g}}}^{0}$, which means that there exists a cluster algebra structure on $K(\mathscr{C}_{\mathfrak{g}}^{0})$ coming from the quantum cluster algebra structure of $\mathcal{T}_{N}$. For the other types $C$ and $D$, $\mathscr{C}_{J}$ is a proper subcategory of $\mathscr{C}_{{\mathfrak{g}}}^{0}$. As for type $A_{N-1}^{(1)}$, we obtain an additional result using our monoidal categorification. In [17], for an untwisted quantum affine algebras, Hernandez and Leclerc studied the cluster algebra $\mathscr{A}$ with initial quiver $G^{-}$ of infinite rank, which is isomorphic to the Grothendieck ring of a half $\mathscr{C}^{-}_{\mathfrak{g}}$ of $\mathscr{C}^{0}_{\mathfrak{g}}$. It was conjectured in [17] that all cluster monomials of $\mathscr{A}$ correspond to the classes of certain simple objects of $\mathscr{C}^{-}_{\mathfrak{g}}$. When $A$ is of type $A_{N-1}^{(1)}$, we prove that this conjecture is true using our monoidal categorification $\mathscr{C}_{{\mathfrak{g}}}^{0}$ (see Theorem 6.15).

This paper is organized as follows. In Section 1, we review quantum groups and quantum affine algebras. In Section 2, we recall quantum cluster algebras and monoidal categorification. In Section 3, we review quiver Hecke algebras for monoidal categorification and generalized quantum Schur-Weyl duality. In Section 4, we study the monoidal categorification of R^{J}\mbox{-\mathrm{gmod}} associated with the infinite sequence ${\widetilde{\textbf{{w}}}}$. In Section 5, we investigate the cluster algebra structure of R^{J}\mbox{-\mathrm{gmod}} and prove that $\mathcal{T}_{N}$ has a monoidal categorification structure induced from R^{J}\mbox{-\mathrm{gmod}} via \Upomega_{N}\colon R^{J}\mbox{-\mathrm{gmod}}\to\mathcal{T}_{N}. In Section 6, we provide the condition on quasi-good $U_{q}^{\prime}({\mathfrak{g}})$-modules to make the generalized Schur-Weyl duality functor $\mathcal{F}\colon\mathcal{A}\to\mathscr{C}_{J}$ factor through the functor $\Upomega_{N}\colon\mathcal{A}\to\mathcal{T}_{N}$, and give an explicit quasi-good modules satisfying the condition for various types.

Acknowledgments

The second, third and fourth authors gratefully acknowledge for the hospitality of RIMS (Kyoto University) during their visits in 2018 and 2019.

1. Quantum groups, quantum coordinate rings and quantum affine algebras

In this section, we shortly recall the basic materials on quantum groups, quantum coordinate rings and quantum affine algebras. We refer to [1, 22, 23, 24, 29] for details.

For simplicity, we use the following convention:

[TABLE]

1.1. Quantum groups

Let $I$ be an index set. A Cartan datum is a quintuple $(\mathsf{A},\mathsf{P},\Pi,\mathsf{P}^{\vee},\Pi^{\vee})$ consisting of (i) a symmetrizable generalized Cartan matrix $\mathsf{A}=(a_{i,j})_{i,j\in I}$, (ii) a free abelian group $\mathsf{P}$, called the weight lattice, (iii) $\Pi=\{\alpha_{i}\in\mathsf{P}\ |\ i\in I\}$, called the set of simple roots, (iv) $\mathsf{P}^{\vee}\mathbin{:=}\operatorname{Hom}_{\mathbb{\mspace{1.0mu}Z}}(\mathsf{P},\mathbb{\mspace{1.0mu}Z})$, called the co-weight lattice, (v) $\Pi^{\vee}=\{h_{i}\ |\ i\in I\}\subset\mathsf{P}^{\vee}$, called the set of simple coroots, satisfying (1) $\langle h_{i},\alpha_{j}\rangle=a_{i,j}$ for all $i,j\in I$, (2) $\Pi$ is linearly independent, (3) for each $i\in I$ there exists a $\Lambda_{i}\in\mathsf{P}$ such that $\langle h_{j},\Lambda_{i}\rangle=\delta_{i,j}$ for all $j\in I$. We call $\Lambda_{i}$ the fundamental weights. Note that there exists a diagonal matrix $\mathsf{D}={\rm diag}(\mathsf{d}_{i}\ |\ i\in I)$ such that $\mathsf{d}_{i}\in\mathbb{\mspace{1.0mu}Z}_{>0}$ and $\mathsf{D}\mathsf{A}$ is symmetric.

We denote by $\mathsf{Q}\mathbin{:=}\bigoplus_{i\in I}\mathbb{\mspace{1.0mu}Z}\alpha_{i}$ the root lattice, $\mathsf{Q}^{+}\mathbin{:=}\bigoplus_{i\in I}\mathbb{\mspace{1.0mu}Z}_{\geq 0}\alpha_{i}$ the positive root lattice and $\mathsf{Q}^{-}\mathbin{:=}\bigoplus_{i\in I}\mathbb{\mspace{1.0mu}Z}_{\leq 0}\alpha_{i}$ the negative root lattice. For $\beta=\sum_{i\in I}m_{i}\alpha_{i}\in\mathsf{Q}^{+}$, we set $|\beta|=\sum_{i\in I}m_{i}$.

Set ${\mathfrak{h}}=\mathbb{Q}\mathop{\otimes}_{\mathbb{\mspace{1.0mu}Z}}\mathsf{P}^{\vee}$. Then there exists a symmetric bilinear form $(\ ,\ )$ on ${\mathfrak{h}}^{*}$ such that

[TABLE]

Let ${\mathfrak{g}}$ be the Kac-Moody algebra associated with a Cartan datum $(\mathsf{A},\mathsf{P},\Pi,\mathsf{P}^{\vee},\Pi^{\vee})$. The Weyl group $W$ of ${\mathfrak{g}}$ is the subgroup of ${\rm GL}({\mathfrak{h}}^{*})$ generated by the set of reflections $S=\{s_{i}\ |\ i\in I\}$, where $s_{i}(\lambda)\mathbin{:=}\lambda-\langle h_{i},\lambda\rangle\alpha_{i}$ for $\lambda\in{\mathfrak{h}}^{*}$.

Let $q$ be an indeterminate and set $q_{i}\mathbin{:=}q^{\mathsf{d}_{i}}$ for each $i\in I$. We denote by $U_{q}({\mathfrak{g}})$ the quantum group associated to ${\mathfrak{g}}$, which is a $\mathbb{Q}(q)$-algebra generated by $e_{i}$, $f_{i}$ $(i\in I)$ and $q^{h}$ $(h\in\mathsf{P}^{\vee})$. We set $U_{q}^{+}({\mathfrak{g}})$ (resp. $U_{q}^{-}({\mathfrak{g}})$) the subalgebra of $U_{q}({\mathfrak{g}})$ generated by $e_{i}$’s (resp. $f_{i}$’s).

Recall that $U_{q}({\mathfrak{g}})$ admits the weight space decomposition U_{q}({\mathfrak{g}})=\mathop{\mbox{\normalsize\bigoplus}}\limits_{\beta\in\mathsf{Q}}U_{q}({\mathfrak{g}})_{\beta}, where $U_{q}({\mathfrak{g}})_{\beta}\mathbin{:=}\{x\in U_{q}({\mathfrak{g}})\ |\ q^{h}xq^{-h}=q^{\langle h,\beta\rangle}\text{ for any }h\in\mathsf{P}^{\vee}\}$. For $x\in U_{q}({\mathfrak{g}})_{\beta}$, we set $\operatorname{wt}(x)=\beta$.

Set $\mathbf{A}=\mathbb{\mspace{1.0mu}Z}[q^{\pm 1}]$. Let us denote by $U_{\mathbf{A}}^{-}({\mathfrak{g}})$ the $\mathbf{A}$-subalgebra of $U_{q}({\mathfrak{g}})$ generated by $f_{i}^{(n)}\mathbin{:=}f_{i}^{n}/[n]_{i}!$, and by $U_{\mathbf{A}}^{+}({\mathfrak{g}})$ the $\mathbf{A}$-subalgebra of $U_{q}({\mathfrak{g}})$ generated by $e_{i}^{(n)}\mathbin{:=}e_{i}^{n}/[n]_{i}!$ ($i\in I$, $n\in\mathbb{\mspace{1.0mu}Z}_{\geq 0}$), where $[n]_{i}=\dfrac{q^{n}_{i}-q^{-n}_{i}}{q_{i}-q^{-1}_{i}}$ and $[n]_{i}!=\displaystyle\prod^{n}_{k=1}[k]_{i}$.

There is a $\mathbb{Q}(q)$-algebra anti-automorphism $\varphi$ of $U_{q}({\mathfrak{g}})$ given as follows$\colon$

[TABLE]

We say that a $U_{q}({\mathfrak{g}})$-module $M$ is called integrable if the actions of $e_{i}$ and $f_{i}$ on $M$ are locally nilpotent for all $i\in I$. We denote by $\mathcal{O}_{{\rm int}}({\mathfrak{g}})$ the category of integrable left $U_{q}({\mathfrak{g}})$-module $M$ satisfying

1. (i)

$M=\bigoplus_{\eta\in\mathsf{P}}M_{\eta}$ where $M_{\eta}=\{m\in M\ |\ q^{h}m=q^{\langle h,\eta\rangle}m\}$, $\dim M_{\eta}<\infty$, 2. (ii)

there exist finitely many weights $\lambda_{1},\ldots,\lambda_{m}$ such that $\operatorname{wt}(M)\subset\cup_{j}(\lambda_{j}+\mathsf{Q}^{-})$, where $\operatorname{wt}(M)=\{\eta\in\mathsf{P}\ |\ \dim M_{\eta}\neq 0\}$.

It is well-known that $\mathcal{O}_{{\rm int}}({\mathfrak{g}})$ is a semisimple category such that every simple object is isomorphic to an irreducible highest weight module $V(\Lambda)$ with the highest weight vector $u_{\Lambda}$ of highest weight $\Lambda$. Here $\Lambda$ is an element of the set $\mathsf{P}^{+}$ of dominant integral weights$\colon$

[TABLE]

1.2. Unipotent quantum coordinate rings and unipotent quantum minors

Note that $U_{q}^{+}({\mathfrak{g}})\mathop{\otimes}U_{q}^{+}({\mathfrak{g}})$ has an algebra structure defined by

[TABLE]

for homogeneous elements $x_{1}$, $x_{2}$, $y_{1}$ and $y_{2}$. Then we have the algebra homomorphism $\Delta_{\mathfrak{n}}\colon U_{q}^{+}({\mathfrak{g}})\to U_{q}^{+}({\mathfrak{g}})\mathop{\otimes}U_{q}^{+}({\mathfrak{g}})$ given by

[TABLE]

Definition 1.1**.**

We define the unipotent quantum coordinate ring $A_{q}({\mathfrak{n}})$ as follows$\colon$

[TABLE]

Note that $A_{q}({\mathfrak{n}})$ has a ring structure given as follows $\colon$

[TABLE]

The $\mathbf{A}$-form of $A_{q}({\mathfrak{n}})$ is defined by $A_{\mathbf{A}}({\mathfrak{n}})\mathbin{:=}\{\psi\in A_{q}({\mathfrak{n}})\ |\ \langle\psi,U_{\mathbf{A}}^{+}({\mathfrak{g}})\rangle\subset\mathbf{A}\}$.

Recall that the algebra $A_{q}({\mathfrak{n}})$ has the upper global basis ([23])

[TABLE]

where $B(A_{q}({\mathfrak{n}}))$ denotes the crystal of $A_{q}({\mathfrak{n}})$.

Let $(\ ,\ )_{\Lambda}$ be the non-degenerate symmetric bilinear form on $V(\Lambda)$ such that $(u_{\Lambda},u_{\Lambda})_{\Lambda}=1$ and $(xu,v)_{\Lambda}=(u,\varphi(x)v)_{\Lambda}$ for $u,v\in V(\Lambda)$ and $x\in U_{q}({\mathfrak{g}})$.

For $\mu,\zeta\in W\Lambda$, the unipotent quantum minor $D(\mu,\zeta)$ is an element in $A_{q}({\mathfrak{n}})$ given by

[TABLE]

for $x\in U_{q}^{+}({\mathfrak{g}})$, where $u_{\mu}$ and $u_{\zeta}$ are the extremal weight vectors in $V(\Lambda)$ of weight $\mu$ and $\zeta$, respectively.

Lemma 1.2** ([29, Lemma 9.1.1]).**

$\mathrm{D}(\eta,\zeta)$* is either contained in $\mathbf{B}^{\mathrm{up}}(A_{q}({\mathfrak{n}}))$ or zero.*

For $\eta,\zeta\in W\Lambda$, we write $\eta\preceq\zeta$ if there exists a sequence $\{\beta_{k}\}_{1\leq k\leq l}$ of positive real roots such that we have $(\beta_{k},\lambda_{k-1})\geq 0$, where $\lambda_{0}\mathbin{:=}\zeta$ and $\lambda_{k}\mathbin{:=}s_{\beta_{k}}\lambda_{k-1}$ for $1\leq k\leq l$. Note that $\eta\preceq\zeta$ implies $\eta-\zeta\in\mathsf{Q}^{-}$.

By [29, Lemma 9.1.4], $\mathrm{D}(\eta,\zeta)\neq 0$ if and only if $\eta\preceq\zeta$, for $\Lambda\in\mathsf{P}^{+}$ and $\eta,\zeta\in W\Lambda$. The behaviors of multiplications among unipotent quantum minors were investigated intensively (see [4, 13, 29]):

Proposition 1.3**.**

Let $\lambda,\mu\in\mathsf{P}^{+}$.

1. (i)

For $u,v\in W$ such that $u\geq v$, we have

[TABLE] 2. (ii)

For $s,t,s^{\prime},t^{\prime}\in W$ satisfying

• •

$\ell(s^{\prime}s)=\ell(s^{\prime})+\ell(s)$* and $\ell(t^{\prime}t)=\ell(t^{\prime})+\ell(t)$,*

• •

$s^{\prime}s\lambda\preceq t^{\prime}\lambda$* and $s^{\prime}\mu\preceq t^{\prime}t\mu$,*

we have

[TABLE]

1.3. The subalgebra $A_{q}({\mathfrak{n}}(w))$ of $A_{q}({\mathfrak{n}})$

In this subsection, we assume that the generalized Cartan matrix $\mathsf{A}$ is symmetric.

Let w be a sequence of the set $S\mathbin{:=}\{s_{i}\ |\ i\in I\}$ of reflections of $W$:

[TABLE]

We set

[TABLE]

For a given sequence w of $S$, $p\in\{1,\ldots,\ell\}$ and $j\in I$, we set

[TABLE]

For a reduced expression ${\widetilde{w}}=s_{i_{1}}s_{i_{2}}\cdots s_{i_{\ell}}$ of $w\in W$ and $0\leq t\leq s\leq\ell$, we set

[TABLE]

The $\mathbb{Q}(q)$-subalgebra of $A_{q}({\mathfrak{n}})$ generated by $\mathrm{D}_{\widetilde{w}}(i,i_{-})$ $(1\leq i\leq\ell)$, is independent of the choice of ${\widetilde{w}}$. We denote it by $A_{q}({\mathfrak{n}}(w))$. Then every $\mathrm{D}_{\widetilde{w}}(s,t)$ is contained in $A_{q}({\mathfrak{n}}(w))$ [13, Corollary 12.4]. The set $\mathbf{B}^{{\mathrm{up}}}(A_{q}({\mathfrak{n}}(w)))\mathbin{:=}\mathbf{B}^{{\mathrm{up}}}(A_{q}({\mathfrak{n}}))\cap A_{q}({\mathfrak{n}}(w))$ forms a $\mathbb{Q}(q)$-basis of $A_{q}({\mathfrak{n}}(w))$ [39, Theorem 4.25]. We call $\mathbf{B}^{{\mathrm{up}}}(A_{q}({\mathfrak{n}}(w)))$ the upper global basis of $A_{q}({\mathfrak{n}}(w))$.

The $\mathbf{A}$-module $A_{\mathbf{A}}({\mathfrak{n}}(w))$ generated by $\mathbf{B}^{{\mathrm{up}}}(A_{q}({\mathfrak{n}}(w)))$ is an $\mathbf{A}$-subalgebra of $A_{q}({\mathfrak{n}})$ ([39, Theorem 4.27]).

1.4. Quantum affine algebras

In this subsection, we briefly review the representation theory of finite-dimensional integrable modules over quantum affine algebras by following [1, 24].

When concerned with quantum affine algebras, we always take the algebraic closure of $\mathbb{C}(q)$ in $\bigcup_{m>0}\mathbb{C}((q^{1/m}))$ as the base field $\mathbf{k}$.

Let $I$ be an index set and let $\mathsf{A}=(a_{ij})_{i,j\in I}$ be a generalized Cartan matrix of affine type. We choose $0\in I$ as the leftmost vertices in the tables in [21, pages 54, 55] except $A^{(2)}_{2n}$-case where we take the longest simple root as $\alpha_{0}$. Set $I_{0}=I\setminus\{0\}$.

We normalize the $\mathbb{Q}$-valued symmetric bilinear form ({\,\raise 1.0pt\hbox{\scriptscriptstyle\bullet}\,},{\,\raise 1.0pt\hbox{\scriptscriptstyle\bullet}\,}) on $\mathsf{P}$ by

[TABLE]

where $\updelta$ denotes the null root and $\mathsf{c}=\sum_{i\in I}\mathsf{c}_{i}h_{i}$ denotes the center. We denote by $\gamma$ the smallest positive integer such that $\gamma\dfrac{(\alpha_{i},\alpha_{i})}{2}\in\mathbb{\mspace{1.0mu}Z}$ for all $i\in I$.

Let us denote by $U_{q}({\mathfrak{g}})$ the quantum group over $\mathbb{Q}(q^{1/\gamma})$ associated with the affine Cartan datum $(\mathsf{A},\mathsf{P},\Pi,\mathsf{P}^{\vee},\Pi^{\vee})$. We denote by $U_{q}^{\prime}({\mathfrak{g}})$ the subalgebra of $U_{q}(\mathfrak{g})$ generated by $e_{i},f_{i},t_{i}^{\pm 1}\mathbin{:=}q^{\pm\frac{(\alpha_{i},\alpha_{i})}{2}h_{i}}$ for $i\in I$ and call it the quantum affine algebra.

We use the comultiplication $\Delta$ of $U_{q}^{\prime}({\mathfrak{g}})$ given by

[TABLE]

Let us denote by \ \bar{}\ the involution of $U_{q}^{\prime}({\mathfrak{g}})$ defined as follows:

[TABLE]

We denote by $\mathscr{C}_{\mathfrak{g}}$ the category of finite-dimensional integrable $U_{q}^{\prime}({\mathfrak{g}})$-modules. For $M\in\mathscr{C}_{g}$, we denote by $M_{\rm{aff}}$ the affinization of $M$ which is $\mathbf{k}[z,z^{-1}]\mathop{\otimes}M$ as a vector space endowed with the $U_{q}^{\prime}({\mathfrak{g}})$-module structure given by

[TABLE]

Here $u_{z}$ denote the element ${\bf{1}}\mathop{\otimes}u\in M_{\rm{aff}}$ for $u\in M$. We also write $M_{z}$ instead of $M_{\rm{aff}}$.

A simple module $M$ in $\mathscr{C}_{\mathfrak{g}}$ contains a non-zero vector $u$ of weight $\lambda\in\mathsf{P}_{\rm{cl}}\mathbin{:=}\mathsf{P}/\mathbb{\mspace{1.0mu}Z}\updelta$ such that (1) $\langle h_{i},\lambda\rangle\geq 0$ for all $i\in I_{0}$, (2) all the weight of $M$ are contained in $\lambda-\sum_{i\in I_{0}}\mathbb{\mspace{1.0mu}Z}_{\geq 0}{\rm{cl}}(\alpha_{i})$, where ${\rm{cl}}\colon\mathsf{P}\to\mathsf{P}_{\rm{cl}}$ denotes the canonical projection. Such a $\lambda$ is unique and $u$ is unique up to a constant multiple. We call $\lambda$ the dominant extremal weight of $M$ and $u$ the dominant extremal weight vector of $M$.

For $x\in\mathbf{k}^{\times}$ and $M\in\mathscr{C}_{\mathfrak{g}}$, we define $M_{x}\mathbin{:=}M_{\rm{aff}}/(z_{M}-x)M_{\rm{aff}}$, where $z_{M}$ denotes the $U_{q}^{\prime}({\mathfrak{g}})$-module automorphism of $M_{\rm{aff}}$ of weight $\updelta$. We call $x$ the spectral parameter.

For each $i\in I_{0}$, we set

[TABLE]

Then there exists a unique simple $U_{q}^{\prime}({\mathfrak{g}})$-module $V(\varpi_{i})$ in $\mathscr{C}_{\mathfrak{g}}$, called the fundamental module of $($level [math]$)$ weight $\varpi_{i}$, satisfying the certain conditions (see [24, §5.2]).

For a $U_{q}^{\prime}({\mathfrak{g}})$-module $M$, we denote by the $U_{q}^{\prime}({\mathfrak{g}})$-module $\overline{M}=\{\bar{u}\mid u\in M\}$ whose module structure is given as $x\bar{u}\mathbin{:=}\overline{\mspace{2.0mu}\overline{x}u\mspace{2.0mu}}$ for $x\in U_{q}^{\prime}({\mathfrak{g}})$. Then we have

[TABLE]

In particular, $\overline{V(\varpi_{i})}\simeq V(\varpi_{i})$ (see [1, Appendix A]).

For a module $M$ in $\mathscr{C}_{\mathfrak{g}}$, let us denote the right and the left dual of $M$ by ${}^{*}M$ and $M^{*}$, respectively. That is, we have isomorphisms

[TABLE]

which are functorial in $U_{q}^{\prime}({\mathfrak{g}})$-modules $X$ and $Y$. In particular, the module $V(\varpi_{i})_{x}$ $(x\in\mathbf{k}^{\times})$ has the left dual and right dual as follows:

[TABLE]

where $p^{*}$ is an element in $\mathbf{k}^{\times}$ depending only on $U_{q}^{\prime}({\mathfrak{g}})$ (see [1, Appendix A]), and $i\mapsto i^{*}$ denotes the involution on $I_{0}$ given by $\alpha_{i}=-w_{0}\,\alpha_{i^{*}}$. Here $w_{0}$ is the longest element of $W_{0}=\langle s_{i}\ |\ i\in I_{0}\rangle\subset W$.

We say that a $U_{q}^{\prime}({\mathfrak{g}})$-module $M$ is good if it has a bar involution, a crystal basis with simple crystal graph, and a global basis (see [24] for the precise definition). For instance, every fundamental module $V(\varpi_{i})$ for $i\in I_{0}$ is a good module. Note that every good module is a simple $U_{q}^{\prime}({\mathfrak{g}})$-module. Moreover the tensor product of good modules is again good. Hence any good module $M$ is real simple, i.e., $M\mathop{\otimes}M$ is simple.

Definition 1.4**.**

We call a $U_{q}^{\prime}({\mathfrak{g}})$ module $M$ quasi-good if

[TABLE]

for some good module $V$ and $c\in\mathbf{k}^{\times}$.

1.5. R-matrices

In this subsection, we briefly review the notion of $R$-matrices for quantum affine algebras following [24, §8].

