April 24, 2026
In the present paper, we study the factorization properties of the generalized minors \(\Delta(w_{\leq k}\Lambda,\, v_{\leq k}\Lambda),\) introduced by Fomin–Zelevinsky, in the coordinate rings of Kac–Moody open Richardson varieties. By analyzing their simple factors in the monoidal category \(\mathscr{C}_{w,v}\), we connect the cluster algebra structure of these varieties with the categorical framework developed by Kashiwara–Kim–Oh–Park.
In particular, we prove that cluster monomials in the coordinate ring of a Kac–Moody open Richardson variety correspond to isomorphism classes of simple modules in \(\mathscr{C}_{w,v}\). As a consequence, we show that the Grothendieck ring \(K(\mathscr{C}_{w,v})\) contains the cluster algebra structure on the coordinate ring constructed by Bao–Ye. In finite type, we further prove that Leclerc’s seeds coincide with Ménard’s seeds for open Richardson varieties, and that the category \(\widetilde{\mathscr{C}}_{w,v}\) provides a monoidal categorification of the cluster structure on the open Richardson variety.
Let \(G\) be a symmetric Kac–Moody group over \(\mathbb{C}\), and let \(B^+\) and \(B^-\) be opposite Borel subgroups. Denote by \(\mathcal{B}=G/B^+\) the flag variety. For Weyl group elements \(v,w\in W\) with \(v\leq w\) in the Bruhat order, define the Schubert cell \(\mathring{\mathcal{B}}_w := B^+ w B^+/B^+\) and the opposite Schubert cell \(\mathring{\mathcal{B}}^{\,v} := B^- v B^+/B^+\). Their intersection \[\mathring{\mathcal{B}}_{v,w} := \mathring{\mathcal{B}}_w \cap \mathring{\mathcal{B}}^{\,v}\] is called an open Richardson variety.
In finite type, Leclerc conjectured in [1] that the coordinate ring \(\mathbb{C}[\mathring{\mathcal{B}}_{v,w}]\) admits a cluster algebra structure whose initial cluster variables are given by the prime factors of the generalized minors \(\Delta(w_{\leq k}^{-1}\lambda,\, v_{\leq k}^{-1}\lambda)\). This conjecture was recently confirmed by Casals–Gorsky–Gorsky–Le–Shen–Simental [2] and independently by Galashin–Lam–Sherman-Bennett–Speyer [3], who constructed explicit cluster structures on \(\mathbb{C}[\mathring{\mathcal{B}}_{v,w}]\).
In type \(A_n\), Serhiyenko and Sherman-Bennett [4] related Leclerc’s seed to the seed constructed by Ingermanson [5]. Moreover, when open Richardson varieties are realized as special cases of braid varieties, the two approaches above yield different descriptions of the factorization of generalized minors. In [2], the authors decompose generalized minors using Demazure waves together with the tropical combinatorics of Lusztig cycles, whereas in [3] the decomposition is obtained via Deodhar geometry. On the other hand, Ménard introduced in [6] a seed \(\mathbf{s}(v,w)\) for open Richardson varieties, and it is conjectured that this seed coincides with Leclerc’s seed.
In symmetric Kac–Moody type, however, the prime factorization of quantum minors remains largely unknown. Determining these prime factors is one of the main goals of the present paper.
On the categorical side, Kashiwara, Kim, Oh, and Park [7], [8] constructed a monoidal subcategory \(\mathscr{C}_{w,v}\) of the module category of quiver Hecke algebras. By introducing a family of determinantial modules \(M(w_{\leq k}\Lambda,\, v_{\leq k}\Lambda)\), they showed that the Grothendieck ring of a suitable localization of \(\mathscr{C}_{v,w}\) is isomorphic to the coordinate ring \(\mathbb{C}[\mathring{\mathcal{B}}_{v,w}]\).
In the symmetrizable Kac–Moody setting, Bao and Ye [9], building on Ménard’s work [6], proved that \(\mathbb{C}[\mathring{\mathcal{B}}_{v,w}]\) carries a natural upper cluster algebra structure. They further conjectured that, after inverting the frozen variables, this upper cluster algebra coincides with the corresponding cluster algebra.
A central difficulty in categorifying \(\mathbb{C}[\mathring{\mathcal{B}}_{v,w}]\) is to determine whether the cluster variables in the initial seed \(\mathbf{s}(v,w)\) correspond to isomorphism classes of simple modules in the category \(\mathscr{C}_{v,w}\). A second difficulty is to understand whether every simple module in \(\mathscr{C}_{v,w}\) can be expressed in terms of the cluster variables of \(\mathbb{C}[\mathring{\mathcal{B}}_{v,w}]\).
To address the first problem, we establish precise relations between the cluster variables in the seed \(\mathbf{s}(v,w)\) and the determinantial modules in \(\mathscr{C}_{v,w}\). This leads to the following result.
Let \(\overline{w}=(i_1,\dots,i_r)\) be a reduced expression of \(w\in W\), and let \(v\leq w\) in the Bruhat order. Denote by \(\overline{v}=(i_{p_1},\dots,i_{p_m})\) the leftmost subexpression of \(\overline{w}\) corresponding to \(v\). One then defines the seed \(\mathbf{s}(\overline{v},\overline{w})\) as in Definition 4. Let \(J\) denote the vertex set of the seed \(\mathbf{s}(\overline{v},\overline{w})\).
Theorem 1 (Theorem 19; Theorem 22). Let \(M_k\) be the simple module in \(\mathscr{C}_w\) corresponding to the cluster variable \(X_k\) in the initial seed \(\mathbf{s}(\overline{v},\overline{w})\). Then there exists a bijection \(\Phi: J \to \{1,\dots,r\}\setminus\{p_1,\dots,p_m\}\) such that \(M_k\) is a multiplicity-one factor of the determinantial module \[M\!\binom{w_{\le \Phi(k)}\varpi_{i_{\Phi(k)}}}{v_{\le \Phi(k)}\varpi_{i_{\Phi(k)}}}.\] Moreover, for any \(l\notin \{p_1,\dots,p_m\}\), the determinantial module \(M\!\binom{w_{\le l}\varpi_{i_l}}{v_{\le l}\varpi_{i_l}}\) is a cluster monomial in the seed \(\mathbf{s}(\overline{v},\overline{w})\). Here \(M\!\binom{w_{\le l}\varpi_{i_l}}{v_{\le l}\varpi_{i_l}}\) corresponds to the quantum minor \(D(w_{\le l}\varpi_{i_l},\,v_{\le l}\varpi_{i_l})\). In particular, in finite type, Leclerc’s seed coincides with Ménard’s seed.
For a more explicit formulation, see Theorem 19, where we write \(w_l\) and \(v_l\) in place of \(w_{\le l}\) and \(v_{\le l}\), respectively. Let \(\overline{\mathcal{A}}_q(\mathbf{s}(\overline{v},\overline{w}))\) denote the quantum cluster algebra associated with the seed \(\mathbf{s}(\overline{v},\overline{w})\) without inverting frozen variables, and let \(\mathcal{A}_q(\mathbf{s}(\overline{v},\overline{w}))\) denote the corresponding quantum cluster algebra in which the frozen variables are inverted. Combining the above results, we obtain the following theorem.
Theorem 2 (Theorem 21; Theorem 23). Let \(v\le w\) be Weyl group elements of symmetric Kac–Moody type. Then the Grothendieck ring \(K(\mathscr{C}_{v,w})\) contains the quantum cluster algebra \(\overline{\mathcal{A}}_q(\mathbf{s}(\overline{v},\overline{w}))\) with initial seed \(\mathbf{s}(\overline{v},\overline{w})\), namely the initial seed for the coordinate ring \(\mathbb{C}[\mathring{\mathcal{B}}_{v,w}]\) constructed by Bao–Ye [9]. Moreover, every cluster monomial in \(\overline{\mathcal{A}}_q(\mathbf{s}(\overline{v},\overline{w}))\) is realized as the isomorphism class of a simple module in the category \(\mathscr{C}_{v,w}\). In finite type, \(K(\widetilde{\mathscr{C}}_{v,w})\) provides a monoidal categorification of the quantum cluster algebra \(\mathcal{A}_q(\mathbf{s}(\overline{v},\overline{w}))\).
The paper is organized as follows. In Section 2, we recall basic material on cluster algebras. In Section 3, we review Kac–Moody open Richardson varieties \(\mathring{\mathcal{B}}_{v,w}\) and the cluster algebra structure on their coordinate rings. In Section 4, we study determinantial modules and their properties. Finally, in Section 5, we prove the main results of the paper.
The author would like to express his sincere gratitude to Masaki Kashiwara and Ryo Fujita for many helpful discussions and valuable suggestions.
In this section, we will recall the notion of cluster algebras and quantum cluster algebras.
Let \(I\) be a finite set together with a partition \[I = I_{\mathrm{uf}} \sqcup I_{\mathrm{fr}}.\] Let \(B=(b_{ij})\) be an \(I\times I_{\mathrm{uf}}\) matrix with entries in \(\mathbb{Q}\) such that the principal part \[B_{I_{\mathrm{uf}}\times I_{\mathrm{uf}}}\] is a skew-symmetric integer matrix. The elements of \(I_{\mathrm{uf}}\) are called mutable vertices, and those of \(I_{\mathrm{fr}}\) are called frozen vertices. We may associate with \(B\) a quiver \(Q_B\) by drawing \(b_{ij}\) arrows from \(i\) to \(j\) whenever \(b_{ij}>0\).
Consider the Laurent polynomial ring \[\mathbb{C}[x_i^{\pm1}\mid i\in I],\] and let \(\mathbb{C}(x_i\mid i\in I)\) be its field of fractions. For simplicity, we write \(\mathbb{C}[x_i^{\pm1}]\) and \(\mathbb{C}(x_i)\), respectively.
A seed is a triple \[\mathbf{t}=\bigl((x_i)_{i\in I},\, B,\, I_{\mathrm{fr}}\bigr).\] For \(\mathbf{a}=(a_i)_{i\in I}\in \mathbb{Z}^I\), we set \[x^\mathbf{a}:=\prod_{i\in I}x_i^{a_i}.\]
Definition 1 (Mutation). Let \(k\in I_{\mathrm{uf}}\). The mutation \(\mu_k\) at \(k\) is defined as follows.
The mutated exchange matrix \(\mu_k(B)=(\mu_k(B)_{ij})\) is given by \[\mu_k(B)_{ij}= \begin{cases} -b_{ij}, & \text{if } i=k \text{ or } j=k,\\[1mm] b_{ij}+(-1)^{\delta(b_{ik}<0)}\max(b_{ik}b_{kj},0), & \text{otherwise}. \end{cases}\]
The mutated cluster variables are \[\mu_k(x_i)= \begin{cases} x^{\mathbf{a}'}+x^{\mathbf{a}''}, & \text{if } i=k,\\ x_i, & \text{if } i\neq k, \end{cases}\] where \(\mathbf{a}'=(a_i')_{i\in I}\) and \(\mathbf{a}''=(a_i'')_{i\in I}\) are defined by \[\label{eq:bfa} a_i'= \begin{cases} -1, & \text{if } i=k,\\ \max(0,b_{ik}), & \text{if } i\neq k, \end{cases} \qquad a_i''= \begin{cases} -1, & \text{if } i=k,\\ \max(0,-b_{ik}), & \text{if } i\neq k. \end{cases}\tag{1}\]
The seed \[\mu_k(\mathbf{t}):=\bigl((\mu_k(x_i))_{i\in I},\,\mu_k(B),\,I_{\mathrm{fr}}\bigr)\] is called the mutation of \(\mathbf{t}\) at \(k\).
For a seed \(\mathbf{t}'=\bigl((x_i')_{i\in I},\,B',\,I_{\mathrm{fr}}\bigr)\), the elements \(x_i'\) are called the cluster variables, and \(B'\) is called the exchange matrix. If \(i\in I_{\mathrm{fr}}\), then \(x_i'\) is called a frozen variable.
Let \(T\) be the set of seeds obtained from \(\mathbf{t}\) by finite sequences of mutations.
Definition 2. The cluster algebra \(\mathcal{A}(\mathbf{t})\) associated with the seed \(\mathbf{t}\) is the \(\mathbb{C}\)-subalgebra of \(\mathbb{C}(x_i)\) generated by all cluster variables appearing in all seeds \(\mathbf{t}'\in T\). In our convention, the frozen variables are not assumed to be invertible.
The corresponding upper cluster algebra is defined by \[U(\mathbf{t}):=\bigcap_{\mathbf{t}'\in T}\mathbb{C}[x_{\mathbf{t}',i}^{\pm1}],\] where \(x_{\mathbf{t}',i}\) denotes the \(i\)th cluster variable of the seed \(\mathbf{t}'\).
By the Laurent phenomenon, one has \[\mathcal{A}(\mathbf{t})\subset U(\mathbf{t}).\]
Let \(\Lambda=(\lambda_{ij})\) be a skew-symmetric \(K\times K\) matrix. The pair \((\Lambda,B)\) is said to be compatible if \[\sum_{k\in K}\lambda_{ik}b_{kj}=2\delta_{ij} \qquad \text{for all } i\in K,\;j\in K^{\mathrm{ex}}.\]
Given such a skew-symmetric matrix \(\Lambda\), we define the quantum torus \(\mathcal{T}_\Lambda\) to be the \(\mathbb{K}\)-algebra generated by \(X_i^{\pm1}\) \((i\in K)\), where \(\mathbb{K}=\mathbb{Z}[q^{\pm1/2}]\), subject to the relations \[X_iX_j=q^{\lambda_{ij}}X_jX_i, \qquad X_iX_i^{-1}=X_i^{-1}X_i=1.\]
For any vector \(\mathbf{a}=(a_1,\dots,a_r)\in \mathbb{Z}^r\), define the monomial \[X^{\mathbf{a}} = q^{\frac{1}{2}\sum_{i>j}a_ia_j\lambda_{ij}}X_1^{a_1}\cdots X_r^{a_r}.\]
A quantum seed is a tuple \[\mathbf{t}:=\bigl((X_i)_{i\in K},\,\Lambda,\,B,\,K^{\mathrm{ex}}\bigr),\] where \((\Lambda,B)\) is a compatible pair.
For \(k\in K^{\mathrm{ex}}\), the mutation at \(k\) is defined as follows. The mutated matrix \(\mu_k(\Lambda)\) has entries \[\label{eq95mutations} \mu_k(\Lambda)_{ij}= \begin{cases} -\lambda_{kj}+\sum_{l\in K}\max\{0,-b_{lk}\}\lambda_{lj}, & \text{if } i=k,\;j\neq k,\\[1mm] -\lambda_{ik}+\sum_{l\in K}\max\{0,-b_{lk}\}\lambda_{il}, & \text{if } i\neq k,\;j=k,\\[1mm] \lambda_{ij}, & \text{otherwise}. \end{cases}\tag{2}\] The mutated cluster variables are \[\mu_k(X_i)= \begin{cases} X_i, & \text{if } i\neq k,\\ X^{\mathbf{a}}+X^{\mathbf{a}'}, & \text{if } i=k. \end{cases}\] where \(\mathbf{a}\) and \(\mathbf{a}'\) are given by eqution 1 .
It is well known that \((\mu_k(\Lambda),\mu_k(B))\) is again a compatible pair. Thus one obtains a new quantum seed \[\mu_k(\mathbf{t}) := \bigl((\mu_k(X_i))_{i\in K},\,\mu_k(\Lambda),\,\mu_k(B),\,K^{\mathrm{ex}}\bigr).\]
Definition 3. For a quantum seed \(\mathbf{t}\), the quantum cluster algebra \(\mathcal{A}_q(\mathbf{t})\) is the \(\mathbb{K}\)-subalgebra generated by all cluster variables \(X_i(\mathbf{t}')\) for all seeds \(\mathbf{t}'\in T\), together with the inverses of the frozen variables. The upper quantum cluster algebra is defined by \[U_q(\mathbf{t})=\bigcap_{\mathbf{t}'\in T}\mathcal{T}(\mathbf{t}'),\] where \(\mathcal{T}(\mathbf{t}')\) denotes the quantum torus associated with the seed \(\mathbf{t}'\). We denote by \(\overline{\mathcal{A}}_q(\mathbf{t})\) the \(\mathbb{K}\)-subalgebra of \(\mathcal{T}(\mathbf{t})\) generated by all cluster variables \(X_i(\mathbf{t}')\) for all seeds \(\mathbf{t}'\in T\).
In this subsection, we recall the cluster algebra structures associated with words in \(I\).
Let \(C\) be a symmetric \(I\times I\) Cartan matrix. A word in \(I\) is a sequence \[\mathbf{i}=(i_1,\dots,i_r),\] where \(i_k\in I\) for all \(1\le k\le r\). For a positive integer \(r\), we write \([r]:=\{1,\dots,r\}\).
For \(k\in [r]\) and \(j\in I\), we say that \(k\) has color \(j\) if \(i_k=j\). For each \(j\in I\), let \(n_j\) denote the number of occurrences of the color \(j\) in the word \(\mathbf{i}\).
For \(k\in [r]\), define \[k^+:=\min\{p>k\mid i_p=i_k\}, \qquad k^-:=\max\{p<k\mid i_p=i_k\},\] with the convention that \(k^+=+\infty\) and \(k^-=-\infty\) when the corresponding sets are empty.
It is often convenient to label the \(l\)th occurrence of the color \(j\) by \((j,l)\). Thus, if \(k=(j,l)\), then \(k^+=(j,l+1)\) and \(k^-=(j,l-1)\). We also set \((j,0)=-\infty\) and \((j,n_j+1)=+\infty\).
For \(j\neq i_k\), define \[k(j)^-:=\max\{p<k\mid i_p=j\}, \qquad k(j)^+:=\min\{p>k\mid i_p=j\},\] again allowing the values \(\pm\infty\).
Let \(\mathbf{i}\) be a word of length \(r>0\). Set \[I:=[r], \qquad I_{\mathrm{fr}}:=\{k\in [r]\mid k^+=+\infty\}.\] We define the exchange matrix \(B_{\mathbf{i}}=(b_{ij})_{i\in I,\,j\in I_{\mathrm{uf}}}\) by \[b_{ij}= \begin{cases} 1, & \text{if } i=j^-,\\ -1, & \text{if } j=i^-,\\ -c_{ij}, & \text{if } j<i<j^+<i^+,\\ c_{ij}, & \text{if } i<j<i^+<j^+,\\ 0, & \text{otherwise}. \end{cases}\]
We denote by \[\mathbf{s}(\mathbf{i}):=\bigl((x_i)_{i\in [r]},\,B_{\mathbf{i}},\,I_{\mathrm{fr}}\bigr)\] the seed associated with the word \(\mathbf{i}\). The corresponding cluster algebra and upper cluster algebra are denoted by \[\mathcal{A}(\mathbf{i}):=\mathcal{A}(\mathbf{s}(\mathbf{i})), \qquad U(\mathbf{i}):=U(\mathbf{s}(\mathbf{i})).\]
For a word \(\mathbf{i}=(i_1,\dots,i_r)\), define \[w_{\le k}=s_{i_1}\cdots s_{i_k} \qquad (k\in [r]).\] Let \(\Lambda_{\mathbf{i}}=(\lambda_{kl})\) be the skew-symmetric matrix determined by \[\lambda_{kl}=-\lambda_{lk}= (w_{\le k}\varpi_{i_k}+\varpi_{i_k},\,\varpi_{i_l}-w_{\le l}\varpi_{i_l}) \qquad \text{for } k>l.\] It is well known that \((\Lambda_{\mathbf{i}},B_{\mathbf{i}})\) is a compatible pair. We denote by \[\mathbf{s}(\mathbf{i}):=\bigl((x_i)_{i\in [r]},\,B_{\mathbf{i}},\,\Lambda_{\mathbf{i}},\,I_{\mathrm{fr}}\bigr)\] the associated quantum seed, and write \[\mathcal{A}_q(\mathbf{i}):=\mathcal{A}_q(\mathbf{s}(\mathbf{i})).\]
Now let \(\mathbf{j}\) be a subword of \(\mathbf{i}\). We write \[\mathbf{j}=(i_{p_1},\dots,i_{p_m})=(j_1,\dots,j_m),\] where \(p_k<p_{k+1}\) for all \(k\in [m-1]\).
For \(k\in [m]\), let \(i:=i_{p_k}\) and define \[a_k:=\#\{s\in [k]\mid i_{p_s}=i\}, \qquad b_k:=\#\{t\in [p_k]\mid i_t=i \text{ and } t\notin \{p_1,\dots,p_k\}\},\] and set \[d_k:=a_k+b_k.\] Then \[p_k=(i,a_k+b_k)=(i,d_k).\] We also define \[\alpha(j,k):=\#\{t\in [k]\mid i_{p_t}=j\}.\]
Definition 4. Let \(\mathbf{i}\) be a word and let \(\mathbf{j}\) be a subword of \(\mathbf{i}\). For each \(k\in [m]\), define a sequence of mutations \(\widetilde{\mu}_k\) by \[\widetilde{\mu}_k= \begin{cases} \mu_{(i,n_i-a_k)}\circ \cdots \circ \mu_{(i,b_k+1)}, & \text{if } a_k+b_k<n_i,\\ \operatorname{Id}, & \text{otherwise}, \end{cases}\] where \(i=i_{p_k}\).
Set \[\widetilde{\mathbf{s}}(\mathbf{j}_k,\mathbf{i}) := \widetilde{\mu}_k\circ \widetilde{\mu}_{k-1}\circ \cdots \circ \widetilde{\mu}_1\bigl(\mathbf{s}(\mathbf{i})\bigr).\] The seed \(\mathbf{s}(\mathbf{j}_k,\mathbf{i})\) is obtained from \(\widetilde{\mathbf{s}}(\mathbf{j}_k,\mathbf{i})\) by deleting the vertices \((j,n_j-\ell+1)\) for all \(0\le \ell\le \alpha(j,k)\), and then freezing the vertices adjacent to the deleted ones in the quiver \(Q_{\widetilde{\mathbf{s}}(\mathbf{j}_k,\mathbf{i})}\).
Denote by \(\widetilde{B}_{\mathbf{j}_k,\mathbf{i}}\) and \(\widetilde{\Lambda}_{\mathbf{j}_k,\mathbf{i}}\) the exchange matrix and skew-symmetric matrix of \(\widetilde{\mathbf{s}}(\mathbf{j}_k,\mathbf{i})\), respectively. Restricting these matrices to the remaining vertices of \(\mathbf{s}(\mathbf{j}_k,\mathbf{i})\) yields a compatible pair \((B_{\mathbf{j}_k,\mathbf{i}},\Lambda_{\mathbf{j}_k,\mathbf{i}})\).
In particular, we write \(\mathbf{s}(\mathbf{j},\mathbf{i}):=\mathbf{s}(\mathbf{j}_m,\mathbf{i})\) and define \[\mathcal{A}(\mathbf{j},\mathbf{i}):=\mathcal{A}(\mathbf{s}(\mathbf{j},\mathbf{i})), \qquad \mathcal{A}_q(\mathbf{j},\mathbf{i}):=\mathcal{A}_q(\mathbf{s}(\mathbf{j},\mathbf{i})),\] \[U(\mathbf{j},\mathbf{i}):=U(\mathbf{s}(\mathbf{j},\mathbf{i})), \qquad U_q(\mathbf{j},\mathbf{i}):=U_q(\mathbf{s}(\mathbf{j},\mathbf{i})).\]
Example 1. Consider type \(A_4\) and the reduced expression \[\overline{w}=(1234123121)\] of \(w_0\). Let \[v=s_2s_3s_1s_2s_1<w_0.\] The corresponding subexpression \(\overline{v}\) of \(v\) inside \(\overline{w}\) is \[(1,\underline{2},\underline{3},4,\underline{1},\underline{2},3,\underline{1},2,1),\] so that \[(p_1,\dots,p_5)=(2,3,5,6,8).\] Moreover, \[a_1=1,\;a_2=1,\;a_3=1,\;a_4=2,\;a_5=2, \qquad b_1=b_2=0,\;b_3=1,\;b_4=0,\;b_5=1.\] It follows that \[\widetilde{\mu}_1=\mu_{(2,2)}\mu_{(2,1)}=\mu_6\mu_2,\qquad \widetilde{\mu}_2=\mu_{(3,1)}=\mu_3,\] \[\widetilde{\mu}_3=\mu_{(1,3)}\mu_{(1,2)}=\mu_8\mu_5,\qquad \widetilde{\mu}_4=\mu_{(2,1)}=\mu_2,\qquad \widetilde{\mu}_5=\mu_{(1,2)}=\mu_5.\]
In this subsection, we introduce the main notation used throughout the paper. This notation is fundamental for the formulation of our results and will be crucial for describing the factors of determinantial modules in \(\mathscr{C}_{v,w}\).
Let \(\overline{w}=(i_1\cdots i_r)\) be a reduced expression of \(w\). Let \(v\leq w\) and \(\overline{v}=(i_{p_1}\cdots i_{p_m})=(j_1\cdots j_m)\) is the left most subexpression of \(v\) in \(\overline{w}\). We define \(w_k=s_{i_1}\cdots s_{i_k}\) and \[\overline{v}_k=\begin{cases} s_{i_{p_1}}\cdots s_{i_{p_t}} &\text{ if } k=p_t\\ v_{p_t} &\text{ if } p_t<k<p_{t+1}. \end{cases}\]
It is easy to see that \(\overline{v}_k\leq w_k\), Hence \(D\binom{w_k\varpi_{i_k}}{\overline{v}_k\varpi_{i_k}}\neq 0\). Following the definition of \(v'_k\) in [1], we define \(v'_k\) inductively by setting \(v'_0=e\) and \[v'_k= \begin{cases} v'_{k-1}s_{i_k} & \text{if } s_{i_k}v_{k-1}^{'-1}v < v_{k-1}^{' -1}v,\\ v'_{k-1} & \text{otherwise}. \end{cases}\]
Lemma 1. Let \(\overline{w}=(i_1\cdots i_r)\) be a reduced expression of \(w\), and let \(v\le w\). Then for all \(k\) we have \[\overline{v}_k = v'_k.\]
Proof. We prove the statement by induction on \(k\).
Step 1: \(k=1\). If \(p_1 \neq 1\), then \(s_{i_1}v > v\). Otherwise, suppose that there exists a reduced expression of \(v\) starting with \(i_1\), and set \(u := s_{i_1}v\).
Since the reduced expression \(\overline{w}\) starts with \(s_{i_1}\), and \[u \le s_{i_1}w.\] there exists a subexpression \(\gamma\) of \((i_2, \dots, i_r)\) which is a reduced expression of \(u\).
It follows that \((i_1,\gamma)\) is a subexpression of \(\overline{w}\) giving a reduced expression of \(v\), which is strictly smaller than \(\overline{v}\). This yields a contradiction. \[\overline{v}_1=v'_1=e.\] If \(p_1=1\), then \(s_{i_1}v<v\), so \[\overline{v}_1=v'_1=s_{i_1}.\]
Induction step. Assume the statement holds for all \(l<k\). We consider two cases.
Case 1: \(k=p_t\) for some \(t\in[m]\).
By induction, \[\overline{v}_{k-1}=v'_{k-1} = s_{i_{p_1}}\cdots s_{i_{p_{t-1}}}.\] Hence \[\overline{v}_{k-1}^{-1}v = s_{i_{p_t}}\cdots s_{i_{p_m}}.\] Since multiplying on the left by \(s_{i_k}=s_{i_{p_t}}\) reduces the length, \[s_{i_k}\overline{v}_{k-1}^{-1}v < \overline{v}_{k-1}^{-1}v.\] Therefore, \[v'_k = v'_{k-1}s_{i_k} = v_{k-1}s_{i_k} = v_k.\]
Case 2: \(p_t<k<p_{t+1}\) for some \(t\in[m]\).
Again by induction, \[\overline{v}_{k-1}=v'_{k-1} = s_{i_{p_1}}\cdots s_{i_{p_t}}.\] We claim that \[s_{i_k}\overline{v}_{k-1}^{-1}v > \overline{v}_{k-1}^{-1}v.\] Suppose instead that \[s_{i_k}\overline{v}_{k-1}^{-1}v < \overline{v}_{k-1}^{-1}v.\] Then \[s_{i_k}\, s_{i_{p_{t+1}}}\cdots s_{i_{p_m}} < s_{i_{p_{t+1}}}\cdots s_{i_{p_m}}.\] Let \(u := s_{i_k}\, s_{i_{p_{t+1}}} \cdots s_{i_{p_m}}\). Since \((i_{p_{t+1}}, \dots, i_{p_m})\) is a subexpression of \((i_{k+1}, \dots, i_r)\), there exists a subexpression \(\gamma\) of \((i_{k+1}, \dots, i_r)\) which gives a reduced expression of \(u\).
Note that \(\overline{v}_{k-1}^{-1}v = s_{i_k}u\). It follows that there exists a reduced expression of \(v\) of the form \[(i_{p_1}, \dots, i_{p_t},\, i_k,\, \gamma),\] which is lexicographically smaller than \(\overline{v}\). This contradicts the minimality of the chosen subexpression. Therefore, \[s_{i_k}\overline{v}_{k-1}^{-1}v > \overline{v}_{k-1}^{-1}v,\] and hence \(v'_k=v'_{k-1}=\overline{v}_k\).
This completes the induction. ◻
For \(l\in [m]\), we denote \[v^l=s_{i_{p_1}}\cdots s_{i_{p_l}} .\] For \(k\in [r]\), define \(\overline{v}_k^l\) by \[\overline{v}_k^l= \begin{cases} \overline{v}_k, & k\le p_l,\\ v^l, & k>p_l . \end{cases}\] This means that \(\overline{v}^l_k\) is given by the leftmost expression of \(v^l\) in \(\overline{w}\). It is straightforward to verify that \(\overline{v}^{\,l-1}_k = \overline{v}^{\,l}_k\) for all \(k < p_l\), and that \[\overline{v}^{\,l-1}_k s_{i_{p_l}} = \overline{v}^{\,l}_k\] for all \(k \ge p_l\).
For each \(j\in I\), we define \[I_j=\{(j,k)\in [r]\mid k\in [n_j]\},\] the subset of \(\overline{w}\) consisting of vertices of color \(j\).
We now introduce a subset of \(I_j\) determined by the positions appearing in \(v\). Define \[I_j^{v} := \{(j,t_k)\in I_j \mid p_k=(j,t_k) \text{ for } k\in [\alpha(j,m)]\} = I_j\cap\{p_1,\dots,p_m\}.\] Thus \(I_j^{v}\) consists of those vertices in \(I_j\) whose positions appear among \(\{p_1,\dots,p_m\}\) in the expression of \(v\). In other words, it records the occurrences of the color \(j\) that contribute to \(v\).
More generally, for \(v^l\) we define the corresponding subset \(I_j^{v^l}\). It is straightforward to see that \[|I_j^{v^l}|=\alpha(j,l),\] that is, the number of vertices of color \(j\) appearing in \(v^l\) equals \(\alpha(j,l)\). For simplicity of notation, we write \(I_j^l := I_j^{v^l}\).
Next we partition the segment \(I_j\) according to the vertices in \(I_j^l\). Recall that \(I_j^l=\{(j,t_1),\dots,(j,t_{\alpha(j,l)})\}\) records the positions of color \(j\) that appear in \(v^l\). These vertices divide the segment \(I_j\) into consecutive subsegments.