For $M,N\in\mathscr{C}_{\mathfrak{g}}$, there is a morphism of $\mathbf{k}[[z_{N}/z_{M}]]\mathop{\otimes}_{\mathbf{k}[z_{N}/z_{M}]}\mathbf{k}[z_{M}^{\pm 1},z_{N}^{\pm 1}]\mathop{\otimes}U_{q}^{\prime}({\mathfrak{g}})$-modules, denoted by $R^{\rm{univ}}_{M_{z_{M}},N_{z_{N}}}$ and called the universal $R$-matrix:

[TABLE]

We say that $R^{\rm{univ}}_{M_{z_{M}},N_{z_{N}}}$ is rationally renormalizable if there exist $a\in\mathbf{k}((z_{N}/z_{M}))$ and a $\mathbf{k}[z_{M}^{\pm 1},z_{N}^{\pm 1}]\mathop{\otimes}U_{q}^{\prime}({\mathfrak{g}})$-module homomorphism

[TABLE]

such that $R^{\rm{ren}}_{M_{z_{M}},N_{z_{N}}}=aR^{\rm{univ}}_{M_{z_{M}},N_{z_{N}}}$. Then we can choose $R^{\rm{ren}}_{M_{z_{M}},N_{z_{N}}}$ so that for any $a_{M},a_{N}\in\mathbf{k}^{\times}$, the specialization of $R^{\rm{ren}}_{M_{z_{M}},N_{z_{N}}}$ at $z_{M}=a_{M}$, $z_{N}=a_{N}$,

[TABLE]

does not vanish provided that $M$ and $N$ are non-zero $U_{q}^{\prime}({\mathfrak{g}})$-modules in $\mathscr{C}_{\mathfrak{g}}$. It is called a renormalized $R$-matrix.

We denote by

[TABLE]

and call it the $R$-matrix. By the definition {\mathbf{r}}_{\mspace{-2.0mu}\raisebox{-1.50694pt}{{\scriptstyle{M,N}}}} never vanishes.

For simple $U_{q}^{\prime}({\mathfrak{g}})$-modules $M$ and $N$ in $\mathscr{C}_{\mathfrak{g}}$, the universal $R$-matrix $R^{\rm{univ}}_{M_{z_{M}},N_{z_{N}}}$ is rationally renormalizable. Then, for dominant extremal weight vectors $u_{M}$ and $u_{N}$ of $M$ and $N$, there exists $a_{M,N}(z_{N}/z_{M})\in\mathbf{k}[[z_{N}/z_{M}]]^{\times}$ such that

[TABLE]

Then $R^{\rm{norm}}_{M_{z_{M}},N_{z_{N}}}\mathbin{:=}a_{M,N}(z_{N}/z_{M})^{-1}R^{\rm{univ}}_{M_{z_{M}},N_{z_{N}}}$ is a unique $\mathbf{k}(z_{M},z_{N})\mathop{\otimes}_{\mathbf{k}[z_{N}/z_{M}]}\mathbf{k}[z_{M}^{\pm 1},z_{N}^{\pm 1}]\mathop{\otimes}U_{q}^{\prime}({\mathfrak{g}})$-module homomorphism sending \big{(}(u_{M})_{z_{M}}\mathop{\otimes}(u_{N})_{z_{N}}\big{)} to \big{(}(u_{N})_{z_{N}}\mathop{\otimes}(u_{M})_{z_{M}}\big{)}.

It is known that $\mathbf{k}(z_{M},z_{N})\mathop{\otimes}_{\mathbf{k}[z_{M}^{\pm 1},z_{N}^{\pm 1}]}(M_{z_{M}}\mathop{\otimes}N_{z_{N}})$ is a simple $\mathbf{k}(z_{M},z_{N})\mathop{\otimes}_{\mathbf{k}[z_{M}^{\pm 1},z_{N}^{\pm 1}]}U_{q}^{\prime}({\mathfrak{g}})$-module ([24, Proposition 9.5]). We call $R^{\rm{norm}}_{M_{z_{M}},N_{z_{N}}}$ the normalized $R$-matrix.

Let us denote by $d_{M,N}(u)\in\mathbf{k}[u]$ a monic polynomial of the smallest degree such that the image of $d_{M,N}(z_{N}/z_{M})R^{\rm{norm}}_{M_{z_{M}},N_{z_{N}}}$ is contained in $N_{z_{N}}\mathop{\otimes}M_{z_{M}}$. We call $d_{M,N}$ the denominator of $R^{\rm{norm}}_{M_{z_{M}},N_{z_{N}}}$. Then,

[TABLE]

is a renormalized $R$-matrix, and the $R$-matrix {\mathbf{r}}_{\mspace{-2.0mu}\raisebox{-1.50694pt}{{\scriptstyle{M,N}}}}\colon M\mathop{\otimes}N\to N\mathop{\otimes}M is equal to d_{M,N}(z_{N}/z_{M})R^{\rm{norm}}_{M_{z_{M}},N_{z_{N}}}\big{|}_{z_{M}=1,z_{N}=1} up to a constant multiple.

1.6. Denominators of normalized $R$-matrices

The denominators of the normalized $R$-matrices $d_{k,l}(z)\mathbin{:=}d_{V(\varpi_{k}),V(\varpi_{l})}(z)$ between $V(\varpi_{k})$ and $V(\varpi_{l})$ were calculated in [1, 6, 26, 44] for classical affine types and in [45] for exceptional affine types (see also [10, 30, 34, 51]). In this subsection, we recall $d_{k,l}(z)$ for quantum affine algebras of type $A$ and $B$. In Table 1, we list the Dynkin diagrams with an enumeration of simple roots and the corresponding fundamental weights for types $A$ and $B$.

Remark 1.5**.**

1. (a)

Note that the convention for Dynkin diagram of type $A^{(2)}_{2n}$ is different from the one in [21, page 54, 55]. However, for each $i\in I_{0}$ the corresponding fundamental modules $V(\varpi_{i})$ are isomorphic to each other, since the corresponding fundamental weights are conjugate to each other under the Weyl group action (see [24, §5.2]). 2. (b)

Note that the Dynkin diagrams of type $B^{(1)}_{2}$ and $A^{(2)}_{3}$ in Table 1 are denoted by $C^{(1)}_{2}$ and $D^{(2)}_{3}$ in [21, page 54, 55], respectively. 3. (c)

Our conventions on quantum affine algebras are different from [17, 18]. To compare, we refer to [18, Remark 3.28].

Theorem 1.6** ([6, 44]).**

We have the following denominator formulas.

1. (a)

For ${\mathfrak{g}}=A^{(1)}_{n-1}$ $(n\geq 2)$, $1\leq k,l\leq n-1$, we have

[TABLE] 2. (b)

For ${\mathfrak{g}}=A^{(2)}_{n-1}$ $(n\geq 3)$, $1\leq k,l\leq\lfloor n/2\rfloor$, we have

[TABLE] 3. (c)

For ${\mathfrak{g}}=B^{(1)}_{n}$ $(n\geq 2)$ $1\leq k,l\leq n-1$, we have

[TABLE]

For $1\leq k\leq n$, we have

[TABLE]

where $q_{s}\mathbin{:=}q_{n}=q^{1/2}$.

1.7. Hernandez-Leclerc category

For each quantum affine algebra $U_{q}^{\prime}({\mathfrak{g}})$, we define a quiver $\upsigma({\mathfrak{g}})$ as follows:

1. (i)

Take the set of equivalence classes $\hat{I}_{{\mathfrak{g}}}\mathbin{:=}(I_{0}\times\mathbf{k}^{\times})/\sim$ as the set of vertices, where the equivalence relation is given by $(i,x)\sim(j,y)$ if and only if $V(\varpi_{i})_{x}\simeq V(\varpi_{j})_{y}$. 2. (ii)

Put $d$-many arrows from $(i,x)$ to $(j,y)$, where $d$ denotes the order of zero of $d_{i,j}(z_{j}/z_{i})$ at $z_{j}/z_{i}={y/x}$.

Note that $(i,x)$ and $(j,y)$ are connected by at least one arrow in $\upsigma({\mathfrak{g}})$ if and only if $V(\varpi_{i})_{x}\mathop{\otimes}V(\varpi_{j})_{y}$ is reducible ([1, Corollary 2.4]).

Let $\upsigma_{0}({\mathfrak{g}})$ be a connected component of $\upsigma({\mathfrak{g}})$. Note that a connected component of $\upsigma({\mathfrak{g}})$ is unique up to a spectral parameter shift and hence $\upsigma_{0}({\mathfrak{g}})$ is uniquely determined up to a quiver isomorphism. For types $A$ and $B$, one can take

[TABLE]

where $q_{\kappa}$ in (1.12) is defined as follows:

[TABLE]

We remark here that

1. (i)

$V(\varpi_{n})_{x}\simeq V(\varpi_{n})_{-x}$ in the $A^{(2)}_{2n-1}$-case, 2. (ii)

$\upsigma_{0}(A^{(1)}_{2n-1})\simeq\upsigma_{0}(A^{(2)}_{2n-1})$ as quivers ([28, (2.7)]).

Let us denote by $\mathscr{C}^{0}_{\mathfrak{g}}$ the smallest abelian subcategory of $\mathscr{C}_{\mathfrak{g}}$ such that

1. (a)

$\mathscr{C}^{0}_{\mathfrak{g}}$ contains $\{V(\varpi_{i})_{x}\ |\ (i,x)\in\upsigma_{0}({\mathfrak{g}})\}$, 2. (b)

it is stable under taking submodules, quotients, extensions and tensor products.

The category $\mathscr{C}^{0}_{\mathfrak{g}}$ for symmetric affine type $U_{q}^{\prime}({\mathfrak{g}})$ was introduced in [16]. Note that every simple module in $\mathscr{C}_{\mathfrak{g}}$ is a tensor product of certain parameter shifts of some simple modules in $\mathscr{C}^{0}_{\mathfrak{g}}$ [16, §3.7]. The Grothendieck ring $K(\mathscr{C}^{0}_{\mathfrak{g}})$ of $\mathscr{C}^{0}_{\mathfrak{g}}$ is the polynomial ring generated by the classes of modules in $\{V(\varpi_{i})_{x}\ |\ (i,x)\in\upsigma_{0}({\mathfrak{g}})\}$ [9].

2. Quantum cluster algebras and monoidal categorification

In this section, we recall the definition of quantum cluster algebras introduced in [5, 7]. Then we review the monoidal categorification of a quantum cluster algebra developed in [29] (see also [16]).

2.1. Quantum cluster algebras

Fix a countable index set ${K}={{K}}^{\mathrm{ex}}\sqcup{{K}}^{\mathrm{f\mspace{0.01mu}r}}$ which is decomposed into the subset ${{K}}^{\mathrm{ex}}$ of exchangeable indices and the subset ${{K}}^{\mathrm{f\mspace{0.01mu}r}}$ of frozen indices. Let $L=(\lambda_{ij})_{i,j\in{K}}$ be a skew-symmetric integer-valued ${K}\times{K}$-matrix.

Let $A$ be a ${\mathbb{\mspace{1.0mu}Z}[q^{\pm 1/2}]}$-algebra. We say that a family $\{x_{i}\}_{i\in{K}}$ of elements in $A$ is $L$-commuting if it satisfies

[TABLE]

We say that an $L$-commuting family $\{x_{i}\}_{i\in{K}}$ is algebraically independent if the family

[TABLE]

is linearly independent over $\mathbb{\mspace{1.0mu}Z}[q^{\pm 1/2}]$. Here $\leq$ is a total order on $K$.

Let $\widetilde{B}=(b_{ij})_{(i,j)\in{K}\times{{K}}^{\mathrm{ex}}}$ be an integer-valued matrix such that

[TABLE]

We extend the definition of $b_{ij}$ for $(i,j)\in K\times K$ by:

[TABLE]

To the matrix $\widetilde{B}$, we associate the quiver $Q_{\widetilde{B}}$ such that the set of vertices is ${K}$ and the number of arrows from $i\in{K}$ to $j\in{K}$ is $\max(0,b_{ij})$. Then, $Q_{\widetilde{B}}$ satisfies that

[TABLE]

Conversely, for a given quiver satisfying (2.2), we can associate a matrix $\widetilde{B}$ by

[TABLE]

Then $\widetilde{B}$ satisfies (2.1).

We say that the pair $(L,\widetilde{B})$ is compatible with a positive integer $d$, if

[TABLE]

Definition 2.1**.**

For a ${\mathbb{\mspace{1.0mu}Z}[q^{\pm 1/2}]}$-algebra $A$, a triple $\mathscr{S}=(\{x_{i}\}_{i\in{K}},L,\widetilde{B})$ consisting of

1. (i)

a compatible pair $(L,\widetilde{B})$, 2. (ii)

an $L$-commuting algebraically independent family $\{x_{i}\}_{i\in{K}}$

is called a quantum seed in $A$. We also call

1. (a)

$\{x_{i}\}_{i\in{K}}$ the cluster of $\mathscr{S}$ and elements in $\{x_{i}\}_{i\in{K}}$ the cluster variables, 2. (b)

$x_{i}$ $(i\in{{K}}^{\mathrm{ex}})$ the exchangeable variables and $x_{i}$ $(i\in{{K}}^{\mathrm{f\mspace{0.01mu}r}})$ the frozen variables, 3. (c)

$x^{{\bf a}}$ \bigl{(}{\bf a}\in\mathbb{\mspace{1.0mu}Z}_{\geq 0}^{\oplus{K}}\bigr{)} the quantum cluster monomials,

where

[TABLE]

for ${\bf a}=(a_{i})_{i\in{K}}$, $\left\{{i\in{K}}\mid{a_{i}\not=0}\right\}\subset\{i_{1},\ldots,i_{r}\}$. Note that $x^{{\bf a}}$ does not depend on the choice of $\{i_{1},\ldots,i_{r}\}$.

For $k\in{{K}}^{\mathrm{ex}}$, the mutation $\mu_{k}(L,\widetilde{B})\mathbin{:=}(\mu_{k}(L),\mu_{k}(\widetilde{B}))$ of a compatible pair $(L,\widetilde{B})$ in direction $k$ is a pair $(\mu_{k}(L),\mu_{k}(\widetilde{B}))$ consisting of a ${K}\times{K}$-matrix $\mu_{k}(L)$ and a ${K}\times{{K}}^{\mathrm{ex}}$-matrix $\mu_{k}(\widetilde{B})$ defined as follows$\colon$

[TABLE]

Then one can check that $\mu_{k}(\widetilde{B})$ satisfies (2.1) and the pair $(\mu_{k}(L),\mu_{k}(\widetilde{B}))$ is compatible with the same integer $d$ as in [5].

We define

[TABLE]

and set ${\bf a}^{\prime}\mathbin{:=}(a_{i}^{\prime})_{i\in{K}}$ and ${\bf a}^{\prime\prime}\mathbin{:=}(a_{i}^{\prime\prime})_{i\in{K}}$ which are contained in $\mathbb{\mspace{1.0mu}Z}^{\oplus{K}}$.

Let $A$ be a ${\mathbb{\mspace{1.0mu}Z}[q^{\pm 1/2}]}$-algebra contained in a skew-field $\mathscr{F}$. Let $\mathscr{S}=(\{x_{i}\}_{i\in{K}},L,\widetilde{B})$ be a quantum seed in $A$. For $k\in{{K}}^{\mathrm{ex}}$, we define the elements $\mu_{k}(x)_{i}$ of $K$ as follows$\colon$

[TABLE]

Then $\{\mu_{k}(x)_{i}\}_{i\in{K}}$ is a $\mu_{k}(L)$-commuting algebraically independent family. We call

[TABLE]

the mutation of $\mathscr{S}$ in direction $k$.

Definition 2.2**.**

Let $\mathscr{S}=(\{x_{i}\}_{i\in{K}},L,\widetilde{B})$ be a quantum seed in $A$. The quantum cluster algebra $\mathscr{A}_{q^{1/2}}(\mathscr{S})$ associated to the quantum seed $\mathscr{S}$ is the ${\mathbb{\mspace{1.0mu}Z}[q^{\pm 1/2}]}$-subalgebra of the skew field $\mathscr{F}$ generated by all the quantum cluster variables in the quantum seeds obtained from $\mathscr{S}$ by any sequence of mutations.

We call $\mathscr{S}$ the initial quantum seed of the quantum cluster algebra $\mathscr{A}_{q^{1/2}}(\mathscr{S})$.

2.2. Quantum cluster algebras $\mathscr{A}_{q^{1/2}}({\mathfrak{n}}(w))$

In this subsection, we assume that the generalized Cartan matrix $\mathsf{A}$ is symmetric. Let ${\widetilde{w}}=s_{i_{1}}\cdots s_{i_{r}}$ be a reduced expression of $w\in W$. By Proposition 1.3, $\mathrm{D}_{\widetilde{w}}(i,0)$ and $\mathrm{D}_{\widetilde{w}}(j,0)$ $q$-commute; i.e., there exists $\lambda_{ij}\in\mathbb{\mspace{1.0mu}Z}$ satisfying

[TABLE]

Hence we have an integer-valued skew-symmetric matrix $L=(\lambda_{ij})_{1\leq i,j\leq r}$.

Set

[TABLE]

Definition 2.3** ([13]).**

We define the quiver $Q$ with the set of vertices $Q_{0}$ and the set of arrows $Q_{1}$ which is associated to ${\widetilde{w}}$ as follows:

1. $(Q_{0})$

$Q_{0}={K}=\{1,\ldots,r\}$. 2. $(Q_{1})$

There are two types of arrows:

• •

ordinary arrows $\ \ \colon$ $s\xrightarrow{\,\ |a_{i_{s},i_{t}}|\ \,}t$       if $1\leq s<t<s_{+}<t_{+}\leq r+1$,

• •

horizontal arrows $\colon$ $s\xrightarrow{\,\hskip 27.12492pt\,}s_{-}$     if $1\leq s_{-}<s\leq r$.

Let $\widetilde{B}=(b_{ij})$ be the integer-valued ${K}\times{{K}}^{\mathrm{ex}}$-matrix associated to the quiver $Q$ by (2.3).

Proposition 2.4** ([13, Proposition 10.1]).**

The pair $(L,\widetilde{B})$ is compatible with $d=2$.

Theorem 2.5** ([13, Theorem 12.3], [29, Corollary 11.2.8]).**

Let $\mathscr{A}_{q^{1/2}}(\mathscr{S})$ be the quantum cluster algebra associated to the initial quantum seed

[TABLE]

where d_{s}\mathbin{:=}\operatorname{wt}\bigl{(}\mathrm{D}_{\widetilde{w}}(s,0)\bigr{)}. Then we have $\mathbb{\mspace{1.0mu}Z}[q^{\pm 1/2}]$-algebra isomorphism

[TABLE]

2.3. Cluster algebras $K(\mathscr{C}_{\mathfrak{g}}^{-})$

In [17], Hernandez and Leclerc introduced a proper subcategory $\mathscr{C}_{\mathfrak{g}}^{-}$ of $\mathscr{C}_{\mathfrak{g}}$ for untwisted quantum affine algebras which contains all simple objects of $\mathscr{C}_{\mathfrak{g}}$ up to parameter shifts. The definitions of $\mathscr{C}_{\mathfrak{g}}^{-}$ for types $A_{n}^{(1)}$ and $B_{n}^{(1)}$ can be taken as follows. Take $\upsigma_{-}({\mathfrak{g}})$ the subset of $I\times\mathbf{k}^{\times}$ as

[TABLE]

The category $\mathscr{C}^{-}_{\mathfrak{g}}$ is the smallest abelian full subcategory of $\mathscr{C}_{\mathfrak{g}}$ such that

1. (a)

$\mathscr{C}^{-}_{\mathfrak{g}}$ contains $\{V(\varpi_{i})_{x}\ |\ (i,x)\in\upsigma_{-}({\mathfrak{g}})\}$, 2. (b)

it is stable under taking submodules, quotients, extensions and tensor products.

Also, they defined the quiver $G_{\mathfrak{g}}^{-}$ of infinite rank, whose vertices are labeled by a certain subset $\{(i,x)\}$ of $I_{0}\times\mathbb{\mspace{1.0mu}Z}_{\leq 0}$ (see [17] for details). Let $\mathbf{z}=\{z_{i,x}\}$ be the indeterminates labeled by the $\{(i,x)\}$.

Theorem 2.6** ([17, Theorem 5.1]).**

There exists an isomorphism between the cluster algebra $\mathscr{A}$ associated with the initial seed $(\mathbf{z},G_{\mathfrak{g}}^{-})$ and the Grothendieck ring $K(\mathscr{C}^{-}_{\mathfrak{g}})$ of $\mathscr{C}^{-}_{\mathfrak{g}}$.

Conjecture 2.7** ([16, Conjecture 13.2], [17, Conjecture 5.2]).**

The cluster monomials of $\mathscr{A}$ can be identified with the real simple modules of $\mathscr{C}^{-}_{\mathfrak{g}}$ under the isomorphism in Theorem 2.6.

We will give a proof of Conjecture 2.7 for $\mathscr{C}^{-}_{A^{(1)}_{N-1}}$ in Section 6.2.1.

2.4. Monoidal categorification of quantum cluster algebras

In this subsection, we fix a base field $\mathbf{k}$ and a free abelian group $\mathsf{Q}$ equipped with a symmetric bilinear form $(\ ,\ )\colon\mathsf{Q}\times\mathsf{Q}$ such that $(\beta,\beta)\in 2\mathbb{\mspace{1.0mu}Z}\text{ for all }\beta\in\mathsf{Q}$.

Let $\mathcal{C}$ be a $\mathbf{k}$-linear abelian monoidal category (see [25, Appendix A.1]) in the sense that it is abelian and the tensor functor $\mathop{\otimes}$ is $\mathbf{k}$-bilinear and exact. A simple object $M$ in $\mathcal{C}$ is called real if $M\otimes M$ is simple.

We assume that $\mathcal{C}$ satisfies the following conditions$\colon$

1. (i)

Any object of $\mathcal{C}$ is of a finite length. 2. (ii)

\mathbf{k}\mathop{\xrightarrow[\raisebox{1.29167pt}[0.0pt][1.29167pt]{\scriptstyle{}}]{{\raisebox{-2.58334pt}[0.0pt][-2.58334pt]{\mspace{2.0mu}\sim\mspace{2.0mu}}}}}\operatorname{Hom}_{\mathcal{C}}(M,M) for any simple object $M$ of $\mathcal{C}$. 3. (iii)

$\mathcal{C}$ admits a direct sum decomposition \mathcal{C}=\mathop{\mbox{\normalsize\bigoplus}}\limits_{\beta\in\mathsf{Q}}\mathcal{C}_{\beta} such that the tensor functor $\mathop{\otimes}$ sends $\mathcal{C}_{\beta}\times\mathcal{C}_{\gamma}\to\mathcal{C}_{\beta+\gamma}$ for every $\beta,\gamma\in\mathsf{Q}$. 4. (iv)

There exists an object $Q\in\mathcal{C}_{0}$ satisfying

• (a)

there is an isomorphism R_{Q}(X)\colon Q\mathop{\otimes}X\mathop{\xrightarrow[\raisebox{1.29167pt}[0.0pt][1.29167pt]{\scriptstyle{}}]{{\raisebox{-2.58334pt}[0.0pt][-2.58334pt]{\mspace{2.0mu}\sim\mspace{2.0mu}}}}}X\mathop{\otimes}Q functorial in $X\in\mathcal{C}$ that

[TABLE]

commutes for any $X,Y\in\mathcal{C}$,

• (b)

the functor $X\mapsto Q\mathop{\otimes}X$ is an equivalence of categories. 5. (v)

For any $M$, $N\in\mathcal{C}$, we have $\operatorname{Hom}_{\mathcal{C}}(M,Q^{\mathop{\otimes}n}\mathop{\otimes}N)=0$ except finitely many integers $n$ (see Remark 2.8 below).

Remark 2.8**.**

Note that, since the functor $X\mapsto Q\mathop{\otimes}X$ is an equivalence of categories, there exists an object $Q^{-1}\in\mathcal{C}_{0}$ such that $Q\mathop{\otimes}Q^{-1}\simeq Q^{-1}\mathop{\otimes}Q\simeq{\bf{1}}$, and the functor $X\mapsto Q^{-1}\mathop{\otimes}X$ give an equivalence of categories also (see [25, Appendix A.1]). Thus, for each $n\in\mathbb{\mspace{1.0mu}Z}$, we define Q^{\mathop{\otimes}n}\mathbin{:=}\underbrace{Q\mathop{\otimes}Q\mathop{\otimes}\cdots\mathop{\otimes}Q}_{\text{n-times}} if $n\geq 0$, \underbrace{Q^{-1}\mathop{\otimes}Q^{-1}\mathop{\otimes}\cdots\mathop{\otimes}Q^{-1}}_{\text{-n-times}} if $n<0$.

We denote by $q$ the auto-equivalence Q\mathop{\otimes}{\,\raise 1.0pt\hbox{\scriptscriptstyle\bullet}\,}, and call it the grading shift functor. With the grading shift functor, the Grothendieck ring $K(\mathcal{C})$ becomes a $\mathsf{Q}$-graded $\mathbb{\mspace{1.0mu}Z}[q^{\pm 1}]$-algebra.

For $M\in\mathcal{C}_{\beta}$, we write $\beta=\operatorname{wt}(M)$ and call it the weight of $M$. Similarly, for $x\in{\mathbb{\mspace{1.0mu}Z}[q^{\pm 1/2}]}\otimes_{\mathbb{\mspace{1.0mu}Z}[q^{\pm 1}]}K(\mathcal{C}_{\beta})$, we write $\beta=\operatorname{wt}(x)$ and call it the weight of $x$.

Definition 2.9**.**

A pair $(\{M_{i}\}_{i\in{K}},\widetilde{B})$ consisting of

1. (1)

a family of real simple objects $\{M_{i}\}_{i\in{K}}$ of $\mathcal{C}$, where $M_{i}\in\mathcal{C}_{d_{i}}$ for some $d_{i}\in\mathsf{Q}$, 2. (2)

an integer valued ${K}\times{{K}}^{\mathrm{ex}}$-matrix $\widetilde{B}$

is called a quantum monoidal seed if it satisfies the following properties:

• (i)

for all $i,j\in{K}$, there exists an integer $\lambda_{ij}$ satisfying

[TABLE]

• (ii)

$M_{i_{1}}\mathop{\otimes}\cdots\mathop{\otimes}M_{i_{t}}$ is simple for any finite sequence $(i_{1},\ldots,i_{t})$ in ${K}$,

• (iii)

the integer valued ${K}\times{{K}}^{\mathrm{ex}}$-matrix $\widetilde{B}$ satisfies (2.1),

• (iv)

$(L,\widetilde{B})$ is compatible with $d=2$, where $L=(\lambda_{ij})_{i,j\in{K}}$,

• (v)

$\lambda_{ij}-(d_{i},d_{j})\in 2\mathbb{\mspace{1.0mu}Z}$ for all $i,j\in{K}$, where $D\mathbin{:=}=\{d_{i}\in\mathsf{Q}\ |\ M_{i}\in\mathcal{C}_{d_{i}}\}_{i\in{K}}\subset\mathsf{Q}$.

• (vi)

$\sum_{i\in{K}}b_{ik}d_{i}=0$ for all $k\in{{K}}^{\mathrm{ex}}$.

Note that for a quantum monoidal seed $(\{M_{i}\}_{i\in{K}},\widetilde{B})$, the matrix $L=(\lambda_{ij})_{i,j\in{K}}$ and the family $D\mathbin{:=}\{d_{i}\}_{i\in{K}}$ is determined by the family $\{M_{i}\}_{i\in{K}}$.

Let $\mathscr{S}=(\{M_{i}\}_{i\in{K}}),\widetilde{B})$ be a quantum monoidal seed in $\mathcal{C}$. For $X\in\mathcal{C}_{\beta}$ and $Y\in\mathcal{C}_{\gamma}$ such that $X\otimes Y\simeq q^{c}\,Y\otimes X$ and $c+(\beta,\gamma)\in 2\mathbb{\mspace{1.0mu}Z}$, we define

[TABLE]

Then we have X\mathop{\mbox{\normalsize\bigodot}}\limits Y\simeq Y\mathop{\mbox{\normalsize\bigodot}}\limits X.

For any finite sequence $(i_{1},\ldots,,i_{\ell})$ in ${K}$, we define

[TABLE]

When the $L$-commuting family $\{[M_{i}]\}_{i\in{K}}$ of elements in ${\mathbb{\mspace{1.0mu}Z}[q^{\pm 1/2}]}\mathop{\otimes}_{\mathbb{\mspace{1.0mu}Z}[q^{\pm 1}]}K(\mathcal{C})$ is algebraically independent, we define a quantum seed $[\mathscr{S}]$ in ${\mathbb{\mspace{1.0mu}Z}[q^{\pm 1/2}]}\mathop{\otimes}_{\mathbb{\mspace{1.0mu}Z}[q^{\pm 1}]}K(\mathcal{C})$ by

[TABLE]

For a given $k\in{{K}}^{\mathrm{ex}}$, we define the mutation $\mu_{k}(D)\in\mathsf{Q}$ of $D$ in direction $k$ with respect to $\widetilde{B}$ by

[TABLE]

Note that, for any $k\in{{K}}^{\mathrm{ex}}$, we have $\mu_{k}(\mu_{k}(D))=D$, and $(\mu_{k}(L),\mu_{k}(\widetilde{B}),\mu_{k}(D))$ satisfies (v) and (vi) in Definition 2.9.

For $k\in{{K}}^{\mathrm{ex}}$, set

[TABLE]

where $\zeta=-d_{k}+\sum_{b_{ik}>0}b_{ik}d_{i}$.

Definition 2.10**.**

We say that a quantum monoidal seed $\mathscr{S}$ in $\mathcal{C}$ admits a mutation in direction $k\in{{K}}^{\mathrm{ex}}$ if there exists a real simple object $M_{k}^{\prime}\in\mathcal{C}_{\mu_{k}(D)_{k}}$ such that

1. (i)

there exist exact sequences in $\mathcal{C}$

[TABLE]

where $m_{k}$ and $m^{\prime}_{k}$ are given in (2.6). 2. (ii)

\mu_{k}(\mathscr{S})\mathbin{:=}\bigl{(}\{M_{i}\}_{i\neq k}\sqcup\{M_{k}^{\prime}\},\mu_{k}(\widetilde{B})\bigr{)} is a quantum monoidal seed in $\mathcal{C}$.

We call $\mu_{k}(\mathscr{S})$ the mutation of $\mathscr{S}$ in direction $k$.

Definition 2.11**.**

Assume that a $\mathbf{k}$-linear abelian monoidal category $\mathcal{C}$ satisfies the conditions (i)–(v) in the beginning of this subsection. The category $\mathcal{C}$ is called a monoidal categorification of a quantum cluster algebra $\mathscr{A}$ over $\mathbb{\mspace{1.0mu}Z}[q^{\pm 1/2}]$ if

1. (i)

$\mathbb{\mspace{1.0mu}Z}[q^{\pm 1/2}]\mathop{\otimes}_{\mathbb{\mspace{1.0mu}Z}[q^{\pm 1}]}K(\mathcal{C})$ is isomorphic to $\mathscr{A}$, 2. (ii)

there exists a quantum monoidal seed $\mathscr{S}=(\{M_{i}\}_{i\in{K}},\widetilde{B})$ in $\mathcal{C}$ such that $[\mathscr{S}]\mathbin{:=}(\{q^{-(d_{i},d_{i})/4}[M_{i}]\}_{i\in{K}},L,\widetilde{B})$ is a quantum seed of $\mathscr{A}$, 3. (iii)

$\mathscr{S}$ admits successive mutations in all the directions.

3. Quiver Hecke algebras

3.1. Quiver Hecke algebras

Now we briefly recall the definition of quiver Hecke algebra associated to a symmetrizable Cartan datum $(\mathsf{A},\mathsf{P},\Pi,\mathsf{P}^{\vee},\Pi^{\vee})$ (see [37, 47] for more detail).

Let $\mathbf{k}$ be a base field. We take a family of polynomials $(Q_{i,j})_{i,j\in I}$ in $\mathbf{k}[u,v]$ satisfying

[TABLE]

where $t_{i,j;p,q}=t_{j,i;q,p}$ and $t_{i,j:-a_{ij},0}\in\mathbf{k}^{\times}$. Then one can check that $Q_{i,j}(u,v)=Q_{j,i}(v,u)$.

For $n\in\mathbb{\mspace{1.0mu}Z}_{\geq 0}$ and $\beta\in\mathsf{Q}^{+}$ such that $|\beta|=n$, we set

[TABLE]

We denote by $\mathfrak{S}_{n}=\langle s_{1},\ldots,s_{n-1}\rangle$ the symmetric group of degree $n$, where $s_{i}\mathbin{:=}(i,i+1)$ is the transposition of $i$ and $i+1$. Then $\mathfrak{S}_{n}$ acts on $I^{\beta}$ by place permutations.

Definition 3.1**.**

For $\beta\in\mathsf{Q}^{+}$ with $|\beta|=n$, the quiver Hecke algebra $R(\beta)$ at $\beta$ associated with a symmetrizable Cartan datum $(\mathsf{A},\mathsf{P},\Pi,\mathsf{P}^{\vee},\Pi^{\vee})$ and a matrix $(Q_{i,j})_{i,j\in I}$ is the $\mathbf{k}$-algebra generated by the elements $\{e(\nu)\}_{\nu\in I^{\beta}}$, $\{x_{k}\}_{1\leq k\leq n}$ and $\{\tau_{m}\}_{1\leq m\leq n-1}$ satisfying the following defining relations$\colon$

[TABLE]

The above relations are homogeneous with

[TABLE]

and $R(\beta)$ is endowed with a $\mathbb{\mspace{1.0mu}Z}$-graded algebra structure.

For a graded $R(\beta)$-module $M=\bigoplus_{k\in\mathbb{\mspace{1.0mu}Z}}M_{k}$, we define $qM=\bigoplus_{k\in\mathbb{\mspace{1.0mu}Z}}(qM)_{k}$, where

[TABLE]

We call $q$ the grading shift functor on the category of graded $R(\beta)$-modules.

For graded $R(\beta)$-modules $M$ and $N$, $\operatorname{Hom}_{R(\beta)}(M,N)$ denotes the space of degree preserving module homomorphisms. We set $\deg(f)\mathbin{:=}k$ for $f\in\operatorname{Hom}_{R(\beta)}(q^{k}M,N)$.

For an $R(\beta)$-module $M$, we set $\operatorname{wt}(M)\mathbin{:=}-\beta\in\mathsf{Q}^{-}$ and call it the weight of $M$.

Let us denote by R(\beta)\mbox{-\mathrm{gmod}} the category of graded $R(\beta)$-modules which are finite-dimensional over $\mathbf{k}$. We set

[TABLE]

For $\beta,\gamma\in\mathsf{Q}^{+}$ with $|\beta|=m$, $|\gamma|=n$, we define an idempotent $e(\beta,\gamma)$ as follows:

[TABLE]

Then we have an injective ring homomorphism

[TABLE]

For an $R(\beta)$-module $M$ and an $R(\gamma)$-module $N$,

• •

the convolution product M\mathop{\mathbin{\mbox{\large\circ}}}N is an $R(\beta+\gamma)$-module defined by

[TABLE]

• •

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

[TABLE]

where $\psi$ denotes the $\mathbf{k}$-algebra anti-involution on $R(\beta)$ fixing the generators.

We denote by $u\boxtimes v$ the image of $u\mathop{\otimes}v$ in M\mathop{\mathbin{\mbox{\large\circ}}}N.

A simple module $M$ in R\mbox{-\mathrm{gmod}} is called self-dual if $M^{*}\simeq M$. Every simple module is isomorphic to a grading shift of a self-dual simple module ([37, §3.2]).

Then R\mbox{-\mathrm{gmod}} has a monoidal category structure with \mathop{\mathbin{\mbox{\large\circ}}} as a tensor product. Let us denote by K(R\mbox{-\mathrm{gmod}}) the Grothendieck ring of R\mbox{-\mathrm{gmod}} which is an algebra over $\mathbf{A}=\mathbb{\mspace{1.0mu}Z}[q^{\pm 1}]$ with the multiplication induced by the convolution product and the $\mathbf{A}$-action induced by the grading shift functor $q$.

In [37, 38, 47], it is shown that a quiver Hecke algebra categorifies the corresponding unipotent quantum coordinate ring. More precisely, we have the following theorem.

Theorem 3.2** ([37, 38, 47]).**

For a given symmetrizable Cartan datum $(\mathsf{A},\mathsf{P},\Pi,\mathsf{P}^{\vee},\Pi^{\vee})$, we take a parameter matrix $(Q_{ij})_{i,j\in I}$ satisfying the conditions in (3.1), and let $A_{q}({\mathfrak{n}})$ and $R(\beta)$ be the associated unipotent quantum coordinate ring and quiver Hecke algebra, respectively. Then there exists an $\mathbf{A}$-algebra isomorphism

[TABLE]

Definition 3.3**.**

We say that a quiver Hecke algebra $R$ is symmetric if $Q_{i,j}(u,v)$ is a polynomial in $u-v$ for all $i,j\in I$.

In particular, the corresponding generalized Cartan matrix $\mathsf{A}$ is symmetric. In symmetric case, we assume $\mathsf{d}_{i}=1$ for all $i\in I$.

Theorem 3.4** ([48, 50]).**

Assume that the quiver Hecke algebra $R$ is symmetric and the base field $\mathbf{k}$ is of characteristic [math]. Then, under the isomorphism (3.2) in Theorem 3.2, the upper global basis $\mathbf{B}^{\mathrm{up}}(A_{q}({\mathfrak{n}}))$ corresponds to the set of the isomorphism classes of self-dual simple $R$-modules.

3.2. R-matrices

For $\beta,\gamma\in\mathsf{Q}^{+}$ with $|\beta|=m$ and $|\gamma|=n$, let $M$ be an $R(\beta)$-module and $N$ an $R(\gamma)$-module. Then, by [25, Lemma 1.5], there exists an $R(\beta+\gamma)$-module homomorphism (up to a grading shift)

[TABLE]

which is defined by intertwiners $\varphi_{k}\in R(\beta+\gamma)$ $(1\leq k\leq m+n-1)$ (see [25, §1.3.1]) and satisfies the Yang-Baxter equation (see [25, (1.9)]).

For the rest of this paper, we assume that quiver Hecke algebra is symmetric and $\mathsf{d}_{i}=1$ for all $i\in I$. We also work always in the category of graded $R$-modules.

Then each $R(\beta)$-module $M$ admits an affinization

[TABLE]

with the action of $R(\beta)$ twisted by the algebra homomorphism $\psi_{z}\colon R(\beta)\to\mathbf{k}[z]\mathop{\otimes}R(\beta)$ (see [25, §1.3.2]) sending $x_{k}$ to $x_{k}+z$, where $z$ is an indeterminate of homogeneous degree $2$.

By [25, Proposition 1.10], the $R(\beta+\gamma)$-module homomorphism (up to a grading shift)

[TABLE]

induces an $R(\beta+\gamma)$-module homomorphism

[TABLE]

which also satisfies the Yang-Baxter equation and is non-zero provided that $M$ and $N$ are non-zero. Here $\Lambda(M,N)$ denotes the degree of {\mathbf{r}}_{\mspace{-2.0mu}\raisebox{-1.50694pt}{{\scriptstyle{M,N}}}} and we call {\mathbf{r}}_{\mspace{-2.0mu}\raisebox{-1.50694pt}{{\scriptstyle{M,N}}}} the $R$-matrix.

Definition 3.5** ([29]).**

For non-zero $R$-modules $M$ and $N$ in R\mbox{-\mathrm{gmod}}, we define integers $\operatorname{\textfrak{d}}(M,N)$ and $\widetilde{\Lambda}(M,N)$ as follows$\colon$

1. (a)

$\operatorname{\textfrak{d}}(M,N)=\dfrac{1}{2}\left(\Lambda(M,N)+\Lambda(N,M)\right)\in\mathbb{\mspace{1.0mu}Z}_{\geq 0}$, 2. (b)

\widetilde{\Lambda}(M,N)=\dfrac{1}{2}\left(\Lambda(M,N)+\bigl{(}\operatorname{wt}(M),\operatorname{wt}(N)\bigr{)}\right)\in\mathbb{\mspace{1.0mu}Z}_{\geq 0}.

A simple module M\in R\mbox{-\mathrm{gmod}} is called real if M\mathop{\mathbin{\mbox{\large\circ}}}M is simple.

Lemma 3.6** ([27]).**

Let $M$ and $N$ be simple modules in R\mbox{-\mathrm{gmod}}, and assume that one of them is real. Then

1. (i)

M\mathop{\mathbin{\mbox{\large\circ}}}N* and N\mathop{\mathbin{\mbox{\large\circ}}}M have simple socles and simple heads.* 2. (ii)

\operatorname{Im}({\mathbf{r}}_{\mspace{-2.0mu}\raisebox{-1.50694pt}{{\scriptstyle{M,N}}}})* is equal to the head of M\mathop{\mathbin{\mbox{\large\circ}}}N and the socle of N\mathop{\mathbin{\mbox{\large\circ}}}M.*

For $R$-modules $M$ and $N$, we denote by M\mathbin{\scalebox{0.9}{\nabla}}N the head of M\mathop{\mathbin{\mbox{\large\circ}}}N and by M\mathbin{\scalebox{0.9}{\Delta}}N the socle of M\mathop{\mathbin{\mbox{\large\circ}}}N.

Proposition 3.7** ([27, Corollary 3.7]).**

For $\beta,\gamma\in\mathsf{Q}^{+}$, let M be a real simple $R(\beta)$-module. Then the map N\mapsto M\mathbin{\scalebox{0.9}{\nabla}}N is injective from the set of the isomorphism classes of simple objects $N$ of R(\gamma)\mbox{-\mathrm{gmod}} to the set of the isomorphism classes of simple objects of R(\beta+\gamma)\mbox{-\mathrm{gmod}}.

Lemma 3.8** ([29, Lemma 3.1.4]).**

Let $M$ and $N$ be self-dual simple modules. If one of them is real, then

[TABLE]

Thus $\widetilde{\Lambda}(M,N)$ indicates the degree shift that makes M\mathbin{\scalebox{0.9}{\nabla}}N self-dual.

Definition 3.9**.**

For non-zero $R$-modules $M_{1}$ and $M_{2}$, we set

[TABLE]

The non-negative integer $\operatorname{\textfrak{d}}(M,N)$ measures the degree of complexity of M\mathop{\mathbin{\mbox{\large\circ}}}N as seen in the following lemma.

Lemma 3.10** ([27, 29]).**

Let $M$ and $N$ be simple modules in R\mbox{-\mathrm{gmod}}, and assume that one of them is real.

1. (i)

M\mathop{\mathbin{\mbox{\large\circ}}}N* is simple if and only if $\operatorname{\textfrak{d}}(M,N)=0$.* 2. (ii)

If $\operatorname{\textfrak{d}}(M,N)=1$, then M\mathop{\mathbin{\mbox{\large\circ}}}N has length $2$, and there exists an exact sequence

[TABLE]

Definition 3.11**.**

For simple $R$-modules $M$ and $N$, we say that

1. (i)

$M$ and $N$ strongly commute if M\mathop{\mathbin{\mbox{\large\circ}}}N is simple, 2. (ii)

$M$ and $N$ are simply-linked if $\operatorname{\textfrak{d}}(M,N)=1$.

3.3. Determinantial modules

The set of self-dual simple $R$-modules corresponds to the upper global basis of $A_{q}({\mathfrak{n}})$ by Theorem 3.4.

Recall the $\mathbf{A}$-algebra isomorphism $\operatorname{ch}$ in (3.2).

Definition 3.12** **([29, § 9.1, § 10.2],

[33, Proposition 4.1]).

For $\Lambda\in\mathsf{P}^{+}$ and $\eta,\zeta\in W\Lambda$ such that $\eta\preceq\zeta$, let $\mathrm{M}(\eta,\zeta)$ be the self-dual simple $R(\zeta-\eta)$-module such that \operatorname{ch}\bigl{(}\mathrm{M}(\eta,\zeta)\bigr{)}=\mathrm{D}(\eta,\zeta).

By Proposition 1.3 (i), the module $\mathrm{M}(\eta,\zeta)$ is real. We call $\mathrm{M}(\eta,\zeta)$ the determinantial module. We also write $\mathrm{M}_{\widetilde{w}}(s,t)$ (see (1.3)) which is the determinantial module such that

[TABLE]

Theorem 3.13** **([29, Theorem 10.3.1],

[33, Proposition 4.6]).

For $\Lambda\in\mathsf{P}^{+}$ and $\eta_{1},\eta_{2},\eta_{3}\in W\Lambda$ with $\eta_{1}\preceq\eta_{2}\preceq\eta_{3}$, we have

[TABLE]

Now we recall the definition of several functors on $R$-modules.

Definition 3.14**.**

Let $\beta\in\mathsf{Q}^{+}$.

1. (i)

For $i\in I$ and $1\leq a\leq|\beta|$, set

[TABLE] 2. (ii)

For M\in R(\beta)\mbox{-\mathrm{gmod}},

[TABLE]

which are functors from R(\beta)\mbox{-\mathrm{gmod}} to R(\beta-\alpha_{i})\mbox{-\mathrm{gmod}}. 3. (iii)

Let $L(i)$ be the $1$-dimensional $R(\alpha_{i})$-module $R(\alpha_{i})/R(\alpha_{i})x_{1}$. For a simple $R$-module $M$, we set

[TABLE]

Here, for an $R$-module $N$, ${\operatorname{soc}}(N)$ denotes the socle of $N$ and ${\operatorname{hd}}(N)$ denotes the head of $N$. 4. (iv)

For $i\in I$ and $n\in\mathbb{\mspace{1.0mu}Z}_{\geq 0}$, we set L(i^{n})=q^{n(n-1)/2}L(i)^{\mathop{\mathbin{\mbox{\large\circ}}}n} which is a self-dual real simple $R(n\alpha_{i})$-module.

Proposition 3.15** ([29, Proposition 10.2.3]).**

Let $\Lambda\in\mathsf{P}^{+}$, $\eta,\zeta\in W\Lambda$ such that $\eta\preceq\zeta$ and $i\in I$.

1. (i)

If $n\mathbin{:=}\langle h_{i},\eta\rangle\geq 0$, then

[TABLE] 2. (ii)

If $\langle h_{i},\eta\rangle\leq 0$ and $s_{i}\eta\preceq\zeta$, then we have $\varepsilon_{i}(\mathrm{M}(\eta,\zeta))=-\langle h_{i},\eta\rangle$ and $\mathrm{M}(s_{i}\eta,\zeta)\simeq\widetilde{E}_{i}^{-\langle{h_{i},\eta}\rangle}\mathrm{M}(\eta,\zeta)$. 3. (iii)

If $m\mathbin{:=}-\langle h_{i},\zeta\rangle\geq 0$, then

[TABLE] 4. (iv)

If $\langle h_{i},\zeta\rangle\geq 0$ and $\eta\preceq s_{i}\zeta$, then $\varepsilon^{*}_{i}(\mathrm{M}(\eta,\zeta))=\langle h_{i},\zeta\rangle$ and $\mathrm{M}(\eta,s_{i}\zeta)\simeq\widetilde{E}_{i}^{\langle{h_{i},\zeta}\rangle}\mathrm{M}(\eta,\zeta)$.