More precisely, for \(s\in[0,\alpha(j,l)]\) we define \[I_{j,s}^l= \begin{cases} \{(j,k)\in I_j \mid k\le t_1-1\}, & s=0,\\[4pt] \{(j,k)\in I_j \mid t_s\le k\le t_{s+1}-1\}, & 0<s<\alpha(j,l),\\[4pt] \{(j,k)\in I_j \mid t_{\alpha(j,l)}\le k\le n_j\}, & s=\alpha(j,l). \end{cases}\] Thus each \(I_{j,s}^l\) consists of the vertices of color \(j\) lying between two consecutive elements of \(I_j^l\) (including the boundary cases before the first and after the last such vertex). In this way the segment \(I_j\) is decomposed into disjoint consecutive parts: \[I_j=\bigsqcup_{s\in[0,\alpha(j,l)]} I_{j,s}^l .\] Suppose that \(v^l=v^{l-1}s_i\). In this case, the number of occurrences of the simple reflection \(s_i\) in \(v^l\) increases by one. Hence \[\alpha(i,l)=\alpha(i,l-1)+1.\] Consequently, the set \(I_i^l\) is obtained from \(I_i^{l-1}\) by adding one more vertex corresponding to this new occurrence of color \(i\), namely \[I^l_i=I^{l-1}_i\cup\{(i,t_{\alpha(i,l)})\}.\]
Recall that the sets \(I_{i,s}^l\) form a partition of the segment \(I_i\) determined by the vertices in \(I_i^l\). When the new vertex \((i,t_{\alpha(i,l)})\) is added, it further subdivides the last interval of the previous partition. More precisely, the intervals for \(v^{l-1}\) are related to those for \(v^l\) by \[\label{eq:Iisl} I^{l-1}_{i,s}= \begin{cases} I^l_{i,s}, & s<\alpha(i,l-1),\\[4pt] I^l_{i,\alpha(i,l-1)}\cup I^l_{i,\alpha(i,l)}, & s=\alpha(i,l-1). \end{cases}\tag{3}\] Thus the last segment of the partition for \(v^{l-1}\) splits into two consecutive segments in the partition for \(v^l\).
For \(j\neq i\), the reflection \(s_i\) does not affect the occurrences of color \(j\). Hence the corresponding partitions remain unchanged: \[I^{l-1}_{j,s}=I^l_{j,s} \quad \text{for all } s\in[0,\alpha(j,l)],\] since \(\alpha(j,l-1)=\alpha(j,l)\).
For a subset \(T\subset I_j\), we introduce the shifted set \[T[-s]:=\{(j,k)\in I_j\mid (j,k+s)\in T\}.\] In other words, \(T[-s]\) is obtained from \(T\) by shifting the second index by \(s\) positions to the left along the segment \(I_j\).
Recall that the sets \(I^l_{i,s}\) form consecutive subsegments of \(I_i\) determined by the vertices in \(I_i^l\). Since two neighboring segments are connected by consecutive vertices, their shifted versions intersect in exactly one point. More precisely, one easily verifies that \[I^l_{i,s}[-s]\cap I^l_{i,s+1}[-s-1]=(i,t_{s+1}-s-1).\] However, if we remove the boundary vertex \((i,t_{s+1})\) from the next segment, the intersection disappears. That is, \[I^l_{i,s}[-s]\cap \left(I^l_{i,s+1}\setminus\{(i,t_{s+1})\}\right)[-s-1] =\emptyset .\] In fact, we have \[\left(I^l_{i,s}\setminus\{(i,t_s)\}\right)[-s] = \begin{cases} \{(i,k)\in I_i \mid k \leq t_1 - 1\}, & \text{if } s = 0,\\[4pt] \{(i,k)\in I_i \mid t_s - s + 1 \leq k \leq t_{s+1} - s - 1\}, & \text{if } 0 < s < \alpha(i,l),\\[4pt] \{(i,k)\in I_i \mid t_{\alpha(i,l)} - \alpha(i,l) + 1 \leq k \leq n_i - \alpha(i,l)\}, & \text{if } s = \alpha(i,l). \end{cases}\]
It follows that \[\label{eq:partitionIi} I_{i,\leq l}:=\{(i,k)\in I_i \mid k \leq n_i - \alpha(i,l)\} = \bigsqcup_{s=0}^{\alpha(i,l)} \left(I^l_{i,s}\setminus\{(i,t_s)\}\right)[-s].\tag{4}\] Finally, recall that \(I_j^l\) records the vertices of color \(j\) appearing in \(v^l\). Therefore the complement \(I_j\setminus I_j^l\) consists of all vertices of color \(j\) that do not correspond to these positions. Using the partition of \(I_j\) by the segments \(I_{j,s}^l\), we obtain the disjoint decomposition \[I_j\setminus I_j^l =\bigsqcup_{s\in[0,\alpha(j,l)]} \left(I_{j,s}^l\setminus\{(j,t_s)\}\right).\] Thus the complement of \(I_j^l\) is obtained by removing the distinguished vertices \((j,t_s)\) from each segment \(I_{j,s}^l\).
We define a map \[\label{eq:Phiil} \begin{align} \Phi_{i}^l:I_{i,\leq l}&\to I_i\setminus I_i^l\\ (i,k)&\mapsto (i,k+s), \text{ if }(i,k)\in \left(I^l_{i,s}\setminus\{(i,t_s)\}\right)[-s] \end{align}\tag{5}\] It is straightforward to verify that \(\Phi_i^l\) is a bijection.
Suppose \(i_{p_l}=i\). Since \(I_{j,\leq l}\subset I_{j,\leq l-1}\) for all \(j\in I\), when we restrict \(\Phi_j^{l-1}\) to \(I_{j,\leq l}\), we have \[\Phi_j^{l-1} ((j,k))=\Phi_j^{l}((j,k)) \text{ for all } (i,k)\in I_{j,\leq l} \text{ but} \notin\left(I^l_{i,\alpha(i,l)}\setminus\{(i,t_{\alpha(i,l)})\}\right)[-\alpha(i,l)]\] Let \[\Phi^l :\, \bigsqcup_{i\in I} I_{i,\leq l} \to [r]\setminus\{p_1,\dots,p_l\}\] be the bijection obtained by gluing the maps \(\Phi_i^l\) for all \(i \in I\).
It is straightforwards to see \[\left(I^l_{i,\alpha(i,l)}\setminus\{(i,t_{\alpha(i,l)})\}\right)[-\alpha(i,l)]=\{(i,k)\in I_i\mid b_l+1\leq k\leq n_i-\alpha(i,l)\}.\]
Example 2. Following Example 1, let us consider the reduced expression \[\overline{w}=(1234123121)\] of \(w_0\). Let \[v=s_2s_3s_1s_2s_1<w_0 .\] The leftmost subexpression \(\overline{v}\) of \(v\) in \(\overline{w}\) is \[(1,\underline{2},\underline{3},4,\underline{1},\underline{2},3,\underline{1},2,1),\] and hence \[(p_1,\dots,p_5)=(2,3,5,6,8).\]
| \(/\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) |
|---|---|---|---|---|---|---|---|---|---|---|
| \(w\) | \(w_1\) | \(w_2\) | \(w_3\) | \(w_4\) | \(w_5\) | \(w_6\) | \(w_7\) | \(w_8\) | \(w_9\) | \(w_{10}\) |
| \(\overline{v}\) | \(e\) | \(s_2\) | \(s_2s_3\) | \(s_2s_3\) | \(s_2s_3s_1\) | \(s_2s_3s_1s_2\) | \(s_2s_3s_1s_2\) | \(v\) | \(v\) | \(v\) |
| \(\overline{v}^4\) | \(e\) | \(s_2\) | \(s_2s_3\) | \(s_2s_3\) | \(s_2s_3s_1\) | \(v^4\) | \(v^4\) | \(v^4\) | \(v^4\) | \(v^4\) |
| \(\overline{v}^3\) | \(e\) | \(s_2\) | \(s_2s_3\) | \(s_2s_3\) | \(v^3\) | \(v^3\) | \(v^3\) | \(v^3\) | \(v^3\) | \(v^3\) |
| \(\overline{v}^2\) | \(e\) | \(s_2\) | \(v^2\) | \(v^2\) | \(v^2\) | \(v^2\) | \(v^2\) | \(v^2\) | \(v^2\) | \(v^2\) |
Let \(l=5=\ell(v)\) and \(j_5=1\). Then \[I_1=\{(1,1),\fbox{(1,2)},\fbox{(1,3)},(1,4)\}=\{1,\fbox{5},\fbox{8},10\},\] and \[I_{1,0}^v=\{1\},\qquad I_{1,1}^v=\{5\},\qquad I_{1,2}^v=\{8,10\}.\] It follows that \[I^v_{1,2}[-2]=\{(1,1),(1,2)\}=\{1,5\}.\] We have \(I^v_{i,\leq 5}=\{1,5\}=I_{1,0}^v[0]\cup I_{1,1}^v\setminus\{5\}[-1]\cup I^v_{1,2}\setminus\{8\}[-2]=\{1,5\}\). The map \[\Phi_i^5:\, I^v_{i,\leq 5}\to \{1, 10\}\] is given by \(1\mapsto 1\) and \(5\mapsto 10\).
Let \(I\) be a finite set, and let \(C=(c_{ij})_{i,j\in I}\) be a symmetric Cartan matrix. A Kac–Moody root datum is a tuple \[\mathcal{D} := (I, C, X, Y, \{\alpha_i\}_{i\in I}, \{\alpha_i^\vee\}_{i\in I}),\] where \(X\) is a free \(\mathbb{Z}\)-module of finite rank with \(\mathbb{Z}\)-dual \(Y\), and \(\alpha_i \in X\), \(\alpha_i^\vee \in Y\) satisfy \[\langle \alpha_i, \alpha_j^\vee \rangle = c_{ij}.\]
In this paper, we assume that \[Y = \bigoplus_{i\in I}\mathbb{Z}\alpha_i^\vee, \qquad X = \bigoplus_{i\in I}\mathbb{Z}\varpi_i,\] where \(\varpi_i\) denotes the fundamental weight corresponding to \(i \in I\). Set \[X^+ := \mathbb{Z}_{\ge 0}[\varpi_i \mid i \in I].\]
The Weyl group \(W\) is generated by the simple reflections \(s_i\) for \(i \in I\), acting on \(X\) via the bilinear pairing \(\langle \cdot, \cdot \rangle\). We denote by \(\ell(\cdot)\) the length function on \(W\). The set of real roots is defined by \[\Delta_{\mathrm{re}} := \{ w(\alpha_i) \mid i \in I,\; w \in W \},\] which decomposes as \[\Delta_{\mathrm{re}} = \Delta_{\mathrm{re}}^+ \sqcup \Delta_{\mathrm{re}}^- .\]
The minimal Kac–Moody group \(G\) associated with the root datum \(\mathcal{D}\) over \(\mathbb{C}\) is the group generated by the torus \[T := Y \otimes_\mathbb{Z}\mathbb{C}^*\] and the root subgroups \(U_\alpha \cong \mathbb{C}\) for each real root \(\alpha \in \Delta_{\mathrm{re}}\), subject to the Tits relations. Let \(U^+\) (resp.\(U^-\)) denote the subgroup of \(G\) generated by the root subgroups \(U_\alpha\) (resp.\(U_{-\alpha}\)) for \(\alpha \in \Delta_{\mathrm{re}}^+\). Define the Borel subgroups \[B^+ := \langle T, U^+ \rangle, \qquad B^- := \langle T, U^- \rangle .\]
For each \(i \in I\), one can define elements \(x_i(t), y_i(t) \in G\) such that the assignments \[\begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix} \mapsto x_i(t), \qquad \begin{pmatrix} t & 0 \\ 0 & t^{-1} \end{pmatrix} \mapsto \alpha_i^\vee(t), \qquad \begin{pmatrix} 1 & 0 \\ t & 1 \end{pmatrix} \mapsto y_i(t)\] give a well-defined group homomorphism \(\operatorname{\mathbf{SL}}_2 \to G\).
Define \[\dot{s}_i := x_i(1) y_i(-1) x_i(1) \in G.\] For any \(w \in W\) with a reduced expression \(w = s_{i_1} \cdots s_{i_\ell}\), we set \[\dot{w} := \dot{s}_{i_1} \cdots \dot{s}_{i_\ell} \in G.\] This element is independent of the choice of reduced expression of \(w\).
Let \(\mathcal{B} := G/B^+\) be the flag variety. For \(w, v \in W\), define the Schubert cell \[\mathring{\mathcal{B}}_w := B^+ \dot{w} B^+/B^+\] and the opposite Schubert cell \[\mathring{\mathcal{B}}^{\,v} := B^- \dot{v} B^+/B^+ .\] By the Bruhat decomposition and the Birkhoff decomposition, one has \[\mathcal{B} = \bigsqcup_{w \in W} \mathring{\mathcal{B}}_w = \bigsqcup_{v \in W} \mathring{\mathcal{B}}^{\,v}.\]
Define the open Richardson cell by \[\mathring{\mathcal{B}}_{v,w} := \mathring{\mathcal{B}}_w \cap \mathring{\mathcal{B}}^{\,v}.\] It is well known that \(\mathring{\mathcal{B}}_{v,w} \neq \varnothing\) if and only if \(v \le w\) in the Bruhat order.
Definition 5. Let \(v \le w\) be elements of the Weyl group \(W\), and let \[\overline{w} := (i_1 \cdots i_r)\] be a reduced expression of \(w\). Among all subwords of \(\overline{w}\) that give reduced expressions of \(v\), there exists a unique one that is minimal with respect to the left-to-right lexicographic order. We call this subword the leftmost subexpression of \(v\) in \(\overline{w}\), denoted it by \(\overline{v}\).
Theorem 3. [9] For \(v\leq w\in W\), the coordinate ring \(\mathbb{C}[\mathring{\mathcal{B}}_{v,w}]\) has a upper cluster structure with initial seed \(\mathbf{s}(\overline{v},\overline{w})\). That is \(\mathbb{C}[\mathring{\mathcal{B}}_{v,w}]\cong U(\overline{v},\overline{w})\). Refers to the notation in Definition 4.
For a symmetric Cartan matrix \(C\), let \(U_q(\mathfrak{g})\) be the quantum group generated by \(e_i\), \(f_i\), and \(q^h\) for \(i \in I\) and \(h \in \mathbb{Z}[\alpha_i^\vee]_{i \in I}\). Let \(U_q(\mathfrak{n})\) be the \(\mathbb{Q}(q)\)-subalgebra generated by the \(e_i\) for all \(i \in I\).
Let \(\phi\) be the \(\mathbb{Q}(q)\)-antiautomorphism of \(U_q(\mathfrak{g})\) defined by \[\phi(e_i)=f_i, \qquad \phi(f_i)=e_i, \qquad \phi(q^h)=q^h .\]
For \(\lambda \in X^+\), let \(V(\lambda)\) be the irreducible highest weight module with highest weight \(\lambda\). The dual space \(V(\lambda)^*\) becomes a \(U_q(\mathfrak{g})\)-module via the antiautomorphism \(\phi\); more precisely, \[x \cdot f (v) = f(\phi(x)v), \qquad f \in V(\lambda)^*,\; v \in V(\lambda),\; x \in U_q(\mathfrak{g}).\]
There exists a natural pairing \[\langle - , - \rangle_\lambda : V(\lambda)^* \otimes V(\lambda) \to \mathbb{Q}(q)\] such that \[\langle u_\lambda^*, u_\lambda \rangle_\lambda = 1, \qquad \langle x v , w \rangle_\lambda = \langle v , \phi(x) w \rangle_\lambda ,\] where \(u_\lambda\) denotes the highest weight vector of \(V(\lambda)\) and \(u_\lambda^*\) denotes the lowest weight vector of \(V(\lambda)^*\).
Let \(U_q(\mathfrak{g})^*\) be the \(\mathbb{Q}(q)\)-linear dual of \(U_q(\mathfrak{g})\). The coproduct on \(U_q(\mathfrak{g})\) induces an algebra structure on \(U_q(\mathfrak{g})^*\). For any dominant weight \(\lambda\) and \((v,w) \in V(\lambda)^* \otimes V(\lambda)\), define the matrix coefficient \[f_{v,w} : U_q(\mathfrak{g}) \to \mathbb{Q}(q), \qquad x \mapsto \langle x v , w \rangle_\lambda .\]
Let \(A_q(\mathfrak{g})\) be the \(\mathbb{Q}(q)\)-subalgebra of \(U_q(\mathfrak{g})^*\) consisting of those linear forms \(f\) for which there exist left and right ideals \(I\) and \(I'\) of finite codimension such that \[f(I)=0 \quad\text{and}\quad f(I')=0 .\] By [10], one has the following isomorphism. \[\Phi:\, \bigoplus_{\lambda\in X^+}V(\lambda)^*\otimes V(\lambda)\cong A_q(\operatorname{\mathfrak{g}}), \quad (v,w)\mapsto f_{v,w}.\]
Moreover, it is easy to see that \(A_q(\operatorname{\mathfrak{g}})\) is a bimodule over \(U_q(\operatorname{\mathfrak{g}})\).