3.4. Admissible pair and $\mathscr{A}_{q^{1/2}}({\mathfrak{n}}(w))$

In this subsection, we review the results in [29] on the monoidal categorification by using graded modules over quiver Hecke algebras. Throughout this subsection, we focus on a category $\mathcal{C}$ which is a full subcategory of R\mbox{-\mathrm{gmod}} which is stable under taking convolution products, subquotients, extensions, and grading shift. Then we have

[TABLE]

Definition 3.16** (cf. Definition 2.9).**

A pair $(\{M_{i}\}_{i\in{K}},\widetilde{B})$ consisting of

1. (i)

a family of self-dual real simple objects $\{M_{i}\}_{i\in{K}}$ of $\mathcal{C}$ strongly commuting with each other, 2. (ii)

an integer valued ${K}\times{{K}}^{\mathrm{ex}}$-matrix $\widetilde{B}$ satisfying (2.1),

is called admissible if, for each $k\in{{K}}^{\mathrm{ex}}$, there exists an object $M^{\prime}_{k}$ of $\mathcal{C}$ such that

1. (a)

there is an exact sequence in $\mathcal{C}$

[TABLE] 2. (b)

$M_{k}^{\prime}$ is self-dual simple and strongly commutes with $M_{i}$ for any $i\neq k$.

For an admissible pair $(\{M_{i}\}_{i\in{K}},\widetilde{B})$, we can take

• •

$-\Lambda=(-\Lambda_{i,j})_{i,j\in{K}}$ the skew-symmetric integer-valued matrix by

[TABLE]

• •

$D=\{d_{i}\}$ the family of elements in $\mathsf{Q}^{-}$ by $d_{i}=\operatorname{wt}(M_{i})$,

as in Section 2.4.

Proposition 3.17** ([29, Proposition 7.1.2]).**

For an admissible pair $(\{M_{i}\}_{i\in{K}},\widetilde{B})$, we have the following properties$\colon$

1. (i)

$\mathscr{S}\mathbin{:=}(\{M_{i}\}_{i\in{K}},\widetilde{B})$* is a quantum monoidal seed in $\mathcal{C}$.* 2. (ii)

The self-dual simple object $M_{k}^{\prime}$ is real for every $k\in{{K}}^{\mathrm{ex}}$. 3. (iii)

The quantum monoidal seed $\mathscr{S}$ admits a mutation in each direction $k\in{{K}}^{\mathrm{ex}}$. 4. (iv)

$M_{k}$* and $M^{\prime}_{k}$ is simply-linked for any $k\in{{K}}^{\mathrm{ex}}$ *$($i.e., $\operatorname{\textfrak{d}}(M_{k},M^{\prime}_{k})=1)$.

For an admissible pair $(\{M_{i}\}_{i\in{K}},\widetilde{B})$, let us denote by

[TABLE]

Theorem 3.18** ([29, Theorem 7.1.3, Corollary 7.1.4]).**

For an admissible pair $(\{M_{i}\}_{i\in{K}},\widetilde{B})$, we assume that

[TABLE]

Then $\mathcal{C}$ is a monoidal categorification of the quantum cluster algebra $\mathscr{A}_{q^{1/2}}([\mathscr{S}])$.

In particular, the following statements holds$\colon$

1. (i)

Each cluster monomial in $\mathscr{A}_{q^{1/2}}([\mathscr{S}])$ corresponds to the isomorphism class of a real simple object in $\mathcal{C}$ up to a power of $q^{1/2}$. 2. (ii)

Each cluster monomial in $\mathscr{A}_{q^{1/2}}([\mathscr{S}])$ is a Laurent polynomial of the initial cluster variables with coefficient in $\mathbb{\mspace{1.0mu}Z}_{\geq 0}[q^{\pm 1/2}]$.

Definition 3.19**.**

For $w\in W$, let $\mathcal{C}_{w}$ be the smallest full subcategory of R\mbox{-\mathrm{gmod}} satisfying the following properties:

1. (i)

$\mathcal{C}_{w}$ is stable by convolution, taking subquotients, extensions, and grading shifts, 2. (ii)

$\mathcal{C}_{w}$ contains $\mathrm{M}_{\widetilde{w}}(s,s_{-})$ $(1\leq s\leq\ell(w))$, where ${\widetilde{w}}$ is a reduced expression of $w$.

By [13], we have an $\mathbf{A}$-algebra isomorphism

[TABLE]

Let $\widetilde{B}$ be the integer-valued ${K}\times{{K}}^{\mathrm{ex}}$-matrix associated to ${\widetilde{w}}$ (see Definition 2.3).

Theorem 3.20** ([29, Theorem 11.2.2, Theorem 11.2.3]).**

The pair $(\{\mathrm{M}_{\widetilde{w}}(s,0)\}_{1\leq s\leq r},\widetilde{B})$ is admissible. Thus $\mathcal{C}_{w}$ is a monoidal categorification of the quantum cluster algebra $\mathscr{A}_{q^{1/2}}({\mathfrak{n}}(w))$, with the quantum monoidal seed $(\{\mathrm{M}_{\widetilde{w}}(s,0)\}_{1\leq s\leq r},\widetilde{B})$.

Furthermore, the self-dual real simple $R$-module $\mathrm{M}_{\widetilde{w}}(k,0)^{\prime}$ for $k\in{{K}}^{\mathrm{ex}}$ in (3.5) is given as follows$\colon$ $($up to a grade shift$)$

[TABLE]

3.5. Generalized quantum affine Schur-Weyl duality functor

In this subsection, we recall the generalized quantum affine Schur-Weyl duality functor in [25].

Let us assume that we are given an index set $J$ and a family $\{V_{j}\}_{j\in J}$ of quasi-good $U_{q}^{\prime}({\mathfrak{g}})$-modules.

We define a quiver $\Gamma^{J}$ associated with the pair $(J,\{V_{j}\}_{j\in J})$ as follows:

[TABLE]

Note that we have $d_{ij}d_{ji}=0$ for $i,j\in J$ (see [25, Theorem 2.2]).

We define a symmetric Cartan matrix $\mathsf{A}^{J}=(a^{J}_{ij})_{i,j\in J}$ by $a^{J}_{ij}=2$ if $i=j$ and $a^{J}_{ij}=-d_{ij}-d_{ji}$ otherwise. Then we choose a family of polynomial $\{Q_{i,j}(u,v)\}_{i,j\in J}$ satisfying (3.1) with the form,

[TABLE]

for some choices of sign $\pm$.

Now we take a family $\{P_{i,j}(u,v)\}_{i,j\in J}$ of elements in $\mathbf{k}[[u,v]]$ satisfying the following conditions$\colon$

[TABLE]

We call such a family $\{P_{i,j}(u,v)\}_{i,j\in J}$ a duality coefficient.

Let $\{\alpha^{J}_{i}\ |\ i\in J\}$ be the set of simple roots associated to $\mathsf{A}^{J}$ and $\mathsf{Q}^{+}_{J}=\sum_{i\in J}\mathbb{\mspace{1.0mu}Z}_{\geq 0}\alpha^{J}_{i}$ be the corresponding positive root lattice. We define the symmetric bilinear form $(\ ,\ )$ satisfying $(\alpha^{J}_{i},\alpha^{J}_{j})=a^{J}_{ij}$.

Let us denote by $R^{J}(\beta)$ $(\beta\in\mathsf{Q}^{+}_{J})$ the symmetric quiver Hecke algebra associated with $\mathsf{A}^{J}$ and $\{Q_{i,j}(u,v)\}_{i,j\in J}$.

In [25, §3], Kang-Kashiwara-Kim constructed a functor, called the generalized quantum affine Schur-Weyl duality functor

[TABLE]

which is a monoidal functor in the following sense: There exist canonical $U_{q}^{\prime}({\mathfrak{g}})$-isomorphisms

[TABLE]

for any M_{1},M_{2}\in R^{J}\mbox{-\mathrm{gmod}} such that the diagrams in  [25, (A.2)] are commutative.

Proposition 3.21** ([25, Proposition 3.2.2]).**

The monoidal functor $\mathcal{F}$ is a unique (up to an isomorphism) functor which satisfies the following properties:

1. (i)

For any $i\in J$, we have

[TABLE]

where $\mathbf{k}[z_{V_{i}}^{\pm 1}]\to\mathbf{k}[[z]]$ is given by $z_{V_{i}}\mapsto 1+z$. In particular, we have

[TABLE] 2. (ii)

For $i,j\in J$, let R_{L(i)_{z},L(j)_{z^{\prime}}}\colon L(i)_{z}\mathop{\mathbin{\mbox{\large\circ}}}L(j)_{z^{\prime}}\rightarrow L(j)_{z^{\prime}}\mathop{\mathbin{\mbox{\large\circ}}}L(i)_{z} be the $R^{J}(\alpha^{J}_{i}+\alpha^{J}_{j})$-module homomorphism in (3.3). Then we have

[TABLE]

Note that the functor $\mathcal{F}$ depends on the choice of $\{P_{i,j}(u,v)\}_{i,j\in J}$ as seen in (3.10).

Lemma 3.22** ([32, Lemma 1.7.8]).**

Let $M$, N\in R^{J}\mbox{-\mathrm{gmod}} be simple modules, and assume that one of them is real. Assume that

1. (a)

the functor $\mathcal{F}$ in (3.9) is exact, 2. (b)

$\operatorname{\textfrak{d}}(M,N)\leq 1$, 3. (c)

$\mathcal{F}(M),\mathcal{F}(N)\neq 0$* and $\mathcal{F}(M)\otimes\mathcal{F}(N)$ is not simple.*

Then we have

1. (i)

\mathcal{F}({\mathbf{r}}_{\mspace{-2.0mu}\raisebox{-1.50694pt}{{\scriptstyle{M,N}}}})={\mathbf{r}}_{\mspace{-2.0mu}\raisebox{-1.50694pt}{{\scriptstyle{\mathcal{F}(M),\mathcal{F}(N)}}}}* up to a non-zero constant multiple,* 2. (ii)

\mathcal{F}(M\mathbin{\scalebox{0.9}{\nabla}}N)\simeq\mathcal{F}(M)\mathbin{\scalebox{0.9}{\nabla}}\mathcal{F}(N)* which is simple.*

Theorem 3.23** ([25, Theorem 3.8]).**

If the Cartan matrix $\mathsf{A}^{J}$ associated with $\Gamma^{J}$ is of type $A_{n}\ (n\geq 1),D_{n}\ (n\geq 4)$ or $E_{n}\ (n=6,7,8)$, then the functor $\mathcal{F}$ is exact.

4. Cluster structure on R^{A_{\infty}}\mbox{-\mathrm{gmod}}

In this section, we study the cluster structure on the monoidal category R^{A_{\infty}}\mbox{-\mathrm{gmod}}. Here Dynkin diagram of type $A_{\infty}$ is depicted as follows:

[TABLE]

We first introduce an infinite sequence ${\widetilde{\textbf{{w}}}}$ of simple reflections whose first $p$-parts is reduced for every $p\in\mathbb{\mspace{1.0mu}Z}_{\geq 0}$. Then we explicitly compute determinantial modules of type $A_{\infty}$ which are associated to ${\widetilde{\textbf{{w}}}}$. In the last part of this section, we will construct a certain quantum monoidal seed $\mathscr{S}_{\infty}$ and prove that the category R^{J}\mbox{-\mathrm{gmod}} of type $A_{\infty}$ gives a monoidal categorification of the quantum cluster algebra $A_{\mathbf{A}}({\mathfrak{n}})$ of type $A_{\infty}$ whose initial quantum seed is $[\mathscr{S}_{\infty}]$.

4.1. Sequence of simple reflections of length $\infty$

Let $J=\mathbb{\mspace{1.0mu}Z}$ be an index set. We also take the weight lattice \mathsf{P}_{J}=\mathop{\mbox{\normalsize\bigoplus}}\limits_{i\in\mathbb{\mspace{1.0mu}Z}}\mathbb{\mspace{1.0mu}Z}\Lambda_{i} with $(\Lambda_{i},\Lambda_{j})=-|i-j|/2$. We set $\epsilon_{a}=\Lambda_{a}-\Lambda_{a-1}$ and $\alpha_{j}=\epsilon_{j}-\epsilon_{j+1}$. We have $(\epsilon_{a},\epsilon_{b})=\delta_{a,b}$ and $(\Lambda_{i},\alpha_{j})=\delta_{i,j}$. The matrix A^{J}\mathbin{:=}\bigl{(}(\alpha_{i},\alpha_{j})\bigr{)}_{i,j\in J} is a Cartan matrix of type $A_{\infty}$. We write the root lattice \mathsf{Q}_{J}=\mathop{\mbox{\normalsize\bigoplus}}\limits_{j\in J}\mathbb{\mspace{1.0mu}Z}\alpha_{j}\subset\mathsf{P}_{J}.

Let $W_{J}$ be the Weyl group of type $A_{\infty}$ generated by the set of simple reflections $S_{J}\mathbin{:=}\{s_{j}\mid j\in J\}$. Note that, for all $i,j\in J$, we have

[TABLE]

For $k\in\mathbb{\mspace{1.0mu}Z}_{>0}$, we sometimes use the notation $\overline{k}$ to denote $-k\in\mathbb{\mspace{1.0mu}Z}_{<0}\subset J$.

A pair of integers $[a,b]$ with $a\leq b$ is called a segment. The length of $[a,b]$ is defined to be the positive integer $b-a+1$. For a segment $[a,b]$, we denote by $W_{[a,b]}$ the subgroup of $W_{J}$ generated by $s_{k}$ for $a\leq k\leq b$.

Recall the convention in (1.1) and (1.2) on sequences of simple reflections. For each $t\in\mathbb{\mspace{1.0mu}Z}_{\geq 0}$, let $\textbf{{w}}^{(t)}$ be the sequence in $S_{J}$ of length $4t+3$ defined by$\colon$

[TABLE]

We set

[TABLE]

where $*$ denotes the concatenation of sequences.

Finally, we define an infinite sequence ${\widetilde{\textbf{{w}}}}$ of $S_{J}$ by$\colon$

[TABLE]

We define $\mathfrak{j}_{p}\in J$ ($p\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$) by

[TABLE]

For $t\in\frac{1}{2}\mathbb{\mspace{1.0mu}Z}$, we set

[TABLE]

Note that, for $t\in\mathbb{\mspace{1.0mu}Z}_{\geq 0}$, $a(t)$ coincides with the length of $\textbf{{w}}^{(0,t-1)}$.

We have for $t\in\mathbb{\mspace{1.0mu}Z}_{\geq 0}$

[TABLE]

For each $p\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$, we will denote the element in the Weyl group $W_{J}$ defined by the sequence ${\widetilde{\textbf{{w}}}}_{\leq p}$ by the same symbol.

Proposition 4.1**.**

The sequence ${\widetilde{\textbf{{w}}}}$ has the following properties$\colon$

1. (i)

For $t\in\mathbb{\mspace{1.0mu}Z}_{\geq 0}$, we have

[TABLE] 2. (ii)

For any $p\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$, ${\widetilde{\textbf{{w}}}}_{\leq p}$ is reduced.

Proof.

Let us first show (i). Note that, for each $t\in\mathbb{\mspace{1.0mu}Z}_{\geq 0}$, we have

1. (a)

$\mathfrak{j}_{a(t)+1}=-t$ and $\mathfrak{j}_{p}>-t$ for all $p\leq a(t)$, 2. (b)

$\mathfrak{j}_{a(t+\frac{1}{2})+1}=t+1$ and $\mathfrak{j}_{p}<t+1$ for all $p<a(t+\frac{1}{2})+1$.

Assume that $a(t)+1\leq p\leq a(t+\frac{1}{2})$. Then we have

[TABLE]

Note that ${\widetilde{\textbf{{w}}}}_{\leq a(t)}$ is contained in $W_{[-t+1,t]}$ as a Weyl group element. Thus, for all $a(t)+1\leq p\leq a(t+\frac{1}{2})$, we have

[TABLE]

Now we claim that

[TABLE]

Note that

• •

${\widetilde{\textbf{{w}}}}_{\leq a(t)}=\textbf{{w}}^{(0,t-1)}=\textbf{{w}}^{(0,t-2)}*\textbf{{w}}^{(t-1)}$,

• •

$\textbf{{w}}^{(t-1)}(-t+1)=t+1$ and $\textbf{{w}}^{(t-1)}(t+1)=-t+1$,

• •

$\textbf{{w}}^{(t-1)}(k)=k$ for all $k\in\mathbb{\mspace{1.0mu}Z}\setminus\{t+1,-t+1\}$.

Thus the claim follows from an induction on $t$. Hence we have ${\widetilde{\textbf{{w}}}}_{\leq p-1}(\alpha_{\mathfrak{j}_{p}})=\epsilon_{-t}-\epsilon_{t+2-p+a(t)}$.

The case when $a(t+\frac{1}{2})+1\leq p\leq a(t+1)$ can be proved in a similar way.

(ii) follows from (i) since ${\widetilde{\textbf{{w}}}}_{\leq p-1}\alpha_{\mathfrak{j}_{p}}$ is a positive root for any $p\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$. ∎

Remark 4.2**.**

1. (a)

For each $k\in\frac{1}{2}\mathbb{\mspace{1.0mu}Z}_{\geq 1}$, $a(k)$ coincides with the length of the longest element of Weyl group of type $A_{2k}$. Thus ${\widetilde{\textbf{{w}}}}_{\leq a(k)}$ is the longest element of

[TABLE] 2. (b)

For $t>0$, the reduced expressions $\textbf{{w}}^{(0,t)}$ are not adapted ([2]) in the sense that there exists no Dynkin quiver $\mathfrak{Q}$ of type $A_{2t+2}$ satisfying

[TABLE]

Here $s_{i}(\mathfrak{Q})$ denotes the quiver obtained by reversing all arrows incident with $i$.

Definition 4.3**.**

For each $p\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$, we assign a pair $c(p)=(\ell,m)$ of positive integers in the following way$\colon$

[TABLE]

where $t$ is a unique non-negative integer satisfying

[TABLE]

Remark 4.4**.**

The map $c\colon\mathbb{\mspace{1.0mu}Z}_{\geq 1}\to\mathbb{\mspace{1.0mu}Z}_{\geq 1}\times\mathbb{\mspace{1.0mu}Z}_{\geq 1}$ in (4.8) satisfies the following properties$\colon$

• •

$\ell+m=\begin{cases}2t+2\equiv 0\mspace{3.0mu}\mathbin{\mathrm{mod}}\mspace{1.0mu}2&\text{ if }a(t)<p\leq a(t+\frac{1}{2}),\\ 2t+3\equiv 1\mspace{3.0mu}\mathbin{\mathrm{mod}}\mspace{1.0mu}2&\text{ if }a(t+\frac{1}{2})<p\leq a(t+1).\end{cases}$

• •

$c$ is a bijective map whose inverse is given as follows$\colon$

[TABLE]

• •

The integer $t$ in (4.9) is equal to $\lfloor(\ell+m-2)/2\rfloor=\lceil(\ell+m-3)/2\rceil$.

Using the lattice point $\mathbb{\mspace{1.0mu}Z}_{\geq 1}\times\mathbb{\mspace{1.0mu}Z}_{\geq 1}$, the pairs $(p,\mathfrak{j}_{p})$ corresponding to $c(p)=(\ell,m)$ can be exhibited as follows:

[TABLE]

Lemma 4.5**.**

For $p\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$ with $c(p)=(\ell,m)$, the index $\mathfrak{j}_{p}$ is given as follows$\colon$

[TABLE]

Proof.

It is enough to show that $\ell-m-2\mathfrak{j}_{p}$ is [math] or $-1$. Assume that $a(t)<p\leq a(t+1/2)$ for $t\in\mathbb{\mspace{1.0mu}Z}_{\geq 0}$. Then by (4.7) and (4.8), we have

[TABLE]

If $a(t+1/2)+2\geq p\leq a(t+1)$, then we have

[TABLE]

If $p=a(t+1/2)+1$, then we have

[TABLE]

4.2. Determinantial modules of type $A_{\infty}$

We keep the notations in the previous subsection. For $i,j\in J$, let us set

[TABLE]

We denote by $R^{J}$ the quiver Hecke algebra of type $A_{\infty}$ which is associated to $(Q^{J}_{i,j})_{i,j\in J}$. In the sequel, we sometimes drop the subscript $J$, if there is no danger of confusion.

A multisegment is a finite sequence of segments. We assign a total order on the set of segments as follows$\colon$

[TABLE]

If a multisegment \bigl{(}[a_{1},b_{1}],\ldots,[a_{r},b_{r}]\bigr{)} satisfies $[a_{k},b_{k}]\geq[a_{k+1},b_{k+1}]$ for each $k$, we call it an ordered multisegment.

For $j\in J$ and a pair of positive integer $(\ell,m)\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}\times\mathbb{\mspace{1.0mu}Z}_{\geq 1}$, we define an ordered multisegment $\llbracket\ell,m\rrbracket^{(j)}$ as follows$\colon$

[TABLE]

For each segment $[a,b]$ of length $\ell$, there exists a graded $1$-dimensional $R(\epsilon_{a}-\epsilon_{b+1})$-module $L[a,b]=\mathbf{k}u[a,b]$ which is generated by a vector $u[a,b]$ of degree [math]. The action of $R(\epsilon_{a}-\epsilon_{b+1})$ is given as follows (see [25, Lemma 1.16])$\colon$

[TABLE]

Then one can check that

[TABLE]

where ${\bf{1}}$ is the trivial $R(0)$-module.

Proposition 4.6** ( [41, Theorem 7.2 and Section 8.4]).**

1. (i)

Let $M$ be a simple module in R^{J}(\beta)\mbox{-\mathrm{gmod}} with $\beta\in\mathsf{Q}_{J}^{+}$. Then there exists a unique pair of an ordered multisegment \big{(}[a_{1},b_{1}],\ldots,[a_{t},b_{t}]\big{)} and $c\in\mathbb{\mspace{1.0mu}Z}$ such that

[TABLE]

where ${\operatorname{hd}}$ denotes the head. 2. (ii)

For an ordered multisegment \big{(}[a_{1},b_{1}],\ldots,[a_{t},b_{t}]\big{)},

[TABLE]

is a simple $R^{J}(\beta)$-module, where $\beta=\sum_{k=1}^{t}(\epsilon_{a_{k}}-\epsilon_{b_{k}+1})$.

We call the ordered multisegment \big{(}[a_{1},b_{1}],\ldots,[a_{t},b_{t}]\big{)} in Proposition 4.6 (i) the multisegment associated with $M$. It is uniquely determined by $M$.

For $j\in J$ and a pair of positive integer $(\ell,m)\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}\times\mathbb{\mspace{1.0mu}Z}_{\geq 1}$, we define a self-dual simple $R^{J}$-module $\mathrm{W}^{(\ell)}_{m,j}$ associated to $\llbracket\ell,m\rrbracket^{(j)}$ as follows$\colon$ (up to a grading shift)

[TABLE]

Remark 4.7**.**

The $R^{J}$-module $\mathrm{W}^{(\ell)}_{m,j}$ is known to be *homogeneous *in the sense that its grading is concentrated in a single degree ([40]).

Since ${\widetilde{\textbf{{w}}}}_{\leq p}$ is reduced for any $p\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$, we can define $\mathrm{M}_{\widetilde{\textbf{{w}}}}(p_{1},p_{2})$ for $0\leq p_{2}\leq p_{1}$ (see (1.3) and (3.4)):

[TABLE]

Note that, for any $w\in W_{J}$ and $\Lambda_{j}$ $(j\in J)$, we have

[TABLE]

Let us find the multisegment corresponding to $\mathrm{M}_{\widetilde{\textbf{{w}}}}(p,0)$.

For $j\in J$ and $(\ell,m)\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}\times\mathbb{\mspace{1.0mu}Z}_{\geq 1}$, we shall define a sequence ${\widetilde{w}}_{\llbracket\ell,m\rrbracket^{(j)}}$ in $S_{J}$ which is reduced (as seen in Lemma 4.9 below), and arises from the ordered multisegment $\llbracket\ell,m\rrbracket^{(j)}$ as follows$\colon$

• •

${\widetilde{w}}_{[a,b]}\mathbin{:=}s_{a}s_{a+1}\cdots s_{b-1}s_{b}$ for $a\leq b$.

• •

{\widetilde{w}}_{\llbracket\ell,m\rrbracket^{(j)}}=\underbrace{{\widetilde{w}}_{[j-\ell+m,j+m-1]}\cdots\cdots{\widetilde{w}}_{[j-\ell+2,j+1]}\;{\widetilde{w}}_{[j-\ell+1,j]}}_{\text{m-times}}.

Proposition 4.8**.**

For every $p\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$,

[TABLE]

Proof.

We shall use the induction on $\ell+m$. Let $t$ be a unique non-negative integer in (4.9).

(a) Assume that $a(t)<p\leq a(t+\frac{1}{2})$. In this case, we have

• •

$\ell+m\equiv 0\mspace{3.0mu}\mathbin{\mathrm{mod}}\mspace{1.0mu}2$, by Remark 4.4,

• •

$\mathfrak{j}_{p}+t+1=\ell$ since $\mathfrak{j}_{a(t)+1}=-t$ and $\mathfrak{j}_{a(t)+u}=-t+u-1$ for $1\leq u\leq a(t+\frac{1}{2})-a(t)+1$.

Then we have

[TABLE]

Using (4.1), we have

[TABLE]

Note that

[TABLE]

Thus we have

[TABLE]

where $c(p^{-}(\mathfrak{j}_{p}+1))=(\ell,m-1)$ by Lemma 4.5. (If $m=1$, i.e. $p=a(t+1/2)$, then $p^{-}(\mathfrak{j}_{p}+1)=0$ and ${\widetilde{\textbf{{w}}}}_{\leq p^{-}(\mathfrak{j}_{p}+1)}=1$.)

Note that, $\textbf{{w}}^{(0,t-1)}\in W_{[-t+1,t]}$ and $W_{[-t+1,t]}$ fixes $\epsilon_{-t}$. Hence we have

[TABLE]

By the induction hypothesis, we have

[TABLE]

and one can check that the following two reduced expressions coincide

[TABLE]

where $*$ denotes the concatenation. Note that ${\widetilde{\textbf{{w}}}}_{\leq p^{-}(\mathfrak{j}_{p}+1)}\in W_{[-t+1,t]}$ also fixes $\epsilon_{-t}$. Then our assertion follows from (4.12).

(b) Assume that $a(t+\frac{1}{2})<p\leq a(t+1)$. Then one can check that

[TABLE]

as in the previous case. Note that

• •

$\ell+m\equiv 1\mspace{3.0mu}\mathbin{\mathrm{mod}}\mspace{1.0mu}2$ by Remark 4.4,

• •

$t+2-\mathfrak{j}_{p}=m$ and $c(p^{-}(\mathfrak{j}_{p}-1))=(\ell-1,m)$, by Lemma 4.5,

• •

${\widetilde{\textbf{{w}}}}_{\leq p^{-}(\mathfrak{j}_{p}-1)}\in W_{[t,-t]}$,

By the induction hypothesis, we have

[TABLE]

and one can check that the following two reduced expressions are in the same equivalence class

[TABLE]

with respect to the commutation relation $s_{i}s_{j}\equiv s_{j}s_{i}$ $(|i-j|>1)$. Then our assertion follows from (4.13). ∎

Lemma 4.9**.**

For $j\in J$ and $(\ell,m)\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}\times\mathbb{\mspace{1.0mu}Z}_{\geq 1}$, we have

1. (i)

${\widetilde{w}}_{\llbracket\ell,m\rrbracket^{(j)}}$* is reduced,* 2. (ii)

${\widetilde{w}}_{\llbracket\ell,m\rrbracket^{(j)}}\Lambda_{j}=\Lambda_{j+m}-\sum_{k=1}^{m}\epsilon_{j-\ell+k}$, 3. (iii)

$\mathrm{M}({\widetilde{w}}_{\llbracket\ell,m\rrbracket^{(j)}}\Lambda_{j},\Lambda_{j})\simeq\mathrm{W}_{m,j}^{(\ell)}$.

Proof.

Let us prove (i) and (ii) by induction on $m$. Write ${\widetilde{w}}\mathbin{:=}{\widetilde{w}}_{\llbracket\ell,m\rrbracket^{(\mathfrak{j}_{p})}}=s_{i_{1}}s_{i_{2}}\cdots s_{i_{r}}$. Now we shall prove that

[TABLE]

By the induction hypothesis, we have

[TABLE]

Hence, we have

[TABLE]

Since

[TABLE]

and

[TABLE]

(ii) and (4.14) follow. (i) follows from (4.14).

(iii) Let us prove (iii) by induction on $m$. By Theorem 3.13, we have

[TABLE]

Hence it is enough to show

[TABLE]

Then Proposition 3.15 implies

[TABLE]

Theorem 4.10**.**

For $p\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$ with $c(p)=(\ell,m)$, we have

[TABLE]

Proof.

The assertion immediately follows from Proposition 4.8 and Lemma 4.9. ∎

Using the lattice points $\mathbb{\mspace{1.0mu}Z}_{\geq 1}\times\mathbb{\mspace{1.0mu}Z}_{\geq 1}$, we can exhibit $\{\mathrm{M}_{\widetilde{\textbf{{w}}}}(p,0)\simeq\mathrm{W}_{m,i_{p}}^{(\ell)}\}$ as follows$\colon$

[TABLE]

Note that we have

[TABLE]

Here we confuse $M$ and the isomorphic class of $M$.

Corollary 4.11**.**

For $p\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$ with $c(p)=(\ell,m)$, we have

[TABLE]

For a pair $(a,b)$ of integers with $a>b$, we denote also by

[TABLE]

the graded $1$-dimensional $R(\epsilon_{b}-\epsilon_{a+1})$-module.

Proposition 4.12**.**

1. (a)

For $p,p^{\prime}\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$ with $c(p)=(\ell,m)$ and with $c(p^{\prime})=(\ell,m+1)$, we have

[TABLE] 2. (b)

For each $p\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$ with $c(p)=(\ell,m)$, we have

[TABLE]

Proof.

(a) is a consequence of Theorem 4.10, since we have

[TABLE]

For (b), if $\ell+m\equiv 0\mspace{3.0mu}\mathbin{\mathrm{mod}}\mspace{1.0mu}2$, then $\mathfrak{j}_{p^{\prime}}=\mathfrak{j}_{p^{+}}$ and hence it is the case of (a). Now let us consider when $\ell+m\equiv 1\mspace{3.0mu}\mathbin{\mathrm{mod}}\mspace{1.0mu}2$. In this case, $c(p^{+})=(\ell+1,m)$ by Lemma 4.5. Then we have

• •

\mathrm{M}_{\widetilde{\textbf{{w}}}}(p,0)\simeq{\operatorname{hd}}\bigl{(}L[\mathfrak{j}_{p}-\ell+m,\mathfrak{j}_{p}+m-1]\mathop{\mathbin{\mbox{\large\circ}}}\cdots\mathop{\mathbin{\mbox{\large\circ}}}L[\mathfrak{j}_{p}-\ell+2,\mathfrak{j}_{p}+1]\mathop{\mathbin{\mbox{\large\circ}}}L[\mathfrak{j}_{p}-\ell+1,\mathfrak{j}_{p}]\bigr{)},

• •

\mathrm{M}_{\widetilde{\textbf{{w}}}}(p^{+},0)\simeq{\operatorname{hd}}\bigl{(}L[\mathfrak{j}_{p}-\ell+m-1,\mathfrak{j}_{p}+m-1]\mathop{\mathbin{\mbox{\large\circ}}}\cdots\mathop{\mathbin{\mbox{\large\circ}}}L[\mathfrak{j}_{p}-\ell+1,\mathfrak{j}_{p}+1]\mathop{\mathbin{\mbox{\large\circ}}}L[\mathfrak{j}_{p}-\ell,\mathfrak{j}_{p}]\bigr{)}.

Note that

[TABLE]

with respect to the commutation relation. By Theorem 3.13, we have

[TABLE]

As in the proof of Lemma 4.9 (iii), we have

[TABLE]

Hence our assertion follows. ∎

Corollary 4.13**.**

For each $p\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$ with $c(p)=(\ell,m)$, we have

[TABLE]

Proof.

By Theorem 3.13 and Proposition 4.12,

[TABLE]

and

[TABLE]

Then our assertion follows from Proposition 3.7. ∎

4.3. Quantum monoidal seed $\mathscr{S}_{\infty}$

Since Proposition 4.1 tells that ${\widetilde{\textbf{{w}}}}_{\leq p}$ is reduced for every $p\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$, we can consider the quiver Q associated to ${\widetilde{\textbf{{w}}}}$ by taking $p\to\infty$ and using Definition 2.3. The set of vertices of Q is $K\mathbin{:=}\mathbb{\mspace{1.0mu}Z}_{\geq 1}$. The set of the frozen vertices of ${K}$ is empty. Note that there is a bijection $c\colon K\to\mathbb{\mspace{1.0mu}Z}_{\geq 1}\times\mathbb{\mspace{1.0mu}Z}_{\geq 1}$.

Proposition 4.14**.**

Each vertex of the quiver Q is of finite degree. Hence the associated matrix $\widetilde{B}$ satisfies the conditions in (2.1).

Proof.

Note that for $p,p^{\prime}\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$ with $c(p)=(\ell,m)$ and $c(p^{\prime})=(\ell-1,m+1)$,

[TABLE]

By Lemma 4.5, for $p\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$ with $c(p)=(\ell,m)$, we have

[TABLE]

[TABLE]

Combining (4.15) and (4.16), we have

[TABLE]

Hence we have the followings for $c(p)=(\ell,m)$:

1. (i)

If $\ell+m\equiv 0\mspace{3.0mu}\mathbin{\mathrm{mod}}\mspace{1.0mu}2$ and $m\neq 1$, then

[TABLE] 2. (ii)

If $\ell+m\equiv 0\mspace{3.0mu}\mathbin{\mathrm{mod}}\mspace{1.0mu}2$ and $m=1$, then

[TABLE] 3. (iii)

If $\ell+m\equiv 1\mspace{3.0mu}\mathbin{\mathrm{mod}}\mspace{1.0mu}2$ and $\ell\neq 1$, then

[TABLE] 4. (iv)

If $\ell+m\equiv 1\mspace{3.0mu}\mathbin{\mathrm{mod}}\mspace{1.0mu}2$ and $\ell=1$, then

[TABLE]

Hence, Definition 2.3 tells that there is only one ordinary arrow with source $p$, which is given by $p\to p^{+}(\mathfrak{j}_{p}+1)_{+}$ or $p\to p^{+}(\mathfrak{j}_{p}+1)$ if $\ell+m\equiv 0\mspace{3.0mu}\mathbin{\mathrm{mod}}\mspace{1.0mu}2$, and $p\to p^{+}(\mathfrak{j}_{p}-1)_{+}$ or $p\to p^{+}(\mathfrak{j}_{p}-1)$ if $\ell+m\equiv 1\mspace{3.0mu}\mathbin{\mathrm{mod}}\mspace{1.0mu}2$, respectively. Hence each vertex $p\in J$ with $c(p)=(\ell,m)$ has two incoming arrows and two outgoing arrow unless $\min(\ell,m)=1\colon$

[TABLE]

In particular, when (i) $\ell+m\equiv 0\mspace{3.0mu}\mathbin{\mathrm{mod}}\mspace{1.0mu}2$ and $m=1$, or (ii) $\ell+m\equiv 1\mspace{3.0mu}\mathbin{\mathrm{mod}}\mspace{1.0mu}2$ and $\ell=1$, the arrows incident with vertex $p$ can be described as follows:

[TABLE]

By replacing $p=c^{-1}(\ell,m)$ with $\mathrm{W}^{(\ell)}_{m,\mathfrak{j}_{p}}$ in the diagrams above, the quiver Q can be exhibited as follows:

[TABLE]

Hence our assertion follows. ∎

Note that the arrows of Q oriented right or above are arrows of horizontal type, and arrows of Q oriented left or below are arrows of ordinary type. Also the quiver Q is known as the square product $Q_{A_{\infty}}\square Q_{A_{\infty}}$ of the bipartite Dynkin quiver $Q_{A_{\infty}}$ of type $A_{\infty}$, which is related to the periodicity conjecture (see [8, 19, 20, 35, 36, 49, 52]). Here $Q_{A_{\infty}}$ is the quiver

\textstyle{\circ}$$\scriptstyle{1\ }$$\textstyle{\circ}$$\scriptstyle{2\ }$$\textstyle{\circ}$$\scriptstyle{3\ }$$\textstyle{\circ}$$\scriptstyle{4\ } .

Remark 4.15**.**

Take $k\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$. When we restrict the full subquiver $Q^{(k)}$ of Q consisting of vertices $(\ell,m)$ with $\ell+m\leq k+1$, the $Q^{(k)}$ is mutation equivalent to the well-known quiver $Q_{k}^{BFZ}$ of the coordinate ring $\mathbb{C}[N]$ of the unipotent group $N$ of type $A_{k}$ (see [3] and [35, Theorem 4.5]). For example $k=3$, $Q^{(3)}$ is given as follows:

[TABLE]

Note that $Q_{k}^{BFZ}$ is isomorphic to the quiver associated to some adapted reduced expression of the longest element of the Weyl group of $A_{k}$ (see Definition 2.3).

Now we take the skew-symmetric integer-valued $\mathbb{\mspace{1.0mu}Z}_{\geq 1}\times\mathbb{\mspace{1.0mu}Z}_{\geq 1}$-matrix $\Lambda$ as follows:

[TABLE]

Note that for each $p\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$, we can take sufficiently large $t$ such that $p$ is an exchangeable index of ${\widetilde{\textbf{{w}}}}_{\leq t}$ (see (2.5)). Thus the following corollary follows from Theorem 3.18 for ${\widetilde{\textbf{{w}}}}_{\leq t}$:

Corollary 4.16**.**

1. (a)

For any finite sequence $\Sigma$ in ${K}$, $\mu_{\Sigma}(-\Lambda,\widetilde{B})$ is compatible with $d=2$, where $\widetilde{B}$ is the matrix associated with Q. 2. (b)

For every pair of positive integers $p=c^{-1}(\ell,m)$ and $p^{\prime}=c^{-1}(\ell^{\prime},m^{\prime})$, $\mathrm{W}^{(\ell)}_{m,\mathfrak{j}_{p}}$ and $\mathrm{W}^{(\ell^{\prime})}_{m^{\prime},\mathfrak{j}_{p^{\prime}}}$ strongly commute. 3. (c)

For each $p\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$, there exists $\mathrm{M}_{\widetilde{\textbf{{w}}}}(p,0)^{\prime}$ in R^{J}\mbox{-\mathrm{gmod}} satisfying (3.5).

Let us denote by

[TABLE]

Then $\mathscr{S}_{\infty}$ becomes a quantum monoidal seed. Furthermore, we have

• •

the pair $(\{\mathrm{M}_{\widetilde{\textbf{{w}}}}(p,0)\}_{p\in K},\widetilde{B})$ is admissible,

• •

$\mathscr{S}_{\infty}$ admits successive mutations in R^{J}\mbox{-\mathrm{gmod}} for all the directions.

Let $\mathscr{A}_{q^{1/2}}(\infty)$ be the quantum cluster algebra associated with the quantum seed

[TABLE]

without frozen variables.

Remark 4.17**.**

Note that the quantum cluster algebras associated with a quiver of infinite rank is the $\mathbb{\mspace{1.0mu}Z}[q^{\pm 1/2}]$-subalgebra of skew field $\mathscr{F}$ generated by all elements obtained from initial cluster variables by finite sequences of mutations. We refer to [12, 17] for the details of the definition of (quantum) cluster algebra of infinite rank.

Theorem 4.18**.**

The category R^{J}\mbox{-\mathrm{gmod}} is a monoidal categorification of the quantum cluster algebra $\mathscr{A}_{q^{1/2}}(\infty)$.

Proof.

Theorem 3.20 implies that any cluster monomial of $\mathscr{A}_{q^{1/2}}(\infty)$ is contained in K(\mathcal{C}_{{\widetilde{\textbf{{w}}}}_{\leq t}})\subset{K}(R^{J}\mbox{-\mathrm{gmod}}) for sufficiently large $t\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$. On the other hand, let $M$ be a simple $R^{J}(\beta)$-module. We write $\beta=\sum_{k=-p}^{p}a_{k}\alpha_{k}$. Then by Remark 4.4 (a), $[M]$ is contained in $K(\mathcal{C}_{{\widetilde{\textbf{{w}}}}_{\leq a(p+1)}})$. Thus we conclude that {K}(R^{J}\mbox{-\mathrm{gmod}})=\mathscr{A}_{q^{1/2}}(\infty), which completes the proof. ∎

5. The category $\mathcal{T}_{N}$ and its cluster structure

5.1. Category $\mathcal{T}_{N}$

We keep the notations in § 4. In this subsection we briefly recall the quotient category and the localizations of R\mbox{-\mathrm{gmod}} introduced in [25, §4.4–§4.5]. For details of the constructions, we refer to [25, Appendix A and B]. Then in the next section we apply one of the main results in [32] to show that a generalized quantum affine Schur-Weyl duality functor $\mathcal{F}$ factors thorough the category $\mathcal{T}_{N}$ defined below.

Set \mathcal{A}_{\beta}\mathbin{:=}R^{J}({\beta})\mbox{-\mathrm{gmod}} and \mathcal{A}\mathbin{:=}\mathop{\mbox{\normalsize\bigoplus}}\limits_{{\beta}\in\mathsf{Q}_{J}^{+}}\mathcal{A}_{\beta}. Let $\mathcal{S}_{N}$ be the smallest Serre subcategory of $\mathcal{A}$ such that

1. (i)

$\mathcal{S}_{N}$ contains $L[a,a+N]$ for any $a\in J$, 2. (ii)

X\mathop{\mathbin{\mbox{\large\circ}}}Y,\ Y\mathop{\mathbin{\mbox{\large\circ}}}X\in\mathcal{S}_{N} for all $X\in\mathcal{A}$ and $Y\in\mathcal{S}_{N}$.

Note that $\mathcal{S}_{N}$ contains $L[a,b]$ if $b-a+1>N$.

Let us denote by $\mathcal{A}/\mathcal{S}_{N}$ the quotient category of $\mathcal{A}$ by $\mathcal{S}_{N}$ and denote by $\mathcal{Q}_{N}\colon\mathcal{A}\rightarrow\mathcal{A}/\mathcal{S}_{N}$ the canonical functor.

Note that $\mathcal{A}$ and $\mathcal{A}/\mathcal{S}_{N}$ are monoidal categories with the convolution as tensor products. The module $R(0)\simeq\mathbf{k}$ is a unit object. Note also that $Q\mathbin{:=}qR(0)$ is an invertible central object of $\mathcal{A}/\mathcal{S}_{N}$ and X\mapsto Q\mathop{\mathbin{\mbox{\large\circ}}}X\simeq X\mathop{\mathbin{\mbox{\large\circ}}}Q coincides with the grading shift functor. Moreover, the functors $\mathcal{Q}_{N}$ is a monoidal functor.

Definition 5.1**.**

For each $a,j\in J$, we define

1. (a)

$L_{a}\mathbin{:=}L[a,a+N-1]$, 2. (b)

an abelian group homomorphism $c_{a}:\mathsf{Q}_{J}\to\mathbb{\mspace{1.0mu}Z}$ as $c_{a}(\alpha_{j})\mathbin{:=}(\epsilon_{a}+\epsilon_{a+N},\alpha_{j})$, 3. (c)

a polynomial $f_{a,j}(z)\mathbin{:=}(-1)^{\delta_{j,a+N}}z^{-\delta(a\leq j<a+N-1)-\delta_{j,a+N}}\in\mathbf{k}[z^{\pm 1}]$.

Definition 5.2**.**

Let $\mathsf{S}_{N}$ be the automorphism of \mathsf{Q}_{J}=\mathop{\mbox{\normalsize\bigoplus}}\limits_{a\in\mathbb{\mspace{1.0mu}Z}}\mathbb{\mspace{1.0mu}Z}\,\alpha_{a} given by $\mathsf{S}_{N}(\alpha_{a})=\alpha_{a+N}$. We define the bilinear form $\mathfrak{B}_{N}$ on $\mathsf{Q}_{J}$ by

[TABLE]

Proposition 5.3** ([25, Proposition 4.17]).**

For any $M\in(\mathcal{A}/\mathcal{S}_{N})_{\beta}$, we have an isomorphism

[TABLE]

in $\mathcal{A}/\mathcal{S}_{N}$ which is functorial, and

[TABLE]

Definition 5.4**.**

We define the new tensor product \mathop{\mbox{\normalsize\star}}\limits\colon\mathcal{A}\times\mathcal{A}\to\mathcal{A} by

[TABLE]

where $X\in\mathcal{A}_{\alpha}$ and $Y\in\mathcal{A}_{\beta}$.

Then, $\mathcal{A}$ as well as $\mathcal{A}/\mathcal{S}_{N}$ is endowed with a new structure of a monoidal category by \mathop{\mbox{\normalsize\star}}\limits as shown in [25, Appendix A.8].

Theorem 5.5** ([25, Theorem 4.21]).**

The following statements hold.

1. (i)

$L_{a}$* is a central object in $\mathcal{A}/\mathcal{S}_{N}$; i.e.,*

1. (a)

$f_{a,j}(z)R_{L_{a},L(j)_{z}}$* induces an isomorphism R_{a}(X)\colon L_{a}\mathop{\mbox{\normalsize\star}}\limits X\mathop{\xrightarrow[\raisebox{1.29167pt}[0.0pt][1.29167pt]{\scriptstyle{}}]{{\raisebox{-2.58334pt}[0.0pt][-2.58334pt]{\mspace{2.0mu}\sim\mspace{2.0mu}}}}}X\mathop{\mbox{\normalsize\star}}\limits L_{a} functorial in $X\in\mathcal{A}/\mathcal{S}_{N}$,* 2. (a)

the diagram

[TABLE]

is commutative in $\mathcal{A}/\mathcal{S}_{N}$ for any $X,Y\in\mathcal{A}/\mathcal{S}_{N}$. 2. (ii)

The isomorphism R_{a}(L_{a})\colon L_{a}\mathop{\mbox{\normalsize\star}}\limits L_{a}\mathop{\xrightarrow[\raisebox{1.29167pt}[0.0pt][1.29167pt]{\scriptstyle{}}]{{\raisebox{-2.58334pt}[0.0pt][-2.58334pt]{\mspace{2.0mu}\sim\mspace{2.0mu}}}}}L_{a}\mathop{\mbox{\normalsize\star}}\limits L_{a} coincides with \operatorname{id}_{L_{a}\mathop{\mbox{\normalsize\star}}\limits L_{a}} in $\mathcal{A}/\mathcal{S}_{N}$. 3. (iii)

For $a,b\in\mathbb{\mspace{1.0mu}Z}$, the isomorphisms

[TABLE]

in $\mathcal{A}/\mathcal{S}_{N}$ are inverse to each other.