Let \(w, v \in W\) be Weyl group elements. Choose reduced expressions \[\overline{w}=(i_1\cdots i_r), \qquad \overline{v}=(j_1\cdots j_m)\] of \(w\) and \(v\), respectively. Define \[w_k=s_{i_1}\cdots s_{i_k}, \qquad v_l=s_{j_1}\cdots s_{j_l}.\] Fix \(\lambda\in X^+\) and set \[b_k=(\lambda,w_{k-1}(\alpha_{i_k})), \qquad c_l=(\lambda,v_{l-1}(\alpha_{j_l})),\] where \((-,-)\) denotes the bilinear form on \(X\) satisfying \((\alpha_i,\alpha_j)=c_{ij}\).
For a dominant weight \(\lambda\), write \[\Delta\binom{\lambda}{\lambda} := f_{u^*_\lambda,u_\lambda}.\] We define the quantum minors \[\Delta\binom{w\lambda}{v\lambda} = f_{i_r}^{(b_r)}\cdots f_{i_1}^{(b_1)} \Delta\binom{\lambda}{\lambda} e_{j_1}^{(c_1)}\cdots e_{j_m}^{(c_m)},\] where \(e_i^{(a)}\) and \(f_i^{(b)}\) denote the \(q\)-divided powers of the Chevalley generators.
Let \(A_q(\mathfrak{n})\) be the \(\mathbb{Z}[\alpha_i]_{i\in I}\)-graded \(\mathbb{Q}(q)\)-dual of \(U_q(\mathfrak{n})\). Define the restriction morphism \[p_{\mathfrak{n}}: A_q(\mathfrak{g}) \longrightarrow A_q(\mathfrak{n})\] by \[p_{\mathfrak{n}}(f)(x)=f(x), \qquad f\in A_q(\mathfrak{g}),\; x\in U_q(\mathfrak{n}).\]
We define the quantum minors \[D\binom{w\lambda}{v\lambda} = p_{\mathfrak{n}}\!\left( \Delta\binom{w\lambda}{v\lambda} \right).\] It is well known that \[\label{eq:Dneq0} D\binom{w\lambda}{v\lambda}\neq 0 \quad\text{if and only if}\quad v\le w.\tag{6}\]
Lemma 2. [11]If \(v<w\), then \(D\binom{w\lambda}{v\lambda}\) is a dual canonical base element in \(A_q(\operatorname{\mathfrak{n}})\).
The following proposition plays an important role in this paper.
Proposition 4 ([11]). Let \(w, v \in W\), and assume \(v s_i > v\) and \(w s_i > w\). Then \[q^A D\binom{ws_i\varpi_i}{vs_i\varpi_i} D\binom{w\varpi_i}{v\varpi_i} = q^{-1+B} D\binom{ws_i\varpi_i}{v\varpi_i} D\binom{w\varpi_i}{vs_i\varpi_i} + \prod_{j\neq i} D\binom{w\varpi_j}{v\varpi_j}^{-c_{ji}},\] where \[A=\left(v s_i(\varpi_i),\, w(\varpi_i)-v(\varpi_i)\right), \qquad B=\left(v(\varpi_i),\, w(\varpi_i)-v s_i(\varpi_i)\right).\]
We remark that our notation for quantum minors differs from that in [11]. One advantage of our notation is that the above identity can be interpreted in terms of a \(2\times 2\) matrix: \[\begin{pmatrix} ws_i\varpi_i & w\varpi_i\\ vs_i\varpi_i & v\varpi_i \end{pmatrix}.\] The left-hand side corresponds to the product of two columns, while the first term on the right-hand side corresponds to the product of the off-diagonal and diagonal entries. This product can be represented diagrammatically as follows: \[\begin{tikzcd} ws_i\varpi_i \ar[dr,dash] & w\varpi_i \ar[dl,dash] \\ vs_i\varpi_i & v\varpi_i \end{tikzcd}\] together with the additional column terms \[\begin{pmatrix} w\varpi_j\\ v\varpi_j \end{pmatrix} \quad (j\ne i).\]
Recall the notations in Section 2.3. We now describe a structural property of the quantum minors associated with the sequence \(\overline{v}^l\). More precisely, we study the relation between the quantum minors \[D\binom{w_p\varpi_{i_p}}{\overline{v}_p^{\,l}\varpi_{i_p}} \quad\text{and}\quad D\binom{w_q\varpi_{i_q}}{\overline{v}_q^{\,l-1}\varpi_{i_q}}.\]
Theorem 5. For \(l\in [m]\), the quantum minor \[D\binom{w_{p_l}\varpi_{i_{p_l}}} {\overline{v}^{\,l}_{p_l}\varpi_{i_{p_l}}}\] is a Laurent monomial in \[\left\{ D\binom{w_k\varpi_{i_k}} {\overline{v}^{\,l-1}_k\varpi_{i_k}} \mid k<p_l\right\} .\]
Proof. Set \(i=i_{p_l}\). Then \[w_{p_l}=w_{p_l-1}s_i, \qquad \overline{v}^{\,l}_{p_l} = \overline{v}^{\,l-1}_{p_l}s_i = \overline{v}^{\,l-1}_{p_l-1}s_i,\] since \(\overline{v}^{\,l-1}_{p_l-1} = \overline{v}^{\,l-1}_{p_l} = v^{l-1}.\)
By Theorem 4, we obtain \[\label{eq:TsystemD} \begin{align} q^A D\binom{w_{p_l}\varpi_i}{\overline{v}^{\,l}_{p_l}\varpi_i} D\binom{w_{p_l-1}\varpi_i}{\overline{v}^{\,l-1}_{p_l}\varpi_i} &= q^{-1+B} D\binom{w_{p_l}\varpi_i}{\overline{v}^{\,l-1}_{p_l}\varpi_i} D\binom{w_{p_l-1}\varpi_i}{\overline{v}^{\,l}_{p_l}\varpi_i} \\ &\quad+ \prod_{j\neq i} D\binom{w_{p_l-1}\varpi_j} {\overline{v}^{\,l-1}_{p_l}\varpi_j}^{-c_{ji}} . \end{align}\tag{7}\]
Step 1: Vanishing of one term.
We claim that \[w_{p_l-1} \not> v^{l}.\] Otherwise, there would exist a leftmost subexpression \((i_{q_1}\cdots i_{q_{l-1}} i_s)\) of \(v^{l}\) inside \((i_1\cdots i_{p_l-1}).\)
We prove inductively that \[q_k=p_k \quad\text{for all } k\le l-1.\]
The key Bruhat relations are \[s_i v^{l-1}<v^{l-1} \;\Longrightarrow\; s_i v^{l}<v^{l},\] and \[s_i v^{l}>v^{l} \;\Longrightarrow\; s_i v^{l-1}>v^{l-1}.\]
Since \(p_1\) is the first index such that \(s_{i_{p_1}}v^{l-1}<v^{l-1}\) (Lemma 1), and \(q_1\) is the first index such that \(s_{i_{q_1}}v^{l}<v^{l}\), the above implications give \(q_1\le p_1\). If \(q_1<p_1\), then we obtain a lexicographically smaller subexpression of \(v\), contradicting the minimality of \(\overline{v}\). Hence \(q_1=p_1\).
Assume \(q_s=p_s\) for all \(s<k\). Set \[u_{k-1}=(v_{k-1}^{\,l-1})^{-1}v^{l-1}.\] A similar Bruhat comparison argument yields \(q_k\le p_k\), and strict inequality again contradicts the leftmost property. Hence \(q_k=p_k\) for all \(k\le l-1\).
This implies that a reduced expression of \(v\) occurs in \(\overline{w}\) strictly before \(p_l\), contradicting the leftmost property. Therefore \[w_{p_l-1} \not> v^{l}.\]
Step 2: Consequence for quantum minors.
By the non-vanishing criterion (6 ), we deduce \[D\binom{w_{p_l-1}\varpi_i} {\overline{v}^{\,l}_{p_l}\varpi_i} =0.\]
Hence equation 7 reduces to \[q^A D\binom{w_{p_l}\varpi_i} {\overline{v}^{\,l}_{p_l}\varpi_i} D\binom{w_{p_l-1}\varpi_i} {\overline{v}^{\,l-1}_{p_l}\varpi_i} = \prod_{j\neq i} D\binom{w_{p_l-1}\varpi_j} {\overline{v}^{\,l-1}_{p_l}\varpi_j}^{-c_{ji}}.\]
Since \(\overline{v}^{\,l-1}_{p_l} = \overline{v}^{\,l-1}_{p_l-1} = v^{l-1},\) we obtain \[q^A D\binom{w_{p_l}\varpi_i} {\overline{v}^{\,l}_{p_l}\varpi_i} D\binom{w_{p_l-1}\varpi_i} {\overline{v}^{\,l-1}_{p_l-1}\varpi_i} = \prod_{j\neq i} D\binom{w_{p_l-1}\varpi_j} {\overline{v}^{\,l-1}_{p_l-1}\varpi_j}^{-c_{ji}}.\]
Finally, since \((p_l-1)(j)^-\) is the last occurrence of \(j\) before \(p_l-1\) in \(\overline{w}\) and \(\overline{v}^{\,l-1}\) is a subexpression, we have \[D\binom{w_{(p_l-1)(j)^-}\varpi_j} {\overline{v}^{\,l-1}_{(p_l-1)(j)^-}\varpi_j} = D\binom{w_{p_l-1}\varpi_j} {\overline{v}^{\,l-1}_{p_l-1}\varpi_j}.\]
Therefore \[D\binom{w_{p_l}\varpi_{i_{p_l}}} {\overline{v}^{\,l}_{p_l}\varpi_{i_{p_l}}}\] is a Laurent monomial in \(\left\{ D\binom{w_k\varpi_{i_k}} {\overline{v}^{\,l-1}_k\varpi_{i_k}} \mid k<p_l\right\}.\) ◻
Proposition 6. Fix \(l\in [m]\) and write \(i_{p_l}=(i,t_{\alpha(i,l)})\). For each \((i,k)\in I_{i,\alpha(i,l)}^l\setminus\{p_l\}\), we have \[\label{eq:Tsystemid} \begin{align} q^A D\binom{w_{(i,k)}\varpi_{i}}{\overline{v}^{\,l}_{(i,k)}\varpi_{i}} D\binom{w_{(i,k-1)}\varpi_i}{\overline{v}^{\,l-1}_{(i,k-1)}\varpi_i} &= q^{-1+B} D\binom{w_{(i,k)}\varpi_i}{\overline{v}^{\,l-1}_{(i,k)}\varpi_i} D\binom{w_{(i,k-1)}\varpi_i}{\overline{v}^{\,l}_{(i,k-1)}\varpi_i} \\ &\quad+ \prod_{j\neq i} D\binom{w_{(i,k-1)}\varpi_j} {\overline{v}^{\,l-1}_{(i,k-1)}\varpi_j}^{-c_{ji}} . \end{align}\qquad{(1)}\]
Proof. Recall that \(p_l=(i,t_{\alpha(i,l)})\) is the last vertex of color \(i\) in \(I_i^l\). Hence for any \((i,k)\in I^l_{i,\alpha(i,l)}\setminus\{p_l\}\) we have \((i,k)>p_l\), which implies \(k>t_{\alpha(i,l)}\).
Since \(v^l=v^{l-1}s_i\), we apply the \(T\)–system relation 7 to the pair of indices \((i,k)\) and \((i,k-1)\). Using the above identities, the relation takes the form \[\begin{align} q^A D\binom{w_{(i,k)}\varpi_{i}}{\overline{v}^{\,l}_{(i,k)}\varpi_{i}} D\binom{w_{(i,k-1)}\varpi_i}{v^{l-1}\varpi_i} &= q^{-1+B} D\binom{w_{(i,k)}\varpi_i}{v^{l-1}\varpi_i} D\binom{w_{(i,k-1)}\varpi_i}{v^{\,l}\varpi_i} \\ &\quad+ \prod_{j\neq i} D\binom{w_{(i,k-1)}\varpi_j} {v^{l-1}\varpi_j}^{-c_{ji}} . \end{align}\]
Substituting \(\overline{v}_{(i,k-1)}^{\,l}=v^l\) and \(\overline{v}_{(i,k)}^{\,l-1}=\overline{v}_{(i,k)}^{\,l}=v^{l-1}\) into this \(T\)–system relation yields the identity ?? . This completes the proof. ◻
We may interpret Proposition ?? pictorially as follows:
\[\label{dia:vl} \scriptsize \begin{tikzcd}[column sep=2.2em,row sep=1.8em] v^{l-1} & (i,d) \arrow[dr, draw=red, -] & (i,d+1) \arrow[dr, draw=red, -] & \cdots & (i,d+k-1) \arrow[dr, draw=red, -] & (i,d+k) \arrow[dr, draw=red, -] & \cdots \\ v^{l} & (i,d) \arrow[ur, draw=blue, -] & (i,d+1) \arrow[ur, draw=blue, -] & \cdots & (i,d+k-1) \arrow[ur, draw=blue, -] & (i,d+k) \arrow[ur, draw=blue, -] & \cdots \end{tikzcd}\tag{8}\]
The pair of terms connected by the red lines corresponds to the left-hand side of ?? . The pair of terms connected by the blue lines corresponds to the first term on the right-hand side of ?? .
Let \(w\in W\) and \(\overline{w}=(i_1\cdots i_r)\) be a reduced expression of \(w\), we define \[E_k^{\overline{w}}:=T_{i_1}\cdots T_{i_{k-1}}(e_{i_k}) \text{ for all }k\in[r],\] where \(T_i\) refers to the braid symmetrizer defined by Lusztig in [12]. Let \(U_q(\operatorname{\mathfrak{n}}(w))\) be the \(\mathbb{Q}(q)\)-subalgebra of \(U_q(\operatorname{\mathfrak{n}})\) generated by \(\{E^{\overline{w}}_k\mid k\in [r]\}\). Let \(A_q(\operatorname{\mathfrak{n}}(w))\) be the dual of \(U_q(\operatorname{\mathfrak{n}}(w))\). Let \(\mathbb{K}= \mathbb{C}[q^{\pm 1}]\). Then one can define the integral form \(A_q(\operatorname{\mathfrak{n}}(w))_{\mathbb{K}}\).
Define a seed by \[\mathbf{s}(\overline{w}) := \left( \left\{ D\binom{w_k\varpi_{i_k}}{\varpi_{i_k}} \right\}_{k\in [r]}, \, B_{\overline{w}}, \, \Lambda_{\overline{w}},\, I_{\operatorname{fr}} \right),\] where we refer to the notation introduced in Section 2.2. When no confusion arises, we simply write \[D_k := D\binom{w_k\varpi_{i_k}}{\varpi_{i_k}}\] for the corresponding quantum minor.
Theorem 7. There exists an algebra isomorphism \[\Psi:\; \mathcal{A}_q(\mathbf{s}(\overline{w})) \xrightarrow{\;\sim\;} A_q(\mathfrak{n}(w)).\] Here \(\mathcal{A}_q(\mathbf{s}(\overline{w}))\) denotes the \(\mathbb{Q}(q)\)-subalgebra of the torus \(\mathbb{Q}(q)[D_k^{\pm1}\mid k\in [r]]\) generated by all cluster variables appearing in seeds obtained from \(\mathbf{s}(\overline{w})\) by finite sequences of mutations.
In this section, we briefly recall the definition of quiver Hecke algebras and the notion of determinantial modules.
Let \(Q=(I,Q_1)\) be a quiver whose associated Cartan matrix \(C\) is symmetric. For \(i,j\in I\), let \(m_{ij}\) denote the number of arrows from \(i\) to \(j\). We define a polynomial \(q_{i,j}(u,v)\in \mathbb{C}[u,v]\) by \[q_{i,j}(u,v)= \begin{cases} 0, & \text{if } i=j,\\ (v-u)^{m_{ij}}(u-v)^{m_{ji}}, & \text{if } i\neq j. \end{cases}\]
Let \[\alpha=\sum_{i\in I}n_i\,\alpha_i\in Q^+ \qquad\text{with}\qquad n=|\alpha|=\sum_{i\in I}n_i.\] We write \(\langle I\rangle_\alpha\) for the set of words \(\mathbf{i}=(i_1,\dots,i_n)\) of weight \(\alpha\), where \(n=|\alpha|\).