By the preceding theorem, $\{(L_{a},R_{a})\}_{a\in J}$ forms a commuting family of central objects in (\mathcal{A}/\mathcal{S}_{N},\mathop{\mbox{\normalsize\star}}\limits) (See [25, Appendix A. 4]). Following [25, Appendix A. 6], we localize (\mathcal{A}/\mathcal{S}_{N},\mathop{\mbox{\normalsize\star}}\limits) by this commuting family. Let us denote by $\mathcal{T}^{\prime}_{N}$ the resulting category (\mathcal{A}/\mathcal{S}_{N})[L_{a}^{\mathop{\mbox{\normalsize\star}}\limits-1}\mid a\in J]. Let $\Upsilon\colon\mathcal{A}/\mathcal{S}_{N}\to\mathcal{T}^{\prime}_{N}$ be the projection functor. We denote by $\mathcal{T}_{N}$ the monoidal category $(\mathcal{A}/\mathcal{S}_{N})\,[L_{a}\simeq{\bf{1}}\mid a\in J]$ and by $\Xi\colon\mathcal{T}^{\prime}_{N}\to\mathcal{T}_{N}$ the canonical functor (see [25, Appendix A.7] and [25, Remark 4.22]). Thus we have a chain of monoidal functors

[TABLE]

We set

[TABLE]

Theorem 5.6** ([25, Theorem 4.25]).**

The categories $\mathcal{T}_{N}$ and $\mathcal{T}^{\prime}_{N}$ are rigid monoidal categories; i.e., every object has a right dual and a left dual.

5.2. Cluster structure on $\mathcal{T}_{N}$

In this and next subsections, we prove that $K(\mathcal{T}_{N})$ has a structure of quantum cluster algebra, and $\mathcal{T}_{N}$ is its monoidal categorification.

Let us recall the quiver Q in (4.51) associated to the infinite sequence ${\widetilde{\textbf{{w}}}}$ of $S_{J}$ in (4.2). The vertices of Q are labeled by ${K}=\mathbb{\mspace{1.0mu}Z}_{\geq 1}$ which is also identified with $\mathbb{\mspace{1.0mu}Z}_{\geq 1}\times\mathbb{\mspace{1.0mu}Z}_{\geq 1}$ under the map $c$ sending $p$ to $(\ell,m)$.

Proposition 5.7**.**

For each $p\in{K}$ with $c(p)=(\ell,m)$, the $R$-module $\mathrm{M}_{\widetilde{\textbf{{w}}}}(p,0)^{\prime}$ in (3.6) is given as follows$\colon$

[TABLE]

Proof.

By (4.29), (4.40) and Proposition 4.8, the neighborhood of $\mathrm{M}_{\widetilde{\textbf{{w}}}}(p,0)$ in Q with $c(p)=(\ell,m)$ and $\ell+m\equiv 0\mspace{3.0mu}\mathbin{\mathrm{mod}}\mspace{1.0mu}2$ can be described as follows:

[TABLE]

Then we have

• •

$\mathrm{M}_{\widetilde{\textbf{{w}}}}(p_{+},p)\simeq L[\mathfrak{j}_{p}-\ell+m+1,\mathfrak{j}_{p}+m]$ by Corollary 4.13,

• •

\displaystyle\mathop{\mbox{\normalsize\bigodot}}\limits_{t<p<t_{+}<p_{+}}\mathrm{M}_{\widetilde{\textbf{{w}}}}(t,0)^{\mbox{\scriptsize\odot}|a_{\mathfrak{j}_{p},\mathfrak{j}_{t}}|} in (3.6) is isomorphic to $\mathrm{W}^{(\ell)}_{m-1,\mathfrak{j}_{p}+1}$.

Thus  (3.6) implies that

[TABLE]

Similarly, we can prove the assertion when $\ell+m\equiv 1\mspace{3.0mu}\mathbin{\mathrm{mod}}\mspace{1.0mu}2$. ∎

Let us take a total order on $\mathbb{\mspace{1.0mu}Z}_{\geq 1}\times\mathbb{\mspace{1.0mu}Z}_{\geq 1}$ as follows:

[TABLE]

Let $\Sigma^{{\mathrm{even}}}$ (resp. $\Sigma^{{\mathrm{odd}}}$) be the ascending sequence on the set of all $(\ell,m)\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}\times\mathbb{\mspace{1.0mu}Z}_{\geq 1}$ with $\ell+m\equiv 0\mspace{3.0mu}\mathbin{\mathrm{mod}}\mspace{1.0mu}2$ (resp. $\ell+m\equiv 1\mspace{3.0mu}\mathbin{\mathrm{mod}}\mspace{1.0mu}2$)$\colon$

[TABLE]

Let $\Sigma^{+}$ be the sequence $\Sigma^{{\mathrm{even}}}$ followed by $\Sigma^{{\mathrm{odd}}}$, and $\Sigma^{-}$ be the sequence $\Sigma^{{\mathrm{odd}}}$ followed by $\Sigma^{{\mathrm{even}}}\colon$

[TABLE]

For a sequence $\Sigma=((\ell_{1},m_{1}),(\ell_{2},m_{2}),\ldots)$ of $\mathbb{\mspace{1.0mu}Z}_{\geq 1}\times\mathbb{\mspace{1.0mu}Z}_{\geq 1}$, we denote by $\mu_{\Sigma}(\mathscr{S})$ be the new quantum monoidal seed after performing the sequence of mutations indexed by $\Sigma$; that is, regarding the sequence $\Sigma$ as a sequence in ${{K}}^{\mathrm{ex}}$ by

[TABLE]

$\mu_{\Sigma}(\mathscr{S})$ is defined as follows:

[TABLE]

For $r\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$, we denote by $\mu^{(r)}_{\Sigma}(\mathscr{S})$ the quantum monoidal seed obtained from $\mathscr{S}$ after $r$-repetitions of the mutation sequence $\mu_{\Sigma}$.

From now on, we sometime write $\mu_{(\ell,m)}$ instead of $\mu_{c^{-1}(\ell,m)}$ for simplicity.

Lemma 5.8**.**

We have

[TABLE]

where $\textbf{{Q}}^{{\rm op}}$ is the quiver obtained by reversing all arrows in Q.

Proof.

The neighborhoods of $(\ell,m)$ with $\ell+m\equiv 0\mspace{3.0mu}\mathbin{\mathrm{mod}}\mspace{1.0mu}2$ in Q and $\mu_{(\ell,m)}(\textbf{{Q}})$ can be described as follows:

[TABLE]

When we apply $\mu_{(\ell,m)}$ in the sequence of mutations $\mu_{\Sigma^{{\mathrm{even}}}}$, there exist

• •

the arrow $(\ell-1,m)\to(\ell,m-1)$ made by $\mu_{(\ell-1,m-1)}$,

• •

the arrow $(\ell-1,m)\to(\ell,m+1)$ made by $\mu_{(\ell-1,m+1)}$.

Thus the arrows $(\ell,m-1)\to(\ell-1,m)$ and $(\ell,m+1)\to(\ell-1,m)$ in (5.15) are removed. Also,

• •

the arrow $(\ell,m-1)\to(\ell+1,m)$ in (5.15) will be removed when we apply $\mu_{(\ell+1,m-1)}$,

• •

the arrow $(\ell,m+1)\to(\ell+1,m)$ in (5.15) will be removed when we apply $\mu_{(\ell+1,m+1)}$,

by the same reason. Hence our assertion for $\Sigma^{{\mathrm{even}}}$ follows. The second assertion can be proved in a similar way. ∎

Proposition 5.9**.**

1. (a)

$\mu_{\Sigma^{+}}(\textbf{{Q}})\simeq\mu_{\Sigma^{-}}(\textbf{{Q}})\simeq\textbf{{Q}}$* as quivers.* 2. (b)

For $p\in{K}$ with $c(p)=(\ell,m)$, we have

[TABLE]

Proof.

(a) The first assertion can be proved by applying the same argument as in Lemma 5.8.

(b) Note that $\mu_{\Sigma^{+}}=\mu_{\Sigma^{{\mathrm{odd}}}}\circ\mu_{\Sigma^{{\mathrm{even}}}}$. For $(\ell,m)>(\ell^{\prime},m^{\prime})$ with $\ell+m\equiv\ell^{\prime}+m^{\prime}\equiv 0\mspace{3.0mu}\mathbin{\mathrm{mod}}\mspace{1.0mu}2$, the arrows incident with $\mathrm{M}_{\widetilde{\textbf{{w}}}}(p,0)$ sitting at the coordinate $(\ell,m)$ are not affected by the mutation $\mu_{(\ell^{\prime},m^{\prime})}$. Thus the first assertion for $p$ with $\ell+m\equiv 0\mspace{3.0mu}\mathbin{\mathrm{mod}}\mspace{1.0mu}2$ follows from Proposition 5.7.

After performing $\mu_{\Sigma^{{\mathrm{even}}}}$, Lemma 5.8 tells that the neighborhood of $\mathrm{M}_{\widetilde{\textbf{{w}}}}(p,0)$ with $\ell+m\equiv 1\mspace{3.0mu}\mathbin{\mathrm{mod}}\mspace{1.0mu}2$ in $\mu_{\Sigma^{{\mathrm{even}}}}(\textbf{{Q}})$ can be described as follows:

[TABLE]

Then we have (up to grading shifts)

[TABLE]

where the composition is non-zero by [29, Lemma 3.1.5]. Since $\mathrm{W}_{m-1,\mathfrak{j}_{p}+1}^{(\ell)}$ and $\mathrm{W}_{m+1,\mathfrak{j}_{p}}^{(\ell)}$ strongly commute (Corollary 4.16), the composition is an epimorphism. Note that

[TABLE]

Since $\mathrm{M}_{\widetilde{\textbf{{w}}}}(p,0)^{\prime}$ is unique, our first assertion follows from Proposition 3.7. The second assertion can be proved in a similar way. ∎

By applying the argument in the proof of Proposition 5.9 repeatedly, we have the following corollary.

Corollary 5.10**.**

For each $p\in{K}$ with $c(p)=(\ell,m)$, we have

[TABLE]

In particular, we have the following exact sequence$\colon$ For any $k\in J$,

[TABLE]

where \widetilde{\Lambda}=\widetilde{\Lambda}\bigl{(}\mathrm{W}_{m,k}^{(\ell)},\mathrm{W}_{m,k+1}^{(\ell)}\bigr{)}.

For $N\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$, let us denote by ${K}_{\leq N}$, ${K}_{\geq N}$ and $K_{=N}$ the subsets of ${K}$ as follows:

[TABLE]

We denote by $\textbf{{Q}}_{N}$ the subquiver of Q obtained by

[TABLE]

Then the set of vertices of $\textbf{{Q}}_{N}$ are labeled by ${K}_{N}\mathbin{:=}{K}_{\leq N}$.

For each $N\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$, we decompose ${K}_{N}$ into the set of exchange vertices ${{K}}^{\mathrm{ex}}_{N}$ and the set of frozen vertices ${{K}}^{\mathrm{f\mspace{0.01mu}r}}_{N}$ by defining

[TABLE]

For any quiver $Q$ with ${K}$ as its set of vertices, we denote by $Q_{N}$ the quiver obtained by applying the procedure  (5.24) to $Q$.

For any sequence $\Sigma$ in ${{K}}^{\mathrm{ex}}_{N}$, we have

[TABLE]

Lemma 5.11**.**

For any sequence $\Sigma$ in ${{K}}^{\mathrm{ex}}_{N}$, there exists no arrow in $\mu_{\Sigma}(\textbf{{Q}})$ between a vertex in ${K}_{\geq N+1}$ and a vertex in ${K}_{\leq N-1}$.

Proof.

For the initial quiver Q, it is obvious. Then our assertion follows from the mutation rule of quivers described in (2.4) (b). ∎

For a quiver $Q$, let $\overline{Q}$ be the full subquiver of $Q$ obtained by deleting all frozen vertices.

Example 5.12**.**

The quivers $\textbf{{Q}}_{3}$ and $\overline{\textbf{{Q}}_{3}}$ can be described as follows:

[TABLE]

where $\boxed{\ast}$ denotes frozen vertices.

Lemma 5.13**.**

For any sequence $\Sigma$ in ${{K}}^{\mathrm{ex}}_{N}$, we have

[TABLE]

Let $\Sigma^{{\mathrm{even}}}_{N}$, $\Sigma^{{\mathrm{odd}}}_{N}$, $\Sigma^{+}_{N}$ and $\Sigma^{-}_{N}$ be the subsequences of $\Sigma^{{\mathrm{even}}}$, $\Sigma^{{\mathrm{odd}}}$, $\Sigma^{+}$ and $\Sigma^{-}$ obtained by removing all the vertices outside ${{K}}^{\mathrm{ex}}_{N}$, respectively. Then the following lemma can be proved by applying the same arguments as in the proofs of Lemma 5.8 and Proposition 5.9.

Lemma 5.14**.**

We have

1. (i)

\mu_{\Sigma^{{\mathrm{odd}}}_{N}}\bigl{(}\overline{\textbf{{Q}}_{N}}\bigr{)}\simeq\mu_{\Sigma^{{\mathrm{even}}}_{N}}\bigl{(}\overline{\textbf{{Q}}_{N}}\bigr{)}\simeq\left(\overline{\textbf{{Q}}_{N}}\right)^{{\rm op}}* as quivers,* 2. (ii)

\mu_{\Sigma^{+}_{N}}\bigl{(}\overline{\textbf{{Q}}_{N}}\bigr{)}\simeq\mu_{\Sigma^{-}_{N}}\bigl{(}\overline{\textbf{{Q}}_{N}}\bigr{)}\simeq\overline{\textbf{{Q}}_{N}}* as quivers.*

5.3. Monoidal categorifications via $\mathcal{T}_{N}$ and $\mathscr{C}_{J}$

Proposition 5.15** ([25, Proposition 4.12, Proposition 4.31]).**

1. (i)

If an object $Y$ is simple in $\mathcal{A}/\mathcal{S}_{N}$, then there exists a simple object $M$ in $\mathcal{A}$ satisfying

1. (a)

$\mathcal{Q}_{N}(M)\simeq Y$, 2. (b)

$b_{k}-a_{k}+1\leq N-1$* for $1\leq k\leq r$, where $([a_{1},b_{1}],\ldots,[a_{r},b_{r}])$ is the multisegment associated with $M$.* 2. (ii)

Let $([a_{1},b_{1}],\ldots,[a_{r},b_{r}])$ be the multisegment associated with a simple object $M$ in $\mathcal{A}$.

1. (a)

if $b_{k}-a_{k}+1\leq N$ for any $1\leq k\leq r$, then $\Upomega_{N}(M)$ is simple in $\mathcal{T}_{N}$, 2. (b)

if $b_{k}-a_{k}+1>N$ for some $k$, then $\Upomega_{N}(M)$ vanishes, 3. (c)

if $b_{k}-a_{k}+1=N$ for any $1\leq k\leq r$, then $\Upomega_{N}(M)\simeq{\bf{1}}$. 3. (iii)

By the correspondence in (ii), the set of isomorphism classes of self-dual simple objects of $\mathcal{T}_{N}$ is isomorphic to the set of multisegments $([a_{1},b_{1}],\ldots,[a_{r},b_{r}])$ with $b_{k}-a_{k}+1<N$ $(1\leq k\leq r)$.

Proposition 5.16**.**

Let $M_{k}$ be a real simple $R$-module for $1\leq k\leq r$. Let $m_{k}$, $n_{k}\in\mathbb{\mspace{1.0mu}Z}_{\geq 0}$ $(k=1,\ldots,r)$. We assume that $\widetilde{\Lambda}(M_{i},M_{j})=0$ if $1\leq i<j\leq r$. Then we have

1. (i)

M\mathbin{:=}{\operatorname{hd}}\bigl{(}M_{1}^{\circ m_{1}}\mathop{\mathbin{\mbox{\large\circ}}}\cdots\mathop{\mathbin{\mbox{\large\circ}}}M_{r}^{\circ m_{r}}\bigr{)}* and N\mathbin{:=}{\operatorname{hd}}\bigl{(}M_{1}^{\circ n_{1}}\mathop{\mathbin{\mbox{\large\circ}}}\cdots\mathop{\mathbin{\mbox{\large\circ}}}M_{r}^{\circ n_{r}}\bigr{)} are simple modules,* 2. (ii)

L\mathbin{:=}{\operatorname{hd}}\bigl{(}M_{1}^{\circ m_{1}+n_{1}}\mathop{\mathbin{\mbox{\large\circ}}}\cdots\mathop{\mathbin{\mbox{\large\circ}}}M_{r}^{\circ m_{r}+n_{r}}\bigr{)}* is isomorphic to a simple subquotient of M\mathop{\mathbin{\mbox{\large\circ}}}N up to a grading shift.*

Proof.

(i) follows from [31, Corollary 2.10, Lemma 2.6] and $\widetilde{\Lambda}(M_{i}^{\circ m_{i}},M_{j}^{\circ m_{j}})=0$ for $i<j$.

(ii) In the course of the proof, we ignore grading shifts. Set $L_{2k-1}=M_{k}^{\circ m_{k}}$ and $L_{2k}=M_{k}^{\circ n_{k}}$. Set $\beta_{k}=-\operatorname{wt}(L_{k})\in\mathsf{Q}^{+}$ and $\ell_{k}=|\beta_{k}|$, $m=|-\operatorname{wt}(M)|=\sum_{k=1}^{r}\ell_{2k-1}$, $n=|-\operatorname{wt}(N)|=\sum_{k=1}^{r}\ell_{2k}$. Then M\simeq{\operatorname{hd}}(L_{1}\mathop{\mathbin{\mbox{\large\circ}}}L_{3}\mathop{\mathbin{\mbox{\large\circ}}}\cdots\mathop{\mathbin{\mbox{\large\circ}}}L_{2r-1}), N\simeq{\operatorname{hd}}(L_{2}\mathop{\mathbin{\mbox{\large\circ}}}L_{4}\mathop{\mathbin{\mbox{\large\circ}}}\cdots\mathop{\mathbin{\mbox{\large\circ}}}L_{2r}) and L\simeq{\operatorname{hd}}(L_{1}\mathop{\mathbin{\mbox{\large\circ}}}L_{2}\mathop{\mathbin{\mbox{\large\circ}}}\cdots\mathop{\mathbin{\mbox{\large\circ}}}L_{2r}). Let {\mathbf{r}}_{\mspace{-2.0mu}\raisebox{-1.50694pt}{{\scriptstyle{L_{i},L_{j}}}}}\colon L_{i}\mathop{\mathbin{\mbox{\large\circ}}}L_{j}\to L_{j}\mathop{\mathbin{\mbox{\large\circ}}}L_{i} be the $R$-matrix. Then if $i<j$ and $i\equiv 0$, $j\equiv 1\mspace{3.0mu}\mathbin{\mathrm{mod}}\mspace{1.0mu}2$, then we have $\widetilde{\Lambda}(L_{i},L_{j})=0$ and hence

[TABLE]

Here, for $\beta\in\mathsf{Q}^{+}$ and $w\in\mathfrak{S}_{0pt{\beta}}$, $\tau_{w}=\tau_{i_{1}}\cdots\tau_{i_{\ell}}\in R(\beta)$ where $s_{i_{1}}\cdots s_{\i_{\ell}}$ is a reduced expression of $w$. The element $w[a,b]\in\mathfrak{S}_{a+b}$ is defined by $w[a,b](k)=k+b$ if $1\leq k\leq a$ and $w[a,b](k)=k-a$ if $a<k\leq a+b$.

Then we have a homomorphism

[TABLE]

by a product of {\mathbf{r}}_{\mspace{-2.0mu}\raisebox{-1.50694pt}{{\scriptstyle{L_{i},L_{j}}}}}’s. Hence we have

[TABLE]

Here $w_{1}\in\mathfrak{S}_{m+n}$ is a product of $w[\ell_{j},\ell_{i}]$’s. Note that $w_{1}$ is given explicitly

[TABLE]

Now recall the shuffle lemma:

[TABLE]

Here \mathfrak{S}_{m,n}=\left\{{w\in\mathfrak{S}_{m+n}}\mid{\text{w(k)<w(k+1)$if$1\leq k<n+m$and$k\not=m}}\right\}. Note that $w_{1}\in\mathfrak{S}_{m,n}$. Since $L_{1}\mathop{\otimes}L_{3}\mathop{\otimes}\cdots\mathop{\otimes}L_{2r-1}\subset M$ and $L_{2}\mathop{\otimes}L_{4}\mathop{\otimes}\cdots\mathop{\otimes}L_{2r}\subset N$, the shuffle lemma (5.25) says that the composition

[TABLE]

is injective. Hence we obtain a non-zero homomorphism

[TABLE]

Since L_{1}\mathop{\mathbin{\mbox{\large\circ}}}L_{2}\mathop{\mathbin{\mbox{\large\circ}}}\cdots\mathop{\mathbin{\mbox{\large\circ}}}L_{2r} has a simple head $L$, the second assertion follows. ∎

Corollary 5.17**.**

Let $[a_{1},b_{1}],\ldots,[a_{r},b_{r}]$ be a sequence of segments such that $[a_{k},b_{k}]>[a_{k+1},b_{k+1}]$. Let $m_{k}$, $n_{k}\in\mathbb{\mspace{1.0mu}Z}_{\geq 0}$ $(k=1,\ldots,r)$. If M\mathbin{:=}{\operatorname{hd}}\bigl{(}(L[a_{1},b_{1}])^{\circ m_{1}}\mathop{\mathbin{\mbox{\large\circ}}}\cdots\mathop{\mathbin{\mbox{\large\circ}}}(L[a_{r},b_{r}])^{\circ m_{r}}\bigr{)} and N\mathbin{:=}{\operatorname{hd}}\bigl{(}(L[a_{1},b_{1}])^{\circ n_{1}}\mathop{\mathbin{\mbox{\large\circ}}}\cdots\mathop{\mathbin{\mbox{\large\circ}}}(L[a_{r},b_{r}])^{\circ n_{r}}\bigr{)} strongly commute, then M\mathop{\mathbin{\mbox{\large\circ}}}N is isomorphic to {\operatorname{hd}}\bigl{(}(L[a_{1},b_{1}])^{\circ m_{1}+n_{1}}\mathop{\mathbin{\mbox{\large\circ}}}\cdots\mathop{\mathbin{\mbox{\large\circ}}}(L[a_{r},b_{r}])^{\circ m_{r}+n_{r}}\bigr{)} up to a grading shift.

Proof.

This follows from Proposition 5.16 and $\widetilde{\Lambda}(L[a_{i},b_{i}],L[a_{j},b_{j}])=0$ if $i<j$. ∎

For $p\in{K}$, let us denote by $\mathrm{M}_{N}(p,0)\in\mathcal{T}_{N}$ the image of $\mathrm{M}_{{\widetilde{\textbf{{w}}}}}(p,0)$ under $\Upomega_{N}$; i.e.,

[TABLE]

Since $\mathrm{M}_{\widetilde{\textbf{{w}}}}(p,0)$ is associated to an ordered multisegment $\llbracket\ell,m\rrbracket^{(j)}$ consisting of segments $[a,b]$ with $b-a+1=\ell$, where $c(p)=(\ell,m)\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}\times\mathbb{\mspace{1.0mu}Z}_{\geq 1}$, Proposition 5.15 implies the following lemma.

Lemma 5.18**.**

1. (i)

If $p\in K_{\leq N}$, then $\mathrm{M}_{N}(p,0)$ is simple. 2. (ii)

If $p\in K_{=N}={{K}}^{\mathrm{f\mspace{0.01mu}r}}_{N}$, then $\mathrm{M}_{N}(p,0)\simeq{\bf{1}}$. 3. (iii)

If $p\in K_{\geq N+1}$, then $\mathrm{M}_{N}(p,0)$ vanishes. 4. (iv)

For $\{m_{p}\}_{p\in{{K}}^{\mathrm{ex}}_{N}}\in\mathbb{\mspace{1.0mu}Z}_{\geq 0}^{\oplus{{K}}^{\mathrm{ex}}_{N}}$ and $\{n_{p}\}_{p\in{{K}}^{\mathrm{ex}}_{N}}\in\mathbb{\mspace{1.0mu}Z}_{\geq 0}^{\oplus{{K}}^{\mathrm{ex}}_{N}}$, assume that

[TABLE]

Then we have $m_{p}=n_{p}$ for all $p\in{{K}}^{\mathrm{ex}}_{N}$.

Proof.

(i)–(iii) follow from Proposition 5.15. (iv) follows from Corollary 5.17 and Proposition 5.15. ∎

Set

[TABLE]

Let $\pi_{N}\colon\mathsf{Q}_{J}\to\mathsf{Q}_{J,N}$ denote the projection. Let $\,\overline{\alpha}_{a}\in\mathsf{Q}_{J,N}$ be the image of $\alpha_{a}$ in $\mathsf{Q}_{J,N}$. Similarly, let $\overline{\epsilon}_{a}$ denote the image $\epsilon_{a}$ in $\mathsf{P}_{J,N}$. Then $\,\overline{\alpha}_{a+N}=\,\overline{\alpha}_{a}$, \mathsf{Q}_{J,N}=\mathop{\mbox{\normalsize\bigoplus}}\limits_{a=1}^{N-1}\mathbb{\mspace{1.0mu}Z}\,\overline{\alpha}_{a} and \mathsf{P}_{J,N}=\mathop{\mbox{\normalsize\bigoplus}}\limits_{a=0}^{N-1}\mathbb{\mspace{1.0mu}Z}\epsilon_{a}.

The category $\mathcal{T}_{N}$ is graded by $\mathsf{Q}_{J,N}$, i.e., it has a decomposition

[TABLE]

The duality functor $*$ of $\mathcal{A}$ induces a duality functor $*$ of $\mathcal{T}_{N}$ which satisfies

[TABLE]

Here $(\cdot,\cdot)_{N}$ is defined by $(\overline{\epsilon}_{a},\overline{\epsilon}_{b})_{N}=\delta(a\equiv b\mspace{3.0mu}\mathbin{\mathrm{mod}}\mspace{1.0mu}N)$.

Let $\widetilde{B}_{N}=(b_{ij})_{(i,j)\in{{K}}^{\mathrm{ex}}_{N}\times{{K}}^{\mathrm{ex}}_{N}}$ be the exchange matrix associated with the quiver $\overline{\textbf{{Q}}_{N}}$. Let $L_{N}=(\lambda_{ij})_{i,j\in{{K}}^{\mathrm{ex}}_{N}}$ be the skew-symmetric matrix defined by

[TABLE]

Theorem 5.19**.**

\mathscr{S}_{N}=\left(\bigl{\{}\mathrm{M}_{N}(p,0)\bigr{\}}_{p\in{{K}}^{\mathrm{ex}}_{N}},\widetilde{B}_{N}\right)* is a quantum monoidal seed of $\mathcal{T}_{N}$ and admits successive mutations in all directions.*

Proof.

Since the category $\mathcal{A}$ gives a monoidal categorification of the quantum cluster algebra $\mathscr{A}_{q^{1/2}}(\infty)$, we can make successive mutation on the initial quantum monoidal seed $\mathscr{S}_{\infty}$ of $\mathcal{A}$. Let $\Sigma$ be a sequence of at vertices in ${{K}}^{\mathrm{ex}}_{N}$. Set $\mathscr{S}\mathbin{:=}\mu_{\Sigma}(\mathscr{S}_{\infty})=(\{M_{i}\}_{i\in{K}},(\tilde{b}_{ij})_{(i,j)\in{K}\times{K}})$.

We shall show that

[TABLE]

by applying the induction of the length of $\Sigma$.

In order to see this, by arguing by induction on the length of $\Sigma$, it is enough to show the following statement:

[TABLE]

Set $\mu_{p}(\mathscr{S})=(\{M^{\prime}_{i}\}_{i\in{K}},(\tilde{b}^{\prime}_{ij})_{(i,j)\in{K}\times{{K}}^{\mathrm{ex}}})$. Then $M^{\prime}_{i}=M_{i}$ for $i\not=p$, and we have an exact sequence

[TABLE]

in $\mathcal{A}$.

Note that we have

[TABLE]

By Lemma 5.11, any index $i$ such that $\tilde{b}_{ip}\neq 0$ satisfies $i\in{K}_{\leq N}$. Hence we obtain

[TABLE]

hold in $\mathcal{T}_{N}$,

Applying the exact functor $\Upomega_{N}$ to (5.27), we obtain an exact sequence in $\mathcal{T}_{N}$. Hence \bigl{(}\{\Upomega_{N}(M^{\prime}_{i})\}_{i\in{{K}}^{\mathrm{ex}}_{N}},(\tilde{b}^{\prime}_{ij})_{i,j\in{{K}}^{\mathrm{ex}}_{N}}\bigr{)} is the mutation of $\Omega_{N}(\mathscr{S})$ at $p$. The conditions (iv), (v), (vi) can be checked by using the similar arguments in [29, Proposition 7.1.2]. Thus our assertion follows. ∎

Proposition 5.20**.**

For each $p\in{{K}}^{\mathrm{ex}}_{N}$ with $c(p)=(\ell,m)$ and $r\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$, we have

[TABLE]

up to grading shifts.

Proof.

We will proceed by induction on $r$. When $r=0$, it is the definition. Assume that $r>0$. Note that every index $(\ell,m)\in c({K})$ appears at most once in the sequence of mutations $\mu_{\Sigma^{+}}$ and $\mu_{\Sigma^{+}_{N}}$. Let $\mu_{\Sigma^{+}}(\ell,m)$ and $\mu_{\Sigma^{+}_{N}}(\ell,m)$ be the subsequences of $\mu_{\Sigma^{+}}$ and $\mu_{\Sigma^{+}_{N}}$, respectively, consisting of the mutations preceding the one at the vertex $(\ell,m)$. By induction, we assume that \mu_{\Sigma^{+}_{N}}(\ell,m)\circ\mu^{(r-1)}_{\Sigma^{+}_{N}}\bigl{(}\mathrm{M}_{N}(p^{\prime},0)\bigr{)}\simeq\Upomega_{N}\bigl{(}\mathrm{W}^{(\ell^{\prime})}_{m^{\prime},\mathfrak{j}_{p^{\prime}}+r-1}\bigr{)} for any $p^{\prime}\in{{K}}^{\mathrm{ex}}_{N}$ such that either $p^{\prime}=p$ or $c(p^{\prime})=(\ell^{\prime},m^{\prime})$ is subsequent to $c(p)$ in $\mu_{\Sigma^{+}_{N}}$, and we assume that \mu_{\Sigma^{+}_{N}}(\ell,m)\circ\mu^{(r-1)}_{\Sigma^{+}_{N}}\bigl{(}\mathrm{M}_{N}(p^{\prime\prime},0)\bigr{)}\simeq\Upomega_{N}\bigl{(}\mathrm{W}^{(\ell^{\prime\prime})}_{m^{\prime\prime},\mathfrak{j}_{p^{\prime\prime}}+r}\bigr{)} for any $p^{\prime\prime}=(\ell^{\prime\prime},m^{\prime\prime})\in{{K}}^{\mathrm{ex}}_{N}$ such that $c(p^{\prime\prime})$ is preceding $c(p)$ in $\mu_{\Sigma^{+}_{N}}$.

Note that we have \overline{(\mu_{\Sigma^{+}}(\ell,m)\bigl{(}\textbf{{Q}}\bigr{)})_{N}}=\mu_{\Sigma^{+}_{N}}(\ell,m)\bigl{(}\overline{\textbf{{Q}}_{N}}\bigr{)} for all $(\ell,m)\in c({{K}}^{\mathrm{ex}}_{N})$. Indeed, the mutation at a vertex in ${K}_{\geq N+1}$ does not affect the arrows between the vertices in ${{K}}^{\mathrm{ex}}_{N}$ by Lemma 5.11, and for any $\tilde{m}$, the quiver $\mu_{\Sigma^{+}}(N,\tilde{m})(\textbf{{Q}})$ has only one arrow connecting the vertex $(N,\tilde{m})$ with the ones in ${{K}}^{\mathrm{ex}}_{N}$ so that the mutation at $(N,\tilde{m})$ does not affect the arrows between the vertices in ${{K}}^{\mathrm{ex}}_{N}$, either.

It follows that

[TABLE]

up to a power of $q$. Note that when $\ell=N-1$, the fourth term should be $\Upomega_{N}(\mathrm{W}_{m,{\mathfrak{j}_{p}+r-1}}^{(\ell-1)})$, but it is isomorphic to $\Upomega_{N}(\mathrm{W}_{m,{\mathfrak{j}_{p}+r-1}}^{(\ell-1)})\mathop{\otimes}\Upomega_{N}(\mathrm{W}_{m,{\mathfrak{j}_{p}+r}}^{(\ell+1)})$, since we have $\Upomega_{N}(\mathrm{W}_{m,{\mathfrak{j}_{p}+r}}^{(\ell+1)})\simeq{\bf{1}}$. By applying $\Upomega_{N}$ to the exact sequence (5.23) and comparing it with the above exact sequence, we get

[TABLE]

as desired. ∎

Let $\mathscr{A}_{q^{1/2}}(N)$ be the quantum cluster algebra whose initial quantum seed is given as follows:

[TABLE]

Note that \bigl{\{}q^{-\frac{1}{4}(d_{p},d_{p})_{N}}[\mathrm{M}_{N}(p,0)]\bigr{\}}_{p\in{{K}}^{\mathrm{ex}}_{N}} is algebraically independent by Lemma 5.18.

By Theorem 5.19, we can conclude the following:

Corollary 5.21**.**

$\mathscr{A}_{q^{1/2}}(N)$ is a subalgebra in ${\mathbb{\mspace{1.0mu}Z}[q^{\pm 1/2}]}\mathop{\otimes}_{\mathbb{\mspace{1.0mu}Z}[q^{\pm 1}]}K(\mathcal{T}_{N})$.

Now, we shall show that indeed

[TABLE]

Remark 5.22** (see Lemma 5.14 and also [17, Remark 3.3]).**

For each $p\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$ with $c(p)=(\ell,m)$ and $r\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$, there exists finite sequences $\Sigma_{1}$ and $\Sigma_{2}$ of mutations satisfying

[TABLE]

By [25, Proposition 4.31], the Grothendieck ring $K(\mathcal{T}_{N})$ of $\mathcal{T}_{N}$ is generated by $[\Upomega_{N}(L[a,b])]$ ($b-a+1<N$) as a $\mathbb{\mspace{1.0mu}Z}[q^{\pm 1}]$-algebra.

On the other hand, since $L[a,b]=\mathrm{W}^{(b-a+1)}_{1,b}$, Proposition 5.20 and Remark 5.22 imply that every $[\Upomega_{N}(L[a,b])]$ with $b-a+1<N$ appears as a cluster variable and hence it is contained in $\mathscr{A}_{q^{1/2}}(N)$.

Thus we have

Theorem 5.23**.**

As ${\mathbb{\mspace{1.0mu}Z}[q^{\pm 1/2}]}$-algebras, we have an isomorphism

[TABLE]

By Theorem 5.19 and Theorem 5.23, we obtain the following result.

Theorem 5.24**.**

The category $\mathcal{T}_{N}$ gives a monoidal categorification of the quantum cluster algebra $\mathscr{A}_{q^{1/2}}(N)$. Hence, we have

1. (i)

each cluster monomial in $\mathscr{A}_{q^{1/2}}(N)$ corresponds to the isomorphism class of a real simple object of $\mathcal{T}_{N}$ up to a power of $q^{1/2}$, 2. (ii)

each cluster monomial in $\mathscr{A}_{q^{1/2}}(N)$ is a Laurent polynomial of the initial cluster variables with coefficient in $\mathbb{\mspace{1.0mu}Z}_{\geq 0}[q^{\pm 1/2}]$.

6. Main result

In this section, we first assume that we have a family of quasi-good modules with certain conditions which induces a generalized Schur-Weyl duality functor $\mathcal{F}$ of type $A_{\infty}$, and investigate properties of $\mathcal{F}$ under the assumption. Then we will prove that some subcategories in R^{J}\mbox{-\mathrm{gmod}} and $\mathscr{C}_{\mathfrak{g}}$ give monoidal categorifications of quantum cluster algebras. In the last part, we give families of distinct quasi-good modules for quantum affine algebras of type $A$, $B$, $C$ and $D$, satisfying the conditions. Note that the families for the types $A$ and $B$ were taken from [25, 28], and [32], but the ones for types $C$ and $D$ are new.

6.1. Schur-Weyl duality functor of type $A_{\infty}$

Let $U^{\prime}_{q}({\mathfrak{g}})$ be a quantum affine algebra as in § 1.4. Let $J=\mathbb{\mspace{1.0mu}Z}$ be the index set as in the previous section. We assume that there exists a family of quasi-good modules $\{V_{a}\}_{a\in J}$ in $\mathscr{C}_{\mathfrak{g}}$ satisfying the following properties:

[TABLE]

With the assumption (LABEL:it:a) in (6.1), we have the exact functor in Section 3.5

[TABLE]

Here $R^{J}(\beta)$ is the quiver Hecke algebra of type $A_{\infty}$ defined in Section 4.2.

Lemma 6.1**.**

The family $\{V_{a}\}_{a\in\mathbb{\mspace{1.0mu}Z}}$ has the following properties.

1. (i)

$V_{a}\not\simeq V_{b}$* if $a\not=b$,* 2. (ii)

${\operatorname{hd}}(V_{a}\mathop{\otimes}V_{a+1}\mathop{\otimes}\cdots\mathop{\otimes}V_{a+\ell})$* is simple if $0\leq\ell\leq N-1$,* 3. (iii)

for any $a\in J$ and $\ell\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$,

[TABLE] 4. (iv)

$\mathcal{F}(L[a,a+N-1])\simeq\mathbf{k}$.

Proof.

(i) Assume $a<b$. Then $V_{a-1}\mathop{\otimes}V_{a}$ is not simple, but $V_{a-1}\mathop{\otimes}V_{b}$ is simple by (6.1) (LABEL:it:a).

(ii) follows from (6.1) (LABEL:it:1).

(iii) We first prove the statement for $\ell\leq N$ by induction on $\ell$. When $\ell=1$, $\mathcal{F}(L(a))\simeq V_{a}$ by Proposition 3.21 (a). By induction on $\ell$, we may assume that $\mathcal{F}(L[a,a+\ell-2])\simeq{\operatorname{hd}}(V_{a}\mathop{\otimes}V_{a+1}\mathop{\otimes}\cdots\mathop{\otimes}V_{a+\ell-2})$ if $\ell\leq N$. Since $\operatorname{\textfrak{d}}(L[a,a+\ell-2],L(a+\ell-1))=1$ by [25, Theorem 4.3] and L[a,a+\ell-1]\simeq L[a,a+\ell-2]\mathbin{\scalebox{0.9}{\nabla}}L(a+\ell-1) the condition (6.1) (LABEL:it:2) and Lemma 3.22 imply

[TABLE]

Hence we have proved (iii) when $\ell\leq N$. In particular, $\mathcal{F}(L[a,a+N-1])\simeq\mathbf{k}$ by (LABEL:it:1). Hence we have

[TABLE]

for all $a\in\mathbb{\mspace{1.0mu}Z}$. If $\mathcal{F}(L[a,a+N])$ did not vanish, then we would have

[TABLE]

which contradicts (i).

Hence we conclude that

[TABLE]

Now assume that $\ell>N+1$. Then we have a surjective homomorphism

[TABLE]

which yields $\mathcal{F}(L[a,a+\ell-1])\simeq 0$. ∎

Let us denote by $\mathscr{C}_{J}$ is the smallest abelian full subcategory of $\mathscr{C}_{\mathfrak{g}}$ such that

1. (a)

$\mathscr{C}_{J}$ contains $\{V_{a}\}_{a\in J}$, 2. (b)

it is stable under taking submodules, quotients, extensions and tensor products.

Then we have an exact functor

[TABLE]

Now we shall show that the functor $\mathcal{F}$ under the assumption (6.1) factors through the category $\mathcal{T}_{N}$ with a suitable choice of duality coefficient $\{P_{i,j}(u,v)\}_{i,j\in J}$. To do that, we employ the framework of [32, §2.6].

Lemma 6.2**.**

For $a,b\in J$, set $\tilde{z}=z_{V_{b}}$,

[TABLE]

and

[TABLE]

where $z=\tilde{z}-1$ and $a_{M,N}(z_{N})$ denotes the ratio of $R^{\rm{univ}}_{M,N}$ and $R^{\rm{norm}}_{M,N}$ in (1.5). Then the following diagram is commutative

[TABLE]

for any surjective homomorphism $\varphi\colon V_{a}\mathop{\otimes}\cdots\mathop{\otimes}V_{a+N-1}\twoheadrightarrow\mathbf{k}$.

Proof.

Let

[TABLE]

By replacing $R^{\rm{norm}}_{a,b}$ and $g_{a,b}(z)$ with $R^{\rm{univ}}_{a,b}$ and $\operatorname{id}$ respectively, the diagram is commutative.

Then our assertion follows from $R^{\rm{univ}}_{a,b}=g_{a,b}(z)R^{\rm{norm}}_{a,b}$. ∎

Now we temporarily fix a duality coefficient $\{P^{\mathrm{T}}_{i,j}(u,v)\}_{i,j\in J}$ satisfying the conditions in (3.8). We denote the corresponding functor from $\mathcal{A}/\mathcal{S}_{N}$ to $\mathscr{C}_{J}$ by $\mathcal{F}^{\mathrm{T}}$.

Recall $L_{a}=L[a,a+N-1]$ and $f_{a,b}(z)$ in Definition 5.1.

Proposition 6.3**.**

For $a,b\in J$, let

[TABLE]

Then the following diagrams are commutative$\colon$

[TABLE]

for any \varphi_{a}\colon\mathcal{F}^{\mathrm{T}}(L_{a})\mathop{\xrightarrow[\raisebox{1.29167pt}[0.0pt][1.29167pt]{\scriptstyle{}}]{{\raisebox{-2.58334pt}[0.0pt][-2.58334pt]{\mspace{2.0mu}\sim\mspace{2.0mu}}}}}\mathbf{k} and \varphi_{b}\colon\mathcal{F}^{\mathrm{T}}(L_{b})\mathop{\xrightarrow[\raisebox{1.29167pt}[0.0pt][1.29167pt]{\scriptstyle{}}]{{\raisebox{-2.58334pt}[0.0pt][-2.58334pt]{\mspace{2.0mu}\sim\mspace{2.0mu}}}}}\mathbf{k}. Furthermore $h_{a,b}(z)$ has no poles and no zeros at $z=0$.

Proof.

The proof is the same as one of [32, Proposition 2.6.2, Corollary 2.6.3, Proposition 2.6.4]. ∎

Corollary 6.4**.**

Set $\lambda_{a,b}\mathbin{:=}\prod_{k=b}^{b+N-1}h_{a,k}(0)$. Then we have

[TABLE]

Proof.

By Theorem 5.5 (ii), we have R_{a}(L_{a})=\operatorname{id}_{L_{a}\mathop{\mbox{\normalsize\star}}\limits L_{a}}. Thus $\mathcal{F}^{\mathrm{T}}(R_{a}(L_{a}))$ is the identity map on $\mathbf{k}$, which implies $\lambda_{a,a}=1$ by the previous proposition. Similarly, Theorem 5.5 (iii) implies the second assertion. ∎

Lemma 6.5** ([32, Lemma 2.6.6, Lemma 2.6.7]).**

Assume that

[TABLE]

satisfies that

[TABLE]

Then there exists a family $\{c_{a,b}(u,v)\ |\ a,b\in J\}$ of elements in $\mathbf{k}[[u,v]]$ satisfying

[TABLE]

By Lemma 6.5, we have a family $\{c_{a,b}(u,v)\ |\ a,b\in J\}$ of elements in $\mathbf{k}[[u,v]]$ satisfying (6.3). Using $\{c_{a,b}(u,v)\ |\ a,b\in J\}$, we define new duality coefficient $\{P^{\mathrm{new}}_{a,b}(u,v)\ |\ a,b\in J\}$ as follows:

[TABLE]

We denote the corresponding functor from $\mathcal{A}/\mathcal{S}_{N}$ to $\mathscr{C}_{J}$ by $\mathcal{F}^{\prime}$. Since $c_{a,b}(u,v)c_{b,a}(v,u)=1$, we have

[TABLE]

Theorem 6.6**.**

Under the assumption of (6.1), there exists a duality coefficient $\{P^{\mathrm{new}}_{i,j}(u,v)\}_{i,j\in J}$ that makes the following diagram commutative$\colon$

[TABLE]

for any $a\in J$, $M\in\mathcal{A}/\mathcal{S}_{N}$ and an isomorphism g_{a}\colon\mathcal{F}^{\prime}(L_{a})\mathop{\xrightarrow[\raisebox{1.29167pt}[0.0pt][1.29167pt]{\scriptstyle{}}]{{\raisebox{-2.58334pt}[0.0pt][-2.58334pt]{\mspace{2.0mu}\sim\mspace{2.0mu}}}}}\mathbf{k}.

Proof.

It is enough to show that our assertion holds when $M=L(b)_{z}$ for some $b\in J$. Equivalently, it is enough to check that $f_{a,b}(z)g_{a,b}(z)\prod_{k=a}^{a+N-1}P^{\rm{new}}_{k,b}(0,z)=1$. By the choice of $\{c_{a,b}(u,v)\ |\ a,b\in J\}$, we have

[TABLE]

by (6.3). Thus our assertion follows. ∎

Now, [25, Proposition A.11, Proposition A.12] imply the following theorem:

Theorem 6.7**.**

Under the assumption of (6.1), there exists an exact monoidal functor $\widetilde{\mathcal{F}}\colon\mathcal{T}_{N}\to\mathscr{C}_{J}$ such that the following diagram quasi-commutes$\colon$

[TABLE]

In particular, $\widetilde{\mathcal{F}}$ induces a surjective ring homomorphism $\phi_{\widetilde{\mathcal{F}}}\colon K(\mathcal{T}_{N})|_{q=1}\twoheadrightarrow K(\mathscr{C}_{J})$, where $K(\mathcal{T}_{N})|_{q=1}\mathbin{:=}K(\mathcal{T}_{N})/(q-1)K(\mathcal{T}_{N})$.

Corollary 6.8**.**

The functor $\widetilde{\mathcal{F}}$ sends a non-zero object in $\mathcal{T}_{N}$ to a non-zero module in $\mathscr{C}_{J}$. In particular, it sends a simple to a simple and $\mathscr{C}_{J}$ is rigid.

Proof.

Let $M$ be a non-zero object in $\mathcal{T}_{N}$. Since $\mathcal{T}_{N}$ is rigid, there exists $M^{*}\in\mathcal{T}_{N}$ such that there is an epimorphism M^{*}\mathop{\mbox{\normalsize\star}}\limits M\twoheadrightarrow\mathbf{k}. Since $\widetilde{\mathcal{F}}$ is exact, we have an epimorphism $\widetilde{\mathcal{F}}(M^{*})\mathop{\otimes}\widetilde{\mathcal{F}}(M)\twoheadrightarrow\mathbf{k}$. Hence $\widetilde{\mathcal{F}}(M)$ is non-zero. ∎

Using the same argument in [32, Lemma 2.6.11, Lemma 2.6.12, Theorem 2.6.13], we can conclude the following:

Theorem 6.9**.**

The functor $\widetilde{\mathcal{F}}$ sends simples to simples bijectively and induces a ring isomorphism

[TABLE]

Recall that we have an exact functor $\mathcal{F}\colon\mathcal{A}\to\mathscr{C}_{J}$. Theorem 6.9 along with Theorem 5.24 implies the following theorem:

Theorem 6.10**.**

Let $\mathscr{A}(N)$ be the cluster algebra whose initial seed is given as follows:

[TABLE]

Then the category $\mathscr{C}_{J}$ gives a monoidal categorification of the cluster algebra $\mathscr{A}(N)$. Hence, we have

1. (i)

any cluster monomial in $\mathscr{A}(N)$ corresponds to the isomorphism class of a real simple object in $\mathscr{C}_{J}$, 2. (ii)

any cluster monomial in $\mathscr{A}(N)$ is a Laurent polynomial of the initial cluster variables with coefficient in $\mathbb{\mspace{1.0mu}Z}_{\geq 0}$.