Definition 6. The quiver Hecke algebra \(R(\alpha)\) is the associative \(\mathbb{C}\)-algebra generated by \[\{1_{\mathbf{i}}\}_{\mathbf{i}\in \langle I\rangle_\alpha} \cup \{x_1,\dots,x_n\} \cup \{\tau_1,\dots,\tau_{n-1}\},\] subject to the following relations: \[\begin{align} &1_{\mathbf{i}}1_{\mathbf{j}}=\delta_{\mathbf{i},\mathbf{j}}1_{\mathbf{i}}, \qquad \sum_{\mathbf{i}\in \langle I\rangle_\alpha}1_{\mathbf{i}}=1_\alpha,\\ &1_{\mathbf{i}}x_k=x_k1_{\mathbf{i}}, \qquad 1_{\mathbf{i}}\tau_k=\tau_k1_{t_k(\mathbf{i})},\\ &x_kx_l=x_lx_k,\\ &(\tau_kx_l-x_{t_k(l)}\tau_k)1_{\mathbf{i}} = \delta_{i_k,i_{k+1}}(\delta_{k+1,l}-\delta_{k,l})1_{\mathbf{i}},\\ &\tau_k^2 1_{\mathbf{i}} = q_{i_k,i_{k+1}}(x_k,x_{k+1})1_{\mathbf{i}},\\ &\tau_k\tau_l=\tau_l\tau_k \qquad\text{if } |k-l|>1,\\ &(\tau_{k+1}\tau_k\tau_{k+1}-\tau_k\tau_{k+1}\tau_k)1_{\mathbf{i}}\\ &\qquad\qquad = \delta_{i_k,i_{k+2}} \frac{ q_{i_k,i_{k+1}}(x_k,x_{k+1})- q_{i_k,i_{k+1}}(x_{k+2},x_{k+1}) }{ x_k-x_{k+2} } \,1_{\mathbf{i}}. \end{align}\] Here \(t_k\in S_n\) denotes the simple transposition exchanging \(k\) and \(k+1\), acting naturally on words.
The algebra \(R(\alpha)\) carries a natural \(\mathbb{Z}\)-grading given by \[\deg(1_{\mathbf{i}})=0, \qquad \deg(x_j)=2, \qquad \deg(\tau_k1_{\mathbf{i}}) =-(\alpha_{i_k},\alpha_{i_{k+1}}),\] where \((\cdot,\cdot)\) is the symmetric bilinear form associated with the Cartan matrix \(C\).
For \(\alpha,\beta\in Q^+\), set \[e(\alpha)=\sum_{\mathbf{i}\in \langle I\rangle_\alpha}1_{\mathbf{i}}, \qquad e(\beta)=\sum_{\mathbf{j}\in \langle I\rangle_\beta}1_{\mathbf{j}}.\] We write \(e(\alpha,\beta)\in R(\alpha+\beta)\) for the idempotent corresponding to the concatenation of words in \(\langle I\rangle_\alpha\) and \(\langle I\rangle_\beta\).
Let \(R(\alpha)\text{-}\operatorname{gmod}\) denote the category of finite-dimensional graded \(R(\alpha)\)-modules. For \[M\in R(\alpha)\text{-}\operatorname{gmod}, \qquad N\in R(\beta)\text{-}\operatorname{gmod},\] their convolution product is defined by \[M\circ N := R(\alpha+\beta)e(\alpha,\beta)\otimes_{R(\alpha)\otimes R(\beta)}(M\boxtimes N).\]
For \(M\in R(\beta)\text{-}\operatorname{gmod}\), the graded dual space \[M^*:=\operatorname{Hom}_\mathbb{C}(M,\mathbb{C})\] admits an \(R(\beta)\)-module structure defined by \[(r\cdot f)(u):=f(\psi(r)u) \qquad (r\in R(\beta),\;u\in M),\] where \(\psi\) is the \(\mathbb{C}\)-algebra anti-involution of \(R(\beta)\) fixing the generators \[1_\nu,\qquad x_m,\qquad \tau_k\] for \(\nu\in \langle I\rangle_\beta\), \(1\le m\le |\beta|\), and \(1\le k<|\beta|\).
A simple \(R(\beta)\)-module \(M\) is called self-dual if \(M^*\cong M\).
Now set \[R\text{-}\operatorname{gmod} := \bigoplus_{\alpha\in Q^+}R(\alpha)\text{-}\operatorname{gmod},\] and denote by \(K(R\text{-}\operatorname{gmod})\) its Grothendieck group. For a module \(M\in R(\alpha)\text{-}\operatorname{gmod}\), we define its weight by \[\mathop{\mathrm{wt}}(M)=\alpha.\]
Theorem 8 ([13]). There is an isomorphism \[\Phi:\,K(R\text{-}\operatorname{gmod})\xrightarrow{\sim} A_q(\mathfrak n)\] under which the classes of self-dual simple modules correspond to the elements of the dual canonical basis.
In particular, by Lemma 2, for each quantum minor \[D\binom{w\lambda}{v\lambda},\] there exists a corresponding self-dual simple module, which we denote by \[M\binom{w\lambda}{v\lambda}.\] We call these modules determinantial modules.
We say that two simple modules \(M\) and \(N\) strongly commute if \(M\circ N\) is simple. A simple module \(M\) is called real if \(M\circ M\) is simple.
For \(M\in R(\alpha)\text{-}\operatorname{gmod}\) and \(N\in R(\beta)\text{-}\operatorname{gmod}\), Kang–Kashiwara–Kim [14] constructed an intertwining \(R(\alpha+\beta)\)-module homomorphism \[\mathbf{r}_{M,N}: M\circ N \longrightarrow N\circ M,\] called the \(R\)-matrix.
We define \[\Lambda(M,N):=\deg(\mathbf{r}_{M,N}), \qquad \mathfrak{d}(M,N) :=\frac{1}{2}\big(\Lambda(M,N)+\Lambda(N,M)\big).\] The following proposition provides a family of strongly commuting pairs of determinantial modules.
Proposition 9 ([15]). Let \(\lambda,\mu\in X^+\) be dominant weights. Let \(s,s',t,t'\) be Weyl group elements such that \[\ell(s's)=\ell(s')+\ell(s), \qquad \ell(t't)=\ell(t')+\ell(t),\] \[s's\lambda \le t'\lambda, \qquad s'\mu \le t't\mu.\] Then the simple modules \[M\binom{s's\lambda}{t'\lambda} \quad\text{and}\quad M\binom{s'\mu}{t't\mu}\] strongly commute.
Following [7], we have the following result.
Proposition 10 ([7]). For any \(l\in [0,\ell(v)]\), the family of determinantial modules \[\mathscr{T}_l:=\left\{ M\binom{w_p\varpi_{i_p}}{\overline{v}^l_p\varpi_{i_p}} \right\}_{p\in [r]}\] mutually strongly commute.
The following lemma shows that, except for the last segment of color \(i\) determined by \(p_l\), the determinantial modules corresponding to \(\overline{v}^{\,l}\) and \(\overline{v}^{\,l-1}\) coincide. This observation will be used repeatedly in the subsequent arguments.
Lemma 3. For \(l\in [\ell(v)]\) with \(i_{p_l}=i\), and for any \(q\notin I^l_{i,\alpha(i,l)}\), we have \[\label{eq:ml61l-1} M\binom{w_q\varpi_{i_q}}{\overline{v}^l_{q}\varpi_{i_q}} = M\binom{w_q\varpi_{i_q}}{\overline{v}^{\,l-1}_{q}\varpi_{i_q}}.\tag{9}\]
Proof. Recall from Section 2.3 that \[\overline{v}^l_q=\overline{v}_q^{\,l-1} \qquad \text{for all } q<p_l .\] Therefore the equality 9 holds immediately for \(q<p_l\).
Now suppose \(q\ge p_l\) but \(q\notin I^l_{i,\alpha(i,l)}\). By definition, \(I^l_{i,\alpha(i,l)}\) consists of all vertices \((i,k)\) with \(k\ge t_{\alpha(i,l)}\), that is, all indices \(p\ge p_l\) of color \(i\). Hence \(q\) must have color different from \(i\), say \(q=(j,k)\) with \(j\neq i\).
Since \(v^l=v^{l-1}s_i\), we have \[\overline{v}_q^{\,l}\varpi_j = \overline{v}_q^{\,l-1}s_i\varpi_j.\] Because \(j\neq i\), the simple reflection \(s_i\) fixes the fundamental weight \(\varpi_j\), and therefore \[\overline{v}_q^{\,l}\varpi_j = \overline{v}_q^{\,l-1}\varpi_j .\] This implies the equality of the corresponding determinantial modules, and hence 9 holds for such \(q\) as well. ◻
Recall that for \(l\in [m]\) we set \((i,d_l)=p_l\) and consider the set \(I_{i,\alpha(i,l)}^l\setminus\{p_l\}\). For \((i,d_l+k)\in I_{i,\alpha(i,l)}^l\setminus\{p_l\}\), we define \[\begin{align} Y_{l,k} &= \left\{{(i,s)\in I^l_{i,\alpha(i,l)}}\mid\,{d_l+1\le s\le d_l+k}\right\},\\ T_{l,k} &= [r]\setminus \left\{(i,s)\in I_i\mid d_l\le s\le d_l+k-1\right\}. \end{align}\]
Thus \(Y_{l,k}\) consists of the vertices of color \(i\) in \(I_{i,\alpha(i,l)}^l\) whose indices lie between \(d_l+1\) and \(d_l+k\). In other words, it records the initial segment of \(I_{i,\alpha(i,l)}^l\) following the vertex \(p_l=(i,d_l)\), up to the vertex \((i,d_l+k)\).
On the other hand, \(T_{l,k}\) is obtained from the full index set \([r]\) by removing the vertices \((i,s)\) with \(d_l\le s\le d_l+k-1\). Equivalently, \(T_{l,k}\) may be viewed as the complement in \([r]\) of the shifted set \(Y_{l,k}[-1]\). Set \[I_{i,\alpha(i,l),\leq k}^l:=\{(i,s)\in I_i \mid b_l + 1 \leq s \leq b_l + k\}\] It follows from \(d_l = b_l + \alpha(i,l)\) that \[(\Phi_i^l)^{-1}(Y_{l,k}) = I_{i,\alpha(i,l),\leq k}^l,\] and \[\Phi_i^{l-1}\bigl(I_{i,\alpha(i,l),\leq k}^l\bigr) = Y_{l,k}[-1].\] Therefore, there exists a well-defined map \[\label{eq:Phijlk} \Phi_j^{l,k} : I_{j,\leq l-1} \to Y_{l,k} \sqcup T_{l,k}\tag{10}\] such that \[\Phi_j^{l,k}(j,s) = \begin{cases} \Phi_j^l(j,s), & \text{if } (j,s)\in I_{i,\alpha(i,l),\leq k}^l,\\ \Phi_j^{l-1}(j,s), & \text{otherwise}. \end{cases}\]
Proposition 11. For each \((i,d_l+k)\in I_{i,\alpha(i,l)}^l\setminus\{p_l\}\), the modules \[\label{eq:commutevlvl-1} M\binom{w_{(i,d_l+k)}\varpi_{i}}{\overline{v}^l_{(i,d_l+k)}\varpi_i} \quad\text{and}\quad M\binom{w_p\varpi_{i_p}}{\overline{v}^{\,l-1}_p\varpi_{i_p}} \text{ strongly commute for all p\in T_{l,k}.}\qquad{(2)}\]
Proof. First consider the case \(p\notin I^l_{i,\alpha(i,l)}\). By Lemma 3, we have \[M\binom{w_p\varpi_{i_p}}{\overline{v}^{\,l-1}_p\varpi_{i_p}} = M\binom{w_p\varpi_{i_p}}{\overline{v}^{\,l}_p\varpi_{i_p}}.\] Therefore equation ?? follows immediately from Proposition 10.
Next assume that \(p\in T_{l,k}\cap I_{i,\alpha(i,l)}^l\). By definition, this means \[p=(i,s)\quad\text{with}\quad d_l+k\le s\le n_i .\] For such \(p\), we have \[\overline{v}_p^{\,l-1}s_i = v^{l-1}s_i = v^l = \overline{v}_p^{\,l}.\] Moreover, since \(p=(i,s)\) with \(s\ge d_l+k\), we can write \[w_p = w_{(i,d_l+k)}\,u\] for some \(u\in W\). Applying Proposition 9 to this factorization yields equation ?? .
This completes the proof. ◻
Definition 7. Let \(v\le w\) be Weyl group elements. For \(l\in [\ell(v)]\), write \(i_{p_l}=(i,d)\). For each \(k\in [n_i-d]\), define \[\mathscr{T}_{l,k} := \left\{ M\binom{w_p\varpi_{i_p}}{\overline{v}^{\,l-1}_p\varpi_{i_p}} \right\}_{p\in T_{l,k}} \;\cup\; \left\{ M\binom{w_q\varpi_{i}}{\overline{v}^{\,l}_q\varpi_i} \right\}_{q\in Y_{l,k}}.\]
One of the motivations for this definition is to relate it to the mutation sequence \(\widetilde{\mu}_l\) introduced in Definition 4. By Lemma 3, the determinantial modules in \(\mathscr{T}_{l,k}\) that differ from those in \(\mathscr{T}_l\) are precisely the elements in \[\left\{ M\binom{w_p\varpi_{i_p}}{\overline{v}^{\,l-1}_p\varpi_{i_p}} \;\middle|\; p=(i,s)\;\text{with}\;d_l+k\le s\le n_i \right\}.\]
Note that \[\label{eq:Tlk-1Ylk} T_{l,k-1}=T_{l,k}\cup\{(i,d_l+k-1)\}, \qquad Y_{l,k}=Y_{l,k-1}\cup\{(i,d_l+k)\}.\tag{11}\] Thus the sets \(T_{l,k}\) and \(Y_{l,k}\) are obtained from \(T_{l,k-1}\) and \(Y_{l,k-1}\) by removing and adding a single vertex, respectively.
Consequently, we obtain the recursive relation \[\label{eq:Tlkrecursion} \mathscr{T}_{l,k} \setminus \left\{ M\binom{w_{(i,d_l+k)}\varpi_{i}}{\overline{v}^{\,l}_{(i,d_l+k)}\varpi_i} \right\} = \mathscr{T}_{l,k-1} \setminus \left\{ M\binom{w_{(i,d_l+k-1)}\varpi_i}{\overline{v}^{\,l-1}_{(i,d_l+k-1)}\varpi_i} \right\},\tag{12}\] for \((i,d_l+k)\in I_{i,\alpha(i,l)}^l\), where by convention we set \(\mathscr{T}_{l,0}=\mathscr{T}_{l-1}\). It is easy to see \[\label{eq:Phil-1Phil} \Phi_i^{l-1} (\Phi_i^{l})^{-1}((i,d_l+k))=(i,d_l+k-1).\tag{13}\]
Combining Lemma 3 with the above discussion, we further obtain \[\label{eq:Tlni-d} \mathscr{T}_{l,n_i-d_l}\setminus \left\{ M\binom{w_{(i,n_i)}\varpi_i}{\overline{v}^{\,l-1}_{(i,n_i)}\varpi_i} \right\} = \mathscr{T}_{l}\setminus \left\{ M\binom{w_{p_l}\varpi_i}{\overline{v}^{\,l}_{p_l}\varpi_i} \right\}.\tag{14}\]
Proposition 12. For each \(k\in [n_i-d_l]\), the family \(\mathscr{T}_{l,k}\) mutually strongly commutes. Moreover, \[\label{eq:dMlml-1} \mathfrak{d}\!\left( M\binom{w_{(i,d_l+k)}\varpi_{i}}{\overline{v}^{\,l}_{(i,d_l+k)}\varpi_i}, M\binom{w_{(i,d_l+k-1)}\varpi_{i}}{\overline{v}^{\,l-1}_{(i,d_l+k-1)}\varpi_i} \right) =1.\qquad{(3)}\]
Proof. The mutual strong commutativity follows from Proposition 11, Proposition 10 together with the inclusion \[T_{l,k}\subset T_{l,k'} \quad \text{whenever } k'<k.\]
By [15], we have \[\mathfrak{d}\!\left( M\binom{w_{(i,d_l+k)}\varpi_{i}}{\overline{v}^{\,l}_{(i,d_l+k)}\varpi_i}, M\binom{w_{(i,d_l+k-1)}\varpi_{i}}{\overline{v}^{\,l-1}_{(i,d_l+k-1)}\varpi_i} \right) \le 1.\]
On the other hand, by Proposition 6, there exist simple modules \(X\) and \(Y\) such that \[M\binom{w_{(i,d_l+k)}\varpi_{i}}{\overline{v}^{\,l}_{(i,d_l+k)}\varpi_i} \circ M\binom{w_{(i,d_l+k-1)}\varpi_{i}}{\overline{v}^{\,l-1}_{(i,d_l+k-1)}\varpi_i} = q^{m}X+q^{n}Y.\] Applying [15], we deduce \[\mathfrak{d}\!\left( M\binom{w_{(i,d_l+k)}\varpi_{i}}{\overline{v}^{\,l}_{(i,d_l+k)}\varpi_i}, M\binom{w_{(i,d_l+k-1)}\varpi_{i}}{\overline{v}^{\,l-1}_{(i,d_l+k-1)}\varpi_i} \right) >0.\] Combining the two inequalities yields ?? . ◻
Proposition 13. For each \(k\in [0, n_i-d_l]\), the family \[\mathscr{T}_{l,k}\cup \left\{ M\binom{w\varpi_j}{v^s\varpi_j} \;\middle|\; s<l-1,\; j\in I \right\}\] mutually strongly commutes.