6.2. The category $\mathscr{C}_{J}$

In this subsection, we shall give explicitly families of quasi-good modules $\{V_{a}\}_{a\in J}$ which satisfy (6.1) for quantum affine algebras of type $A$, $B$, $C$ and $D$. Thus one can apply the results in the previous subsections. In particular, for quantum affine algebras of type $A$ and $B$, the category $\mathscr{C}_{J}$ coincides with the Hernandez-Leclerc category $\mathscr{C}_{\mathfrak{g}}^{0}$ in Section 1.7.

6.2.1. Type A

Throughout this subsection, ${\mathfrak{g}}^{(t)}$ denotes the affine Kac-Moody algebra of type $A^{(t)}_{N-1}$ $(t=1,2)$. Here $N\geq 2$ if $t=1$ and $N\geq 3$ if $t=2$. Let us denote by $V^{(t)}(\varpi_{i})$ the fundamental module over ${\mathfrak{g}}^{(t)}$. For each $a\in J=\mathbb{\mspace{1.0mu}Z}$, define $V^{(t)}_{a}$ as follows(see [25, §4.1] and [28, §3.1]):

[TABLE]

Then, by Theorem 1.6, [1, Lemma B.1] and [44, Theorem 3.5, Theorem 3.9], we have the followings (see [25, 28] for details):

1. (i)

The family of quasi-good modules $\{V^{(t)}_{a}\}_{a\in J}$ satisfies the conditions in (6.1). 2. (ii)

The subcategory $\mathscr{C}_{J}$ coincides with the category $\mathscr{C}^{0}_{{\mathfrak{g}}^{(t)}}$ in § 1.7.

Then, by Theorem 6.7, we have an exact functor

[TABLE]

which factors thorough $\mathcal{T}_{N}$ via $\widetilde{\mathcal{F}}^{(t)}$

[TABLE]

For $i\in I_{0}$, $k\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$ and $r\in\mathbf{k}^{\times}$, let us denote by $W^{(i)}_{k,r}$ the Kirillov-Reshetikhin module

[TABLE]

which is known as a quasi-good module.

Proposition 6.11** ([25, Proposition 3.9], [28, Proposition 3.3]).**

For a segment $[a,b]$ with $\ell\mathbin{:=}b-a+1<N$, we have

[TABLE]

Thus we can conclude that the image of an $R^{J}$-module $\mathrm{W}_{m,j}^{(\ell)}$ in (4.11) by $\mathcal{F}^{(t)}$ is a Kirillov-Reshetikhin module:

Corollary 6.12**.**

For an $R^{J}$-module $\mathrm{W}_{m,j}^{(\ell)}$ $(\ell<N)$, we have

[TABLE]

Then the exact sequence (5.23) in R^{J}\mbox{-\mathrm{gmod}} can be translated into the exact sequence

[TABLE]

in $\mathscr{C}_{{\mathfrak{g}}^{(1)}}^{0}$ if $t=1$, and the exact sequence

[TABLE]

in $\mathscr{C}_{{\mathfrak{g}}^{(2)}}^{0}$ if $t=2$, where

[TABLE]

Here we understand $W^{(0)}_{k,r}$ and $W^{(N)}_{k,r}$ as trivial modules. Note that the exact sequences (6.4) and (6.5) are well-known as (twisted) T-system of $U_{q}^{\prime}(A^{(t)}_{N-1})$ [14, 15, 42, 43].

Since ${\mathbb{\mspace{1.0mu}Z}[q^{\pm 1/2}]}\mathop{\otimes}_{\mathbb{\mspace{1.0mu}Z}[q^{\pm 1}]}K(\mathcal{T}_{N})$ has a quantum cluster algebra structure, the isomorphism $\phi_{\widetilde{\mathcal{F}}^{(t)}}$ in Theorem 6.9 endows the cluster algebra structure on K\bigl{(}\mathscr{C}^{0}_{{\mathfrak{g}}^{(t)}}\bigr{)} whose initial seed can be identified with

[TABLE]

Now we can conclude the following theorem:

Theorem 6.13**.**

The category $\mathscr{C}^{0}_{{\mathfrak{g}}^{(t)}}$ $(t=1,2)$ gives a monoidal categorification of the cluster algebra $\mathscr{A}(N)$. Hence we have the followings$\colon$

1. (i)

Each cluster monomial in $\mathscr{A}(N)$ corresponds to an isomorphism class of a real simple object in $\mathscr{C}^{0}_{{\mathfrak{g}}^{(t)}}$. 2. (ii)

Each cluster monomial in $\mathscr{A}(N)$ is a Laurent polynomial of the initial cluster variables with coefficient in $\mathbb{\mspace{1.0mu}Z}_{\geq 0}$.

As we have mentioned in Section 2.3, Hernandez and Leclerc give a cluster algebra structure $\mathscr{A}$ on the Grothendieck ring of a “half” of $\mathscr{C}_{A_{N-1}^{(1)}}^{0}$, denoted by $\mathscr{C}_{A^{(1)}_{N-1}}^{-}$ in [17], which is associated to the initial quiver $G^{-}_{A^{(1)}_{N-1}}$ with infinite rank. For example, the quiver $G^{-}_{A^{(1)}_{3}}$ is given as follows:

[TABLE]

Note that the quiver $G^{-}_{A^{(1)}_{N-1}}$ satisfies (2.1) and does not have frozen vertex either.

The quiver $G^{-}_{A^{(1)}_{N-1}}$ is mutation equivalent to $\overline{\textbf{{Q}}}_{N}$ via a sequence of mutation of infinite length. To show that, we need to introduce sequences of mutations: For each $s\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$, let ${}_{s}{{K}}^{\mathrm{ex}}_{0}$ (resp. ${}_{s}{{K}}^{\mathrm{ex}}_{1}$) be the vertices $(\ell,m)$ in ${{K}}^{\mathrm{ex}}_{N}$ with $m=s$ and $\ell\equiv 0\mspace{3.0mu}\mathbin{\mathrm{mod}}\mspace{1.0mu}2$ (resp. $\ell\equiv 1\mspace{3.0mu}\mathbin{\mathrm{mod}}\mspace{1.0mu}2$). Note that ${}_{s}{{K}}^{\mathrm{ex}}_{0}$ and ${}_{s}{{K}}^{\mathrm{ex}}_{1}$ are finite subsets of ${{K}}^{\mathrm{ex}}_{N}$. Let $\Sigma^{(s,0)}_{N}$ (resp. $\Sigma^{(s,1)}_{N}$) be the ascending sequence on ${}_{s}{{K}}^{\mathrm{ex}}_{0}$ (resp.  ${}_{s}{{K}}^{\mathrm{ex}}_{1}$). Finally, we set the sequence $\Sigma_{N-1}^{{\rm HL}}$ as follows:

[TABLE]

Then we have

[TABLE]

as quivers (see also [11, Exercise 2.6.5, Exercise 2.6.6]).

We remark here that

• •

each mutation in $\mu_{\Sigma_{N-1}^{{\rm HL}}}\circ\mu_{\Sigma^{{\mathrm{even}}}_{N}}$ corresponds to the T-system described in (6.4),

• •

for $p\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$ with $c(p)=(\ell,m)$ and $\ell<N$, $\mu_{\Sigma_{N-1}^{{\rm HL}}}\circ\mu_{\Sigma^{{\mathrm{even}}}_{N}}(W^{(\ell)}_{m,(-q)^{2\mathfrak{j}_{p}-\ell+1}})$ is a Kirillov-Reshetikhin module. More precisely,

[TABLE]

Remark 6.14**.**

As Remark 5.22, for each $k\in\mathbb{\mspace{1.0mu}Z}_{\geq 1}$, there exists a finite subsequence of mutations $\mu_{\Sigma}$ of $\mu_{\Sigma_{N-1}^{{\rm HL}}}\circ\mu_{\Sigma^{{\mathrm{even}}}_{N}}$ such that

[TABLE]

for any $(\ell,m)\in I_{0}\times\mathbb{\mspace{1.0mu}Z}_{\geq 1}$ with $\ell+m\leq k$, where $\delta_{\ell}\mathbin{:=}\delta(\ell\equiv 0\mspace{3.0mu}\mathbin{\mathrm{mod}}\mspace{1.0mu}2)$.

As a corollary of Theorem 6.13, we can give a proof of Conjecture 2.7 for $\mathscr{C}^{-}_{A^{(1)}_{N-1}}$:

Theorem 6.15**.**

The cluster monomials of $\mathscr{A}$ associated with ${\mathfrak{g}}$ of type $A_{N-1}^{(1)}$ in Theorem 2.6 can be identified with the real simple modules in $\mathscr{C}^{-}_{A_{N-1}^{(1)}}$.

Proof.

By (1.4) and [1, 24], one can check that

[TABLE]

Note that the cluster variables $\{z_{\ell,\delta_{\ell}-2m+1}\}_{(\ell,m)\in I_{0}\times\mathbb{\mspace{1.0mu}Z}_{\geq 1}}$ of the initial seed $\mathscr{S}_{HL}$ of $\mathscr{A}$ are identified with $\{W^{(\ell)}_{m,(-q)^{\delta_{\ell}-2m+1}}\}_{(\ell,m)\in I_{0}\times\mathbb{\mspace{1.0mu}Z}_{\geq 1}}$ via truncated $q$-characters (see [17, (3), §3.2.3]). Thus cluster monomials of $\mathscr{S}_{HL}$ can be identified with the tensor product of modules in $\left\{W^{(\ell)}_{m,(-q)^{\delta_{\ell}-2m+1}}\right\}_{(\ell,m)\in I_{0}\times\mathbb{\mspace{1.0mu}Z}_{\geq 1}}$, which can be obtained from the set of modules

[TABLE]

by taking parameter shift by $-q$.

Thus, by taking the finite sequence of mutations $\mu_{\Sigma}$ in Remark 6.14 for $k\gg 0$, (6.6) implies that, for $M\in\{W^{(\ell)}_{m,(-q)^{2-\delta_{\ell}}}\}_{(\ell,m)\in I_{0}\times\mathbb{\mspace{1.0mu}Z}_{\geq 1}}$ and a finite sequence ((\ell_{i},m_{i}))_{1\leq i\leq r}\in\bigl{(}I_{0}\times\mathbb{\mspace{1.0mu}Z}_{\geq 1}\bigr{)}^{r}, we have

[TABLE]

Hence Theorem 6.13 tells that any cluster monomial of $\mathscr{A}$ can be identified with a real simple in $\mathscr{C}^{-}_{A_{N-1}^{(1)}}$. ∎

6.2.2. Type B

Now ${\mathfrak{g}}$ denotes the affine Kac-Moody algebra of type $B^{(1)}_{n}$ ($n\geq 2$) and set $N=2n$. For each $a\in J=\mathbb{\mspace{1.0mu}Z}$, define $i_{a}\in I_{0}$ as follows (see [32, §2.2]):

[TABLE]

We define a map $X\colon J\to\mathbf{k}^{\times}$ as follows:

[TABLE]

where $q_{\kappa}=(-1)^{n+1}q^{n+1/2}$, and extend it by

[TABLE]

For each $a\in J=\mathbb{\mspace{1.0mu}Z}$, define $V_{a}$ as follows:

[TABLE]

By (1.6) and (1.7), the quiver $\Gamma^{J}$ in (3.7) is of type $A_{\infty}$. Furthermore, (6.7) tells that the family of quasi-good modules $\{V_{a}\}$ satisfies ${\rm(iii)}$ in (6.1).

Proposition 6.16** ([32, Proposition 2.4.2],[44, §4]).**

1. (i)

The family of quasi-good modules $\{V_{a}\}_{a\in J}$ satisfies the conditions in (6.1). 2. (ii)

The subcategory $\mathscr{C}_{J}$ coincides with the category $\mathscr{C}^{0}_{{\mathfrak{g}}}$ in (1.12).

Then we have an exact functor

[TABLE]

which factors thorough $\mathcal{T}_{N}$ via $\widetilde{\mathcal{F}}\colon\mathcal{T}_{N}\to\mathscr{C}^{0}_{{\mathfrak{g}}}$.

Proposition 6.17** ([32, Proposition 3.2.1]).**

Let $0\leq a\leq N-1$ and $1\leq b-a+1\leq N-1$. Then we have

1. (i)

$\mathcal{F}(L[0,b])\simeq V(\varpi_{n})_{q^{2b}}$* for $0\leq b\leq N-2$.* 2. (ii)

$\mathcal{F}(L[a,N-1])\simeq V(\varpi_{n})_{q^{2a+N-3}}$* for $1\leq a\leq N-1$.* 3. (iii)

\mathcal{F}(L[a,b])\simeq\begin{cases}V(\varpi_{b-a+1})_{(-1)^{b-a}q_{\kappa}q^{a+b-2}}&\text{for}\ 1\leq a\leq b\leq N-2,\ b-a+1<n,\\ V(\varpi_{n})_{q^{2b}}\mathbin{\scalebox{0.9}{\nabla}}V(\varpi_{n})_{q^{2a+N-3}}&\text{for}\ 1\leq a\leq b\leq N-2,\ b-a+1\geq n,\\ V(\varpi_{n})_{q^{2a+N-3}}\mathbin{\scalebox{0.9}{\nabla}}V(\varpi_{n})_{q^{2b-2}}&\text{for}\ 1\leq a\leq N-1<b,\ b-a+1\leq n,\\ V(\varpi_{N-b+a-1})_{(-1)^{b-a}q_{\kappa}q^{a+b-3}}&\text{for}\ 1\leq a\leq N-1<b,\ b-a+1>n.\end{cases}**

In this case, the exact sequence (5.23) in R^{J}\mbox{-\mathrm{gmod}} is not translated into the known $T$-system of $U_{q}^{\prime}(B_{n}^{(1)})$ in [14, 42, 43].

Now we have a $U_{q}^{\prime}(B_{n}^{(1)})$-version of Theorem 6.13 as follows:

Theorem 6.18**.**

The category $\mathscr{C}^{0}_{{\mathfrak{g}}}$ gives a monoidal categorification of the cluster algebra $\mathscr{A}(N)$. In particular, we have an isomorphism $(t=1,2)$ of cluster algebras

[TABLE]

6.2.3. Type C

Let ${\mathfrak{g}}$ be the affine Kac-Moody algebra of type $C^{(1)}_{n}$ $(n\geq 3)$ and set $N=n+1$. Recall the denominator formulas $d_{k,l}(z)$ for $U_{q}^{\prime}(C^{(1)}_{n})$.

Theorem 6.19** ([1]).**

For ${\mathfrak{g}}=C^{(1)}_{n}$ $(n\geq 3)$, we have

[TABLE]

where $q_{s}\mathbin{:=}q^{1/2}$.

For each $a\in J=\mathbb{\mspace{1.0mu}Z}$, define $i_{a}\in I_{0}$ as follows:

[TABLE]

Note that $p^{*}=q_{s}^{2n+2}$. Set

[TABLE]

We define a map $X\colon J\to\mathbf{k}^{\times}$ as follows:

[TABLE]

and extend it by

[TABLE]

For each $a\in J=\mathbb{\mspace{1.0mu}Z}$, define $V_{a}$ as follows:

[TABLE]

Proposition 6.20** ([1, Proposition C.2]).**

For each $i,j\in I_{0}$ with $i+j\leq n$, there exists a $U^{\prime}_{q}(C_{n}^{(1)})$-homomorphism given as follows:

[TABLE]

In particular, from (6.10), we have a $U^{\prime}_{q}(C_{n}^{(1)})$-homomorphism as follows:

[TABLE]

for $1\leq u\leq n-1$.

By (6.8), the quiver $\Gamma^{J}$ in (3.7) is of type $A_{\infty}$. Then we have an exact functor

[TABLE]

Proposition 6.21**.**

Let $0\leq a\leq N-1$ and $b-a+1\leq N-1$. Then we have

1. (i)

$\mathcal{F}(L[0,b])\simeq V(\varpi_{n-b})_{(-q_{s})^{b}}$* for $0\leq b\leq N-1$.* 2. (ii)

$\mathcal{F}(L[a,b])\simeq V(\varpi_{b-a+1})_{q_{\kappa}(-q_{s})^{a+b-2}}$* for $1\leq a\leq b\leq N-1$.*

Here we understand $V(\varpi_{0})$ and $V(\varpi_{N})$ as the trivial module $\mathbf{k}$.

Proof.

(i) When $b=0$, it is obvious by Proposition 3.21 (a). By induction on $b\leq N-1$, we may assume that $\mathcal{F}(L[0,b-1])\simeq V(\varpi_{n-b+1})_{(-q_{s})^{b-1}}$.

On the other hand, Theorem 6.19 and Proposition 6.20 imply that

[TABLE]

By Lemma 3.22, we have \mathcal{F}(L[0,b])\simeq\mathcal{F}(L[0,b-1])\mathbin{\scalebox{0.9}{\nabla}}\mathcal{F}(L(b)). Hence, we obtain the desired result.

(ii) When $a=N-1$, it is obvious by Proposition 3.21 (a). By descending induction on $a$, we may assume that $\mathcal{F}(L[a+1,N-1])\simeq V(\varpi_{N-a-1})_{q_{\kappa}(-q_{s})^{N-2+a}}$.

On the other hand, Theorem 6.19 and Proposition 6.20 imply that

[TABLE]

By Lemma 3.22 again, we have \mathcal{F}(L[a,N-1])\simeq\mathcal{F}(L(a))\mathbin{\scalebox{0.9}{\nabla}}\mathcal{F}(L[a+1,N-1]) and we obtain the desired result.

(iii) By fixing $a$ or $b$, one can apply the same argument as in (i) or (ii). ∎

Proposition 6.22**.**

The family of quasi-good modules $\{V_{a}\}_{a\in J}$ satisfies condition (6.1).

Proof.

By Proposition 6.21, condition (6.1) (LABEL:it:b) for $a=0$ is guaranteed. Note that for a segment $[a,b]$ with $b-a+1=N$, $[a,b]$ is one of the following forms:

[TABLE]

for some $k\in\mathbb{\mspace{1.0mu}Z}$ and $1\leq t\leq N-1$. By (6.9), one can conclude that

[TABLE]

which implies (6.1) (LABEL:it:1). Using similar argument to a segment $[a,b]$ with $b-a+1\leq N$, one can prove  (6.1) (LABEL:it:2). Hence our assertion follows. ∎

By Theorem 6.7 and Proposition 6.22, the functor $\mathcal{F}$ factors thorough $\mathcal{T}_{N}$.

Remark 6.23**.**

The subcategory $\mathscr{C}_{J}$ is a proper subcategory of $\mathscr{C}_{\mathfrak{g}}^{0}$. For example, the quasi-good module $V(\varpi_{1})_{(-q_{s})^{-2}}$ is not contained in $\mathscr{C}_{J}$, since $V(\varpi_{1})_{(-q_{s})^{-2}}$ does not strongly commute with infinitely many $\mathcal{F}(\mathrm{W}_{m,\mathfrak{j}_{p}}^{(1)})$’s.

Theorem 6.24**.**

The category $\mathscr{C}_{J}$ gives a monoidal categorification of the cluster algebra $\mathscr{A}(N)$.

6.2.4. Type D

Let ${\mathfrak{g}}^{(t)}$ be the affine the Kac-Moody algebra of type $D^{(t)}_{n}$ $(n\geq 4$ if $t=1,2$, and $n=4$ if $t=3$) and set $N=n$. Let us denote by $V^{(t)}(\varpi_{i})$ the fundamental module over $U_{q}^{\prime}(D_{n}^{(t)})$ $(t=1,2,3)$. The denominator formulas for these types were calculated in [26, 44, 45].

Note that

[TABLE]

For each $a\in J=\mathbb{\mspace{1.0mu}Z}$, define $i_{a}\in I_{0}$ as follows:

[TABLE]

Set

[TABLE]

where $\omega$ is a primitive third root of unity.

We define a map $X\colon J\to\mathbf{k}^{\times}$ as follows:

[TABLE]

and extend it by

[TABLE]

For each $a\in J=\mathbb{\mspace{1.0mu}Z}$, define $V^{(t)}_{a}$ as follows:

[TABLE]

Now we shall present the denominator formulas only related to the choice of the family of distinct quasi-good modules $\{V^{(t)}_{a}\}\colon$

Theorem 6.25** ([26, 34, 44, 45]).**

1. (a)

For $t=1$, we have

[TABLE] 2. (b)

For $t=2$, we have

[TABLE] 3. (c)

For $t=3$, we have

[TABLE]

Then one can check the following holds:

Corollary 6.26**.**

The graph $\Gamma^{J}$ associated to $\{V^{(t)}_{a}\}$ is of type $A_{\infty}$.

By the morphisms in [26, 44, 45], we have the following propositions by applying the same arguments as in Proposition 6.21 and Proposition 6.22:

Proposition 6.27**.**

Let $0\leq a\leq N-1$ and $b-a+1\leq N-1$. Then we have

1. (i)

$\mathcal{F}^{(t)}(L[0,b])\simeq V^{(t)}(\varpi_{N+1-t-b-\delta})_{q^{b}}$* for $0\leq b\leq N-2$, *

where \delta=\begin{cases}1&\text{if b\neq 0$and$t=1},\\ 0&\text{ otherwise.}\end{cases}

In particular, $\mathcal{F}^{(t)}(L[0,N-1])\simeq\mathbf{k}$. 2. (ii)

\mathcal{F}^{(t)}(L[1,b])\simeq\begin{cases}V^{(1)}(\varpi_{n-\epsilon})_{q^{2(b-1)}}&\text{ if t=1},\\ V^{(2)}(\varpi_{n-1})_{(-1)^{\epsilon}q^{2(b-1)}}&\text{ if t=2},\\ V^{(3)}(\varpi_{1})_{\omega^{\epsilon}q^{2(b-1)}}&\text{ if t=3},\end{cases}\quad\text{ for }b\leq N-1,* *

where $\epsilon\in\{0,1\}$ such that $b-1\equiv\epsilon\mspace{3.0mu}\mathbin{\mathrm{mod}}\mspace{1.0mu}2$. 3. (iii)

$\mathcal{F}^{(t)}(L[a,b])\simeq V(\varpi_{b-a+1})_{q_{\kappa}(-q)^{a+b-2}}$* for $2\leq a\leq b\leq N-1$.*

Proposition 6.28**.**

The family of distinct quasi-good modules $\{V_{a}\}_{a\in J}$ satisfies the conditions in (6.1).

Now we have functors $\mathcal{F}^{(t)}$ and $\widetilde{\mathcal{F}}^{(t)}$ onto $\mathscr{C}_{J}^{(t)}$ as in the previous subsections.

Theorem 6.29**.**

The category $\mathscr{C}^{(t)}_{J}$ gives a monoidal categorification of the cluster algebra $\mathscr{A}(N)$.

Remark 6.30**.**

The subcategory $\mathscr{C}^{(t)}_{J}$ is a proper subcategory of $\mathscr{C}_{{\mathfrak{g}}^{(t)}}^{0}$. For example, the quasi-good module $V(\varpi_{1})_{q_{\kappa}}$ is not contained in $\mathscr{C}_{J}$ of type $D_{n}^{(1)}$ since $V(\varpi_{1})_{q_{\kappa}}$ does not strongly commute with infinitely many $\mathcal{F}(\mathrm{W}_{m,\mathfrak{j}_{p}}^{(1)})$’s.