Proof. Let \(s<l-1\) and \(j\in I\). We first show that the determinantial modules \(M\binom{w\varpi_j}{v^s\varpi_j}\) strongly commute with all modules in \(\mathscr{T}_{l,k}\).
If \(q\le p_s\), then \[\overline{v}^{\,l-1}_q=\overline{v}^{\,l}_q=\overline{v}^{\,s}_q.\] Hence, by Proposition 10, the module \[M\binom{w_q\varpi_{i_q}}{\overline{v}^{\,l-1}_q\varpi_{i_q}}\] strongly commutes with \(M\binom{w\varpi_j}{v^s\varpi_j}\).
Next consider the case \(q>p_s\). In this situation we can write \[\overline{v}_q^{\,l-1}=v^s u_1 \quad\text{and}\quad \overline{v}_q^{\,l}=v^s u_2\] for some \(u_1,u_2\in W\). On the other hand, since \(w_q\le w\), we have \(w_q w'=w\) for some \(w'\in W\). Therefore Proposition 9 implies that \(M\binom{w\varpi_j}{v^s\varpi_j}\) strongly commutes with any determinantial module in \(\mathscr{T}_{l,k}\).
Finally, applying Proposition 9 once again shows that \[M\binom{w\varpi_j}{v^s\varpi_j} \quad\text{and}\quad M\binom{w\varpi_j}{v^t\varpi_j}\] strongly commute for all \(s,t<l-1\). Hence all modules in the stated family mutually strongly commute. ◻
Definition 8. For two strongly commuting familys \(\mathscr{T}=\{T_i\}_{i\in K}\) and \(\mathscr{S}=\{S_j\}_{j\in J}\) in \(R-\operatorname{gmod}\), by we call \(\mathscr{S}\) is generated by \(\mathscr{T}\), if for any module \(S_j\in \mathscr{S}\) the element \(\Phi(S_j)\) is a Laurent monomial of \(\{\Phi(T_i)\}_{i\in K}\), refers to 8 for the notation \(\Phi\).
Proposition 14. For \(l\in [\ell(v)]\), the family \[\mathscr{T}_{l,n_i-d_l} \setminus \left\{ M\binom{w_{(i,n_i)}\varpi_i}{\overline{v}^{\,l-1}_{(i,n_i)}\varpi_i} \right\}\] generates the commuting family \(\mathscr{T}_l\).
Proof. By 14 , it suffices to show that \[\Phi\!\left( M\binom{w_{p_l}\varpi_i}{\overline{v}^{\,l}_{p_l}\varpi_i} \right)\] is a Laurent monomial in the elements of \(\Phi(\mathscr{T}_{l,n_i-d})\).
This follows directly from Theorem 5 together with the definition of \(\mathscr{T}_{l,n_i-d_l}\), which expresses \[\Phi\!\left( M\binom{w_{p_l}\varpi_i}{\overline{v}^{\,l}_{p_l}\varpi_i} \right)\] as a Laurent monomial in the generators \(\Phi(\mathscr{T}_{l,n_i-d})\). ◻
In this section, we introduce the monoidal categories associated with quiver Hecke algebras and state our main results.
For a module \(M\) of \(R(\alpha)\), we define \[\mathbf{W}(M)=\left\{{\beta\in Q^+}\mid\,{e(\beta,\alpha-\beta)M\neq 0}\right\})\] and \[\mathbf{W}^*(M)=\left\{{\beta\in Q^+}\mid\,{e(\alpha-\beta,\beta)M\neq 0}\right\}.\]
Let \(w\) be a Weyl group element, and \(\overline{w}=(i_1\cdots i_r)\) be a reduced exrpession of \(w\). We have \[\Delta^+\cap w\Delta^-=\left\{{\beta_k=s_{i_1}\cdots s_{i_{k-1}}\alpha_{i_k}}\mid\,{k\in [r]}\right\}.\] There exists an convex order \(\prec_{\overline{w}}\) on \(\Delta^+\) such that \[\beta_1\prec_{\overline{w}}\beta_2\prec_{\overline{w}}\cdots\prec_{\overline{w}}\beta_r\prec_{\overline{w}}\beta, \text{ for all }\beta\in w\Delta^+\cap \Delta^+\] Here convex means for any two positive roots \(\alpha,\beta\) such that \(\alpha+\beta\in \Delta^+\) then \(\alpha\prec\alpha+\beta\prec \beta\) or \(\beta\prec\alpha+\beta\prec \alpha\).
Definition 9. Let \(v,w\in W\) with \(v\le w\).
The full subcategory \(\mathscr{C}_w\) of \(R\operatorname{-gmod}\) consists of modules \(M\) such that \[\mathbf{W}(M) \subset \operatorname{span}_{\mathbb{R}_{\ge 0}} \bigl(\Delta^+ \cap w\Delta^- \bigr).\]
The full subcategory \(\mathscr{C}_{*,v}\) of \(R\operatorname{-gmod}\) consists of modules \(M\) such that \[\mathbf{W}^*(M) \subset \operatorname{span}_{\mathbb{R}_{\ge 0}} \bigl(\Delta^+ \cap v\Delta^+ \bigr).\]
We define \[\mathscr{C}_{w,v} := \mathscr{C}_w \cap \mathscr{C}_{*,v}.\]
Proposition 15. [7] The category \(\mathscr{C}_w, \mathscr{C}_{*,v}\), and \(\mathscr{C}_{w,v}\) are under taking subquotients, extensions, convolution products and grading shifts.
We will use the following lemma throughout the remainder of the paper.
Lemma 4. Let \(L=M\circ N\).
If \(L\in \mathscr{C}_w\), then \(M\in \mathscr{C}_w\).
If \(L\in \mathscr{C}_{*,v}\), then \(N\in \mathscr{C}_{*,v}\).
Proof. Let \(\operatorname{wt}(M)=\alpha\) and \(\operatorname{wt}(N)=\beta\). Suppose that \(e(\gamma,\alpha-\gamma)M\neq 0\). We claim that \[e(\gamma,\alpha+\beta-\gamma)L\neq 0.\]
Observe that the idempotent \(e(\gamma,\alpha-\gamma,\beta)\) appears as a summand of \(e(\gamma,\alpha+\beta-\gamma)\). Moreover, \[e(\gamma,\alpha-\gamma,\beta)\, R(\alpha+\beta)\, e(\alpha,\beta)\] contains \(e(\gamma,\alpha-\gamma,\beta)\). Hence \[\begin{align} e(\gamma,\alpha-\gamma,\beta)L &= e(\gamma,\alpha-\gamma,\beta) R(\alpha+\beta) e(\alpha,\beta) \otimes (M\boxtimes N) \\ &\supset e(\gamma,\alpha-\gamma,\beta)(M\boxtimes N). \end{align}\] Since \(e(\gamma,\alpha-\gamma)M\neq 0\), the right-hand side is nonzero, and thus \[e(\gamma,\alpha+\beta-\gamma)L\neq 0.\]
This proves that \(\mathbf{W}(M)\subset \mathbf{W}(L)\). Therefore, if \(L\in \mathscr{C}_w\), then \[\mathbf{W}(M) \subset \mathbf{W}(L) \subset \operatorname{span}_{\mathbb{R}_{\ge 0}}(\Delta^+\cap w\Delta^-),\] and hence \(M\in \mathscr{C}_w\).
The second statement is proved similarly. Indeed, one shows that \[\mathbf{W}^*(N)\subset \mathbf{W}^*(L),\] and thus \(L\in \mathscr{C}_{*,v}\) implies \(N\in \mathscr{C}_{*,v}\). ◻
Theorem 16 ([15]). The Grothendieck ring \(K(\mathscr{C}_w)\) provides a monoidal categorification of \[A_q(\mathfrak{n}(w))_{\mathbb{K}}.\] In particular, every cluster monomial corresponds to the class of a self-dual simple object in \(\mathscr{C}_w\). The initial monoidal seed is \[\label{eq:initialseed} \mathbf{s}(\overline{w}) := \left( \left\{ M\binom{w_k\varpi_{i_k}}{\varpi_{i_k}} \right\}_{k\in [r]}, \, B_{\overline{w}}, \,\Lambda_{\overline{w}},\, I_{\operatorname{fr}} \right),\tag{15}\] We denote by \(M_k\) the simple module \(M\binom{w_k\varpi_{i_k}}{\varpi_{i_k}}\).
For the category \(\mathscr{C}_{w,v}\), we have the following result.
Theorem 17. [7], [8], [1] Let \(v\le w\) and let \(\overline{w}=(i_1\cdots i_r)\) be a reduced expression of \(w\). Then:
The determinantial modules \[M\binom{w_k\varpi_{i_k}}{v_k\varpi_{i_k}}\] belong to \(\mathscr{C}_{w,v}\).
The nongraded Grothendieck ring of the localized category \[\widetilde{\mathscr{C}}_{w,v} := \mathscr{C}_w \left[ M\binom{w\varpi_j}{v\varpi_j}^{-1} \;\middle|\; j\in I \right]\] is isomorphic to \[\mathbb{C}[\mathring{\mathcal{B}}_{v,w}].\]
To determine which determinantial modules do not belong to \(\mathscr{C}_{w,v}\), we shall use the following lemma.
Lemma 5. Let \(v\leq vs_i\leq w\). Then \[M\binom{w\varpi_i}{v\varpi_i}\notin \mathscr{C}_{w,vs_i}.\]
Proof. By [15], we have \[M\binom{w\varpi_i}{v\varpi_i} = M\binom{w\varpi_i}{vs_i\varpi_i} \bigtriangledown M\binom{vs_i\varpi_i}{v\varpi_i}.\] Here \(N\bigtriangledown L\) refers to the head of the module \(N\circ L\). Moreover, \[\mathbf{W}^*\!\left(M\binom{w\varpi_i}{vs_i\varpi_i}\right) \subset \operatorname{span}_{\mathbb{R}_{\geq 0}}(\Delta^+\cap vs_i\Delta^+),\] and \(M\binom{vs_i\varpi_i}{v\varpi_i}\) is the root module of weight \(v\alpha_i\), hence is cuspidal. Therefore, \[\mathbf{W}\!\left(M\binom{vs_i\varpi_i}{v\varpi_i}\right) \subset \operatorname{span}_{\mathbb{R}_{\geq 0}}(\Delta^+\cap vs_i\Delta^-).\] In particular, \[\mathbf{W}^*\!\left(M\binom{w\varpi_i}{vs_i\varpi_i}\right) \cap \mathbf{W}\!\left(M\binom{vs_i\varpi_i}{v\varpi_i}\right) =0.\] It follows from [16] that \[e(vs_i\varpi_i-w\varpi_i,\,v\alpha_i) \Bigl( M\binom{w\varpi_i}{vs_i\varpi_i} \circ M\binom{vs_i\varpi_i}{v\varpi_i} \Bigr) = M\binom{w\varpi_i}{vs_i\varpi_i} \boxtimes M\binom{vs_i\varpi_i}{v\varpi_i}.\]
Now let \(M\) be a proper submodule of \[M\binom{w\varpi_i}{vs_i\varpi_i} \circ M\binom{vs_i\varpi_i}{v\varpi_i}.\] We claim that \[e(vs_i\varpi_i-w\varpi_i,\,v\alpha_i)M=0.\] Indeed, if this were not the case, then \[e(vs_i\varpi_i-w\varpi_i,\,v\alpha_i)M = M\binom{w\varpi_i}{vs_i\varpi_i} \boxtimes M\binom{vs_i\varpi_i}{v\varpi_i},\] since the latter is simple. This would imply \[M= M\binom{w\varpi_i}{vs_i\varpi_i} \circ M\binom{vs_i\varpi_i}{v\varpi_i},\] contradicting the assumption that \(M\) is proper. Hence \[e(vs_i\varpi_i-w\varpi_i,\,v\alpha_i) \Bigl( M\binom{w\varpi_i}{vs_i\varpi_i} \bigtriangledown M\binom{vs_i\varpi_i}{v\varpi_i} \Bigr) = M\binom{w\varpi_i}{vs_i\varpi_i} \boxtimes M\binom{vs_i\varpi_i}{v\varpi_i}.\] It follows that \[v\alpha_i\in \mathbf{W}^*\!\left(M\binom{w\varpi_i}{v\varpi_i}\right).\] Since \[v\alpha_i\notin \Delta^+\cap vs_i\Delta^+,\] we conclude that \[M\binom{w\varpi_i}{v\varpi_i}\notin \mathscr{C}_{w,vs_i}.\] ◻
The following proposition will be used repeatedly in the rest of the paper.
Proposition 18 ([17]). Let \(\mathscr{T}=(N_i)_{i\in [r]}\) be a monoidal seed of \(\mathscr{C}_w\), and let \(M\) be a simple module in \(\mathscr{C}_w\). Assume that \(M\) strongly commutes with \(N_j\) for all \(j\neq k\). Then \(M\) is a cluster monomial with respect to either the seed \(\mathscr{T}\) or the mutated seed \(\mu_k\mathscr{T}\).
Definition 10. Let \(M\) and \(N\) be simple modules. We say that \(M\) is a factor of \(N\) if there exists a simple module \(L\) such that \[N \cong M\circ L,\] and \(M\) and \(L\) strongly commute. In this case, we write \(M\mid N\).
Lemma 6. If a simple module \(M\) is a factor of a simple module \(N\), then \[N\in \mathscr{C}_{w,v}\quad \Longrightarrow\quad M\in \mathscr{C}_{w,v}.\] Equivalently, \[M\notin \mathscr{C}_{w,v}\quad \Longrightarrow\quad N\notin \mathscr{C}_{w,v}.\]
Proof. By Lemma 4, the relation \[N=M\circ L\in \mathscr{C}_w\] implies that \(M\in \mathscr{C}_w\).
Since \(M\) and \(L\) strongly commute, we have \[M\circ L \cong L\circ M.\] Hence, if \(N\in \mathscr{C}_{*,v}\), then Lemma 4 again implies that \(M\in \mathscr{C}_{*,v}\).
Therefore, \[M\in \mathscr{C}_w\cap \mathscr{C}_{*,v} = \mathscr{C}_{w,v}.\] ◻
For \(v\leq w\), let \(\overline{v}=(i_{p_1}\cdots i_{p_m})\) be the leftmost subexpression of a reduced expression \(\overline{w}=(i_1\cdots i_r)\) of \(w\). For each \(l\in [m]\), consider the seed \[\widetilde{\mathbf{s}}(\overline{v}_l,\overline{w}),\] defined in Definition 4. By construction, this seed is obtained from the initial seed \(\mathbf{s}(\overline{w})\) in 15 by a sequence of mutations. We denote by \(M_s^l\) the cluster variable at the vertex \(s\) of the seed \(\widetilde{\mathbf{s}}(\overline{v}_l,\overline{w})\). If \(s=(i,d)\), we also write \(M_{(i,d)}^l\).
We begin with the seed \(\widetilde{\mathbf{s}}(\overline{v}_1,\overline{w})\). Recall that \(p_1=(i,d_1), \qquad p_1^{\max}=(i,n_i).\) The mutation sequence \(\widetilde{\mu}_1\) consists of the mutations \(\mu_{(i,k)}\) for \((i,k)\in \bigl(I^1_{i,1}\setminus\{p_1\}\bigr)[-1].\) For simplicity, we write \(\mu_{(i,k)}\) simply as \(\mu_k\). Then \(\widetilde{\mu}_1=\mu_{n_i-1}\circ \cdots \circ \mu_{b_1+1}.\)
Lemma 7. For every \((i,k)\in \bigl(I^1_{i,1}\setminus\{p_1\}\bigr)[-1]\), one has \[\label{eq:M1ik} M^1_{(i,k)} \mid M\binom{w_{(i,k+1)}\varpi_i}{s_i\varpi_i} \qquad\text{with multiplicity }1.\tag{16}\] Moreover, the determinantal modules in \[\mathscr{T}_1\setminus \left\{ M\binom{w_{p_1}\varpi_{i_{p_1}}}{s_i\varpi_{p_1}} \right\}\] is a cluster monomial in the seed \(\mathbf{s}(\overline{v}_1,\overline{w})\).
Proof. We argue inductively along the mutation sequence \(\widetilde{\mu}_1\).
First consider the initial step, corresponding to the family \(\mathscr{T}_{1,1}\). One checks directly that \[Y_{1,1}=\{(i,d_1+1)\}, \qquad T_{1,1}=[r]\setminus\{(i,d_1)\}.\] By Proposition 12, the module \[M\binom{w_{(i,d_1+1)}\varpi_i}{s_i\varpi_i}\] strongly commutes with \(M_s\) for all \(s\neq (i,d_1)\). Hence, by Proposition 18, it is a cluster monomial with respect to either the seed \(\mathbf{s}(\overline{w})\) or the seed \(\mu_{d_1}\mathbf{s}(\overline{w})\).
On the other hand, Proposition 12 also shows that \[M\binom{w_{(i,d_1+1)}\varpi_i}{s_i\varpi_i}\] does not strongly commute with \(M_{(i,d_1)}\). Therefore it cannot be a cluster monomial in the seed \(\mathbf{s}(\overline{w})\), and hence must be a cluster monomial in the mutated seed \(\mu_{d_1}\mathbf{s}(\overline{w})\). It follows that \(M^1_{(i,d_1)}\) occurs as a factor.
Let \(n\) be the multiplicity of \(M^1_{(i,d_1)}\) in this factorization. Then \[\mathfrak{d}\left( M\binom{w_{(i,d_1+1)}\varpi_i}{s_i\varpi_i}, M\binom{w_{(i,d_1)}\varpi_i}{\varpi_i} \right) = n\,\mathfrak{d}\left( M^1_{(i,d_1)}, M\binom{w_{(i,d_1)}\varpi_i}{\varpi_i} \right) \ge n.\] By Proposition 12, we conclude that \(n=1\). Thus \[M^1_{(i,d_1)} \mid M\binom{w_{(i,d_1+1)}\varpi_i}{s_i\varpi_i}\] with multiplicity one.
Moreover, no other determinantial module in \(\mathscr{T}_{1,1}\) can contain \(M_{(i,d_1)}\) as a factor; otherwise the modules in \(\mathscr{T}_{1,1}\) would fail to commute pairwise. Hence \(\mathscr{T}_{1,1}\) is generated by the seed \(\mu_{d_1}\mathbf{s}(\overline{w})\). In particular, for every cluster variable \(X_{(j,q)}\) of \(\mu_{d_1}\mathbf{s}(\overline{w})\), there is a determinantial module in \(\mathscr{T}_{1,1}\) indexed by \(\Phi_j^{1,1}(j,q)\) having \(X_{(j,q)}\) as a factor, and every determinantial module in \(\mathscr{T}_{1,1}\) is a cluster monomial in the seed \(\mu_{d_1}\mathbf{s}(\overline{w})\).
We now proceed by induction. Suppose that, for some \(t\ge 2\), the statement has been proved up to step \(t-1\), and that \(\mathscr{T}_{1,t-1}\) is generated by the seed \[\mu_{d_1+t-2}\cdots\mu_{d_1}\mathbf{s}(\overline{w}).\] Equivalently, every cluster variable of this seed is a multiplicity-one factor of the determinantial module in \(\mathscr{T}_{1,t-1}\) indexed by the corresponding map \(\Phi^{1,t-1}\).
Consider the family \(\mathscr{T}_{1,t-1}\). By 12 , we have \[\mathscr{T}_{1,t}\setminus \left\{ M\binom{w_{(i,d_1+t)}\varpi_i}{s_i\varpi_i} \right\} = \mathscr{T}_{1,t-1}\setminus\{M_{(i,d_1+t-1)}\}.\] Hence, by the induction hypothesis, every cluster variable of \[\mu_{d_1+t-2}\cdots\mu_{d_1}\mathbf{s}(\overline{w})\] other than \(M_{(i,d_1+t-1)}\) already appears as a factor of a determinantial module in \(\mathscr{T}_{1,t}\setminus\{M\binom{w_{(i,d_1+t)}\varpi_i}{s_i\varpi_i}\}\).
Now apply the same argument as in the initial step. Using Proposition 12, Proposition 18, and the fact that \(M_{(i,d_1+t-1)}\) is a cluster variable of the seed \[\mu_{d_1+t-2}\cdots\mu_{d_1}\mathbf{s}(\overline{w}),\] we obtain that \[M\binom{w_{(i,d_1+t)}\varpi_i}{s_i\varpi_i}\] is a cluster monomial in the seed \[\mu_{d_1+t-1}\cdots\mu_{d_1}\mathbf{s}(\overline{w}),\] and that \(M^1_{(i,d_1+t-1)}\) appears in it as a factor with multiplicity one.
By the definition of \(\Phi_j^{1,t}\) and equations 11 and 13 , it follows that every cluster variable \(X_{(j,s)}\) in the seed \[\mu_{d_1+t-1}\cdots\mu_{d_1}\mathbf{s}(\overline{w})\] is a multiplicity-one factor of a determinantial module in \(\mathscr{T}_{1,t}\) indexed by \(\Phi_j^{1,t}(j,s)\).
Furthermore, no other determinantial module in \(\mathscr{T}_{1,t}\) can contain \(M_{(i,d_1+t-1)}\) as a factor; otherwise the modules in \(\mathscr{T}_{1,t}\) would fail to commute pairwise. Therefore \(\mathscr{T}_{1,t}\) is generated by the seed \[\mu_{d_1+t-1}\cdots\mu_{d_1}\mathbf{s}(\overline{w}).\]
Iterating this argument up to \(t=n_i-d_1\), we conclude that \(\mathscr{T}_{1,n_i-d_1}\) is generated by \[\widetilde{\mu}_1\mathbf{s}(\overline{w}).\] By 14 , for every \((j,k)\neq (i,n_i)\), the cluster variable \(M^1_{(j,k)}\) of \(\widetilde{\mu}_1\mathbf{s}(\overline{w})\) is a multiplicity-one factor of the determinantial module in \(\mathscr{T}_1\) indexed by \(\Phi_j^1(j,k)\). Moreover, every determinantial module in \[\mathscr{T}_1\setminus \left\{ M\binom{w_{p_1}\varpi_{i_{p_1}}}{s_i\varpi_{p_1}} \right\}\] is a cluster monomial in the seed \(\widetilde{\mu}_1\mathbf{s}(\overline{w})\).
Finally, Lemma 5 shows that \[M\binom{w\varpi_{i_{p_1}}}{\varpi_{i_{p_1}}}\notin \mathscr{C}_{w,v^1}.\] Hence this module cannot occur as a factor of any determinantial module in \(\mathscr{T}_1\). Therefore every determinantial module in \[\mathscr{T}_1\setminus \left\{ M\binom{w_{p_1}\varpi_{i_{p_1}}}{s_i\varpi_{p_1}} \right\}\] is already a cluster monomial in the seed \(\mathbf{s}(\overline{v}^1,\overline{w})\). This proves the lemma. ◻
Theorem 19. For \(l\in [m]\) and \(j\in I\), let \((j,k)\in I_{j,\le l}\), and write \[\Phi_j^l((j,k))=(j,k+s),\] where \(\Phi_j^l\) is the map defined in 5 . Then \[\label{eq:Mjsl} M_{(j,k)}^l \mid M\!\binom{w_{(j,k+s)}\varpi_j} {\overline{v}^{\,l}_{(j,k+s)}\varpi_j} \qquad\text{with multiplicity }1.\tag{17}\] Moreover, for every \(q\notin \{p_1,\dots,p_l\}\), the determinantial module \[M\!\binom{w_q\varpi_{i_q}}{\overline{v}^{\,l}_q\varpi_{i_q}}\] is a cluster monomial in the seed \(\mathbf{s}(\overline{v}^{\,l},\overline{w})\).
Proof. We first prove 17 by induction on \(l\).
Step 1: the case \(l=1\). Let \(i=i_{p_1}\). Then \[I_i=I_{i,0}^1\sqcup I_{i,1}^1, \qquad I_j=I_{j,0}^1 \quad (j\neq i),\] and, since \(p_1=(i,b_1+1)\), we have \(t_1=b_1+1\). More precisely, \[I_{i,0}^1=\{(i,k)\in I_i\mid (i,k)<p_1\}, \qquad I_{j,0}^1=I_j \quad (j\neq i).\]
For every \(q\in I_{i,0}^1\cup I_{j,0}^1\), we have \[M_q^1=M_q^0 = M\!\binom{w_q\varpi_{i_q}}{\overline{v}^{\,0}_q\varpi_{i_q}},\] and Lemma 3 gives \[M_q^1 = M\!\binom{w_q\varpi_{i_q}}{\overline{v}^{\,1}_q\varpi_{i_q}}.\]
Next, \[I_{i,1}^1\setminus\{p_1\} = \{(i,k)\in I_i\mid b_1+2\le k\le n_i\}.\] Hence Lemma 7 yields \[M_{(i,k)}^1 \mid M\!\binom{w_{(i,k+1)}\varpi_i}{\overline{v}^{\,1}_{(i,k+1)}\varpi_i} \qquad \text{for }k\in [b_1+1,n_i-1].\] Therefore the theorem holds for \(l=1\).
Step 2: the induction step for the divisibility statement. Assume that the theorem holds for \(l-1\), and set \(i=i_{p_l}\).
We first treat the case \(j\neq i\). By the induction hypothesis, \[\label{eq:induction-jneqi} M_{(j,k)}^{\,l-1} \mid M\!\binom{w_{(j,k+s)}\varpi_j} {\overline{v}^{\,l-1}_{(j,k+s)}\varpi_j} \qquad \text{for all }(j,k+s)\in I_{j,s}^{\,l-1}\setminus\{(j,t_s)\}.\tag{18}\] Since \(j\neq i\), we have \(\alpha(j,l)=\alpha(j,l-1)\), and hence \[I_{j,s}^{\,l}=I_{j,s}^{\,l-1} \qquad \text{for all } s\in [0,\alpha(j,l)].\] Together with Lemma 3, this implies \[M_{(j,k)}^{\,l}=M_{(j,k)}^{\,l-1} \mid M\!\binom{w_{(j,k+s)}\varpi_j} {\overline{v}^{\,l}_{(j,k+s)}\varpi_j} \qquad \text{for all } (j,k+s)\in I_{j,s}^{\,l}\setminus\{(j,t_s)\}.\]
Now consider the color \(i=i_{p_l}\). Note that \[d_l=b_l+\alpha(i,l)\] and \[\bigl(I_{i,\alpha(i,l)}^l\setminus\{p_l\}\bigr)[-\alpha(i,l)] = \{(i,b_l+1),\dots,(i,n_i-\alpha(i,l))\}.\] Thus the mutation sequence \(\widetilde{\mu}_l\) is precisely the sequence of mutations at these vertices.
For \(q\in \bigl(I_{i,s}^l\setminus\{(i,t_s)\}\bigr)[-s]\) with \(s<\alpha(i,l)\), we therefore have \[M_q^l=M_q^{l-1}.\] Moreover, \[\overline{v}^{\,l-1}_{(i,k+s)}=\overline{v}^{\,l}_{(i,k+s)} \qquad \text{for } (i,k+s)\in I_{i,s}^l\setminus\{(i,t_s)\},\;s<\alpha(i,l),\] and \(I_{i,s}^l\subset I_{i,s}^{\,l-1}\) by 3 . Hence the induction hypothesis and Lemma 3 give \[M_{(i,k)}^l = M_{(i,k)}^{\,l-1} \mid M\!\binom{w_{(i,k+s)}\varpi_i} {\overline{v}^{\,l}_{(i,k+s)}\varpi_i} \qquad \text{for all } (i,k+s)\in I_{i,s}^l\setminus\{(i,t_s)\},\;s<\alpha(i,l).\]
It remains to analyze the new layer corresponding to \(s=\alpha(i,l)\). By 12 , \[\label{eq:Tl1-rec} \mathscr{T}_{l,1}\setminus \left\{ M\!\binom{w_{(i,d_l+1)}\varpi_i} {\overline{v}^{\,l}_{(i,d_l+1)}\varpi_i} \right\} = \mathscr{T}_{l-1}\setminus \left\{ M\!\binom{w_{(i,d_l)}\varpi_i} {\overline{v}^{\,l-1}_{(i,d_l)}\varpi_i} \right\}.\tag{19}\] Since \[(i,d_l)\in I_{i,\alpha(i,l-1)}^{\,l-1}, \qquad \alpha(i,l-1)=\alpha(i,l)-1,\] the induction hypothesis shows that \[M_{(i,b_l+1)}^{\,l-1} \mid M\!\binom{w_{(i,d_l)}\varpi_i} {\overline{v}^{\,l-1}_{(i,d_l)}\varpi_i}.\]
On the other hand, every other cluster variable of \(\widetilde{\mathbf{s}}(\overline{v}^{\,l-1},\overline{w})\) appears as a factor of some determinantial module in the family \[\mathscr{T}_{l-1}\cup \left\{ M\!\binom{w\varpi_j}{\overline{v}^{\,s}\varpi_j} \;\middle|\; j\in I,\;s<l-1 \right\}.\] Indeed, let us consider a cluster variable \(M_{(j,k)}^{\,l-1}\) such that \[k+\alpha(j,l-1)>n_j.\] By the induction hypothesis, this module is not a factor of any determinantial module in \(\mathscr{T}_{l-1}\). For such a pair \((j,k)\), there exists an integer \(s<l-1\) such that \[k+\alpha(j,s)=n_j.\] Since the cluster variable remains unchanged up to stage \(s\), we have \[M_{(j,k)}^{\,l-1}=M_{(j,k)}^{\,s}.\] Applying the induction hypothesis again, we see that \(M_{(j,k)}^{\,s}\) appears as a factor of \[M\binom{w\varpi_j}{\overline{v}^{\,s}\varpi_j}.\]
Consequently, every cluster variable \(M_q^{\,l-1}\) of \(\widetilde{\mathbf{s}}(\overline{v}_{l-1},\overline{w})\) is a factor of some determinantial module in the family \[\mathscr{T}_{l-1} \cup \left\{ M\binom{w\varpi_j}{\overline{v}^{\,s}\varpi_j} \;\middle|\; j\in I,\;s<l-1 \right\}.\] Hence, by 19 , the only cluster variable of \(\widetilde{\mathbf{s}}(\overline{v}^{\,l-1},\overline{w})\) which is not already represented in \[\mathscr{T}_{l,1}\cup \left\{ M\!\binom{w\varpi_j}{\overline{v}^{\,s}\varpi_j} \;\middle|\; j\in I,\;s<l-1 \right\}\] is \(M_{(i,b_l+1)}^{\,l-1}\).
Now Proposition 13 together with Proposition 18 implies that \[M\!\binom{w_{(i,d_l+1)}\varpi_i} {\overline{v}^{\,l}_{(i,d_l+1)}\varpi_i}\] is a cluster monomial in the cluster variables of either \[\widetilde{\mathbf{s}}(\overline{v}^{\,l-1},\overline{w}) \qquad\text{or}\qquad \mu_{(i,b_l+1)}\widetilde{\mathbf{s}}(\overline{v}^{\,l-1},\overline{w}).\] If it were a cluster monomial in \(\widetilde{\mathbf{s}}(\overline{v}^{\,l-1},\overline{w})\), then, by the induction hypothesis, \[M\!\binom{w_{(i,d_l)}\varpi_i} {\overline{v}^{\,l-1}_{(i,d_l)}\varpi_i}\] would also be a cluster monomial in that seed. This would imply \[\mathfrak d\!\left( M\!\binom{w_{(i,d_l+1)}\varpi_i} {\overline{v}^{\,l}_{(i,d_l+1)}\varpi_i}, \, M\!\binom{w_{(i,d_l)}\varpi_i} {\overline{v}^{\,l-1}_{(i,d_l)}\varpi_i} \right)=0,\] contrary to ?? . Therefore \[M_{(i,b_l+1)}^{\,l} \mid M\!\binom{w_{(i,d_l+1)}\varpi_i} {\overline{v}^{\,l}_{(i,d_l+1)}\varpi_i},\] and the multiplicity-one assertion follows from ?? .
By 13 , every cluster variable \(X_{(i,s)}\) of \[\mu_{(i,b_l+1)}\widetilde{\mathbf{s}}(\overline{v}^{\,l-1},\overline{w})\] is a multiplicity-one factor of the determinantial module in \(\mathscr{T}_{l,1}\) indexed by \(\Phi_i^{l,1}(i,s)\).
Repeating the same argument for the successive mutations \(\mu_{(i,b_l+2)},\dots,\mu_{(i,n_i-\alpha(i,l))}\), we obtain that for every \(k\ge 1\), \[\label{eq:Mliblk-new} M_{(i,b_l+k)}^{\,l} \mid M\!\binom{w_{(i,d_l+k)}\varpi_i} {\overline{v}^{\,l}_{(i,d_l+k)}\varpi_i},\tag{20}\] and that \[M\!\binom{w_{(i,d_l+k)}\varpi_i} {\overline{v}^{\,l}_{(i,d_l+k)}\varpi_i}\] is a cluster monomial in the seed \[\mu_{(i,b_l+k)}\cdots\mu_{(i,b_l+1)} \widetilde{\mathbf{s}}(\overline{v}^{\,l-1},\overline{w}).\]
Furthermore, no determinantial module in \[\mathscr{T}_{l,k}\setminus \left\{ M\!\binom{w_{p_t}\varpi_{i_{p_t}}} {\overline{v}^{\,l-1}_{p_t}\varpi_{i_{p_t}}} \;\middle|\; t\in [l-1] \right\}\] can contain \(M_{(i,b_l+s)}^{\,l-1}\) as a factor for any \(1\le s\le k\). Otherwise it would fail to strongly commute with \[M\!\binom{w_{(i,d_l+s)}\varpi_i} {\overline{v}^{\,l}_{(i,d_l+s)}\varpi_i},\] which also belongs to \(\mathscr{T}_{l,k}\), contradicting Proposition 12. Hence every such determinantial module is a cluster monomial in the seed \[\mu_{(i,b_l+k)}\cdots\mu_{(i,b_l+1)} \widetilde{\mathbf{s}}(\overline{v}^{\,l-1},\overline{w}).\]
Taking \(k=n_i-d_l\) and using 14 , we deduce that \[M\!\binom{w_q\varpi_{i_q}}{\overline{v}^{\,l}_q\varpi_{i_q}}\] is a cluster monomial in \(\widetilde{\mathbf{s}}(\overline{v}^{\,l},\overline{w})\) for every \(q\notin \{p_1,\dots,p_l\}\).
Finally, since \(\Phi_i^{l,n_i-d_l}=\Phi_i^l\) on \(I_{i,\le l}\) and \(\Phi_j^l=\Phi_j^{l-1}\) for \(j\neq i\), equations 20 and 14 show that, for every \((j,k)\in I_{j,\le l}\), the determinantial module in \(\mathscr{T}_l\) indexed by \(\Phi_j^l(j,k)\) contains \(M_{(j,k)}^l\) as a multiplicity-one factor. This proves 17 .
Step 3: exclusion from \(\mathscr{C}_{w,v^l}\). We next prove that \[\label{eq:notincwv-new} M_{(i,k)}^l\notin \mathscr{C}_{w,v^l} \qquad \text{whenever } k+\alpha(i,l)>n_i.\tag{21}\]
We argue by induction on \(l\). For \(l=1\) with \(i_{p_1}=i\), the only such variable is \(M_{(i,n_i)}^1\), and \[M_{(i,n_i)}^1=M\!\binom{w\varpi_i}{\varpi_i}.\] Hence 21 follows from Lemma 5.
Assume now that 21 holds for \(l-1\), and set \(i=i_{p_l}\). If \(j\neq i\), then \(\alpha(j,l)=\alpha(j,l-1)\), so \[M_{(j,k)}^l=M_{(j,k)}^{\,l-1} \qquad \text{whenever } k+\alpha(j,l)>n_j.\] Similarly, if \(j=i\) and \(k+\alpha(i,l)>n_i\) with \(k>n_i-\alpha(i,l)+1\), then \[M_{(i,k)}^l=M_{(i,k)}^{\,l-1},\] and hence these variables do not belong to \(\mathscr{C}_{w,v^l}\), since \[\mathscr{C}_{w,v^l}\subset \mathscr{C}_{w,v^{l-1}}.\]
It remains to treat the boundary term \[M_{(i,n_i-\alpha(i,l)+1)}^l=M_{(i,n_i-\alpha(i,l)+1)}^{l-1}.\] By 20 , this module is a factor of \[M\!\binom{w\varpi_i} {\overline{v}^{\,l-1}_{(i,n_i)}\varpi_i}.\] Note that the determinantial module \[M\!\binom{w\varpi_i}{\overline{v}^{\,l-1}_{(i,n_i)}\varpi_i}\] belongs to \(\mathscr{T}_{l,n_i-d_l}\), and therefore is a cluster monomial in the seed \(\widetilde{\mathbf{s}}(\overline{v}^{\,l},\overline{w})\). Since \[M\!\binom{w\varpi_i}{\overline{v}^{\,l-1}_{(i,n_i)}\varpi_i}\in \mathscr{C}_{w,v^{l-1}},\] the induction hypothesis 21 implies that all of its factors are of the form \(M^l_{(j,k)}\) with \(k+\alpha(j,l-1)\le n_j\) for every \(j\in I\).
On the other hand, for every \((j,k)\) with \(k+\alpha(j,l)\le n_j\), the cluster variable \(M_{(j,k)}^l\) is a factor of some determinantial module \[M\!\binom{w_t\varpi_{i_t}}{\overline{v}^{\,l}_t\varpi_{i_t}} \in \mathscr{C}_{w,v^l}, \qquad t=\Phi^l(j,k).\] Hence all such \(M_{(j,k)}^l\) belong to \(\mathscr{C}_{w,v^l}\).
Now Lemma 5 shows that \[M\!\binom{w\varpi_i} {\overline{v}^{\,l-1}_{(i,n_i)}\varpi_i} \notin \mathscr{C}_{w,v^l}.\] Therefore at least one of its factors must lie outside \(\mathscr{C}_{w,v^l}\), and the preceding discussion shows that the only possibility is \[M_{(i,n_i-\alpha(i,l)+1)}^l.\] Thus \[M_{(i,n_i-\alpha(i,l)+1)}^l\notin \mathscr{C}_{w,v^l},\] which completes the proof of 21 .
Step 4: conclusion. By Theorem 17, we have \[M\!\binom{w_q\varpi_{i_q}}{\overline{v}^{\,l}_q\varpi_{i_q}} \in \mathscr{C}_{w,v^l} \qquad \text{for all } q\notin \{p_1,\dots,p_l\}.\] Since the cluster variables removed in passing from \(\widetilde{\mathbf{s}}(\overline{v}^{\,l},\overline{w})\) to \(\mathbf{s}(\overline{v}^{\,l},\overline{w})\) are precisely those excluded by 21 , it follows that every such determinantial module is in fact a cluster monomial in the seed \[\mathbf{s}(\overline{v}^{\,l},\overline{w}).\] This proves the theorem. ◻
Remark 20. For any \(l \in [m]\), it is straightforward to verify that the vertex set of the seed \(\mathbf{s}(\overline{v}^l,\overline{w})\) coincides with \(\bigsqcup_{i\in I} I_{i,\leq l}\). Hence, \(\Phi^{\ell(v)}\) induces a bijection from the vertex set \(J\) of the seed \(\mathbf{s}(\overline{v},\overline{w})\) to the set \([r]\setminus\{p_1,\dots,p_m\}\).
Theorem 21. Any simple module corresponding to a cluster variable of \(\mathcal{A}(\mathbf{s}(\overline{v},\overline{w}))\) is contained in \(\mathscr{C}_{w,v}\). In particular, we have \[\overline{\mathcal{A}}_q(\mathbf{s}(\overline{v},\overline{w})) \subset K(\mathscr{C}_{w,v}).\]
Proof. By Theorem 19, each simple module \(M^{\ell(v)}_k\) in the initial seed satisfies \[M^{\ell(v)}_k \mid M\binom{w_{\Phi^{\ell(v)}(k)}\varpi_{i_{\Phi^{\ell(v)}(k)}}}{\overline{v}_{\Phi^{\ell(v)}(k)}\varpi_{i_{\Phi^{\ell(v)}(k)}}}.\] By Theorem 17, for all \(s\in [r]\), the determinantial module \(M\binom{w_{s}\varpi_{i_s}}{\overline{v}_s\varpi_{i_s}}\) belongs to \(\mathscr{C}_{w,v}\). Hence, by Lemma 6, we obtain \(M^{\ell(v)}_k \in \mathscr{C}_{w,v}\). Therefore, all cluster variables in the initial seed lie in \(\mathscr{C}_{w,v}\).
It remains to show that the property is preserved under mutation. Let \(X_j\) be a cluster variable in a fixed monoidal seed of \(\mathcal{A}(\mathbf{s}(\overline{v},\overline{w}))\), and assume that \(X_j \in \mathscr{C}_{w,v}\) for all \(j\). We prove that \(\mu_k(X_j) \in \mathscr{C}_{w,v}\).
If \(k \neq j\), then \(\mu_k(X_j)=X_j\), so there is nothing to prove. Let \(j=k\). In \(\mathscr{C}_w\), the mutation relation takes the form \[X_k \circ \mu_k(X_k) = q^m X + q^n Y,\] where \(X\) and \(Y\) are monomials in the cluster variables of the seed.
Observe that the frozen vertices in \(\mathbf{s}(\overline{v},\overline{w})\) correspond to the vertices connected with the deleted vertices in \(\widetilde{\mathbf{s}}(\overline{v},\overline{w})\), which is a monoidal seed in \(\mathscr{C}_w\). Hence, for all mutable vertices, the exchange relations in \(\mathcal{A}(\mathbf{s}(\overline{v},\overline{w}))\) coincide with those in \(\mathscr{C}_w\).
Since the cluster variables in the seed are assumed to lie in \(\mathscr{C}_{w,v}\), it follows that the monomials \(X\) and \(Y\) belong to \(\mathscr{C}_{w,v}\), because \(\mathscr{C}_{w,v}\) is stable under convolution products. Therefore, \(\mu_k(X_k)\) appears as a factor of the simple module \(X_k \circ \mu_k(X_k)\), which lies in \(\mathscr{C}_{w,v}\). As the proof of Lemma 6, we conclude that \(\mu_k(X_k) \in \mathscr{C}_{w,v}\).
By induction on the length of mutation sequences, all monoidal cluster variables of \(\mathbf{s}(\overline{v},\overline{w})\) lie in \(\mathscr{C}_{w,v}\). ◻
In finite type, Leclerc introduced the generalized minor \(\Delta_{v^{-1}_{\leq l}\varpi_{i_l},\,w^{-1}_{\leq l}\varpi_{i_l}}\) in [1]. We recall that, in Leclerc’s convention, one starts from the reduced expression \((i_r,\dots,i_1)\) of \(w\) and uses the rightmost subexpression corresponding to \(v\). By our definition of the quantum minor \(D\binom{w_l\varpi_{i_l}}{v_l\varpi_{i_l}},\) we have \[\label{eq:Ddelta} D\binom{w_l\varpi_{i_l}}{v_l\varpi_{i_l}}\bigg|_{q=1} = \Delta_{v^{-1}_{\leq l}\varpi_{i_l},\,w^{-1}_{\leq l}\varpi_{i_l}},\tag{22}\] where the right-hand side is understood in the sense of Leclerc.
Theorem 22. In finite type, Leclerc’s seed of \(\mathbb{C}[\mathring{\mathcal{B}}_{v,w}]\) coincides with Ménard’s seed \(\mathbf{s}(\overline{v},\overline{w})\).
Proof. Recall that Leclerc’s seed is defined by the irreducible factors of \(\prod_{l=1}^r \Delta_{v^{-1}_{\leq l}\varpi_{i_l},\,w^{-1}_{\leq l}\varpi_{i_l}}.\) By Theorem 19, every cluster variable of \(\mathbf{s}(\overline{v},\overline{w})\) appears among the irreducible factors of this product. On the other hand, the seed \(\mathbf{s}(\overline{v},\overline{w})\) has exactly \(\ell(w)-\ell(v)\) cluster variables. It follows that the set of irreducible factors of \(\prod_{l=1}^r \Delta_{v^{-1}_{\leq l}\varpi_{i_l},\,w^{-1}_{\leq l}\varpi_{i_l}}\) coincides with the set of cluster variables of \(\mathbf{s}(\overline{v},\overline{w})\).
Therefore, the maximal rigid module in \(\mathcal{C}_{v,w}\) defined by Leclerc in [1], corresponding to the cluster variables in Leclerc’s seed, coincides with the maximal rigid module in \(\mathcal{C}_{v,w}\) associated with the cluster variables of \(\mathbf{s}(\overline{v},\overline{w})\). In particular, the two modules have the same endomorphism algebra and the same injective direct summands. Consequently, they determine the same exchange matrix \(B\) and the same frozen variables. This proves the theorem. ◻
Next, we show that \(K(\widetilde{\mathscr{C}}_{w,v})\) is isomorphic to the quantum cluster algebra \(\mathcal{A}_q(\overline{v},\overline{w})\). We begin with the following lemma.
Lemma 8. For any \(v\leq w\), the quantum cluster algebra \(\mathcal{A}_q(\overline{v},\overline{w})\) coincides with the upper quantum cluster algebra \(U_q(\overline{v},\overline{w})\).
Proof. By [2], the seed \(\mathbf{s}(\overline{v},\overline{w})\) agrees with the right inductive seed \[\overleftarrow{\mathfrak{m}}(\overline{w}\,\overline{v^{-1}w_0}),\] where \(\overline{v^{-1}w_0}\) is a reduced expression of \(v^{-1}w_0\). Moreover, the skew-symmetric matrix \(\Lambda\) of the seed \(\mathbf{s}(\overline{v},\overline{w})\) is induced from the skew-symmetric matrix of \(\mathbf{s}(\overline{w})\). Hence, by [18], we obtain \[\mathcal{A}_q(\overline{v},\overline{w})=U_q(\overline{v},\overline{w}).\] ◻
Theorem 23. Let \(v\leq w\in W\), and let \(\overline{w}\) be a reduced expression of \(w\). Then \[K(\widetilde{\mathscr{C}}_{w,v})=\mathcal{A}_q(\overline{v},\overline{w}).\] In particular, \(\widetilde{\mathscr{C}}_{w,v}\) is a monoidal categorification of the quantum cluster algebra \(\mathcal{A}_q(\overline{v},\overline{w})\).
Proof. For each \(l\in [\ell(v)]\), let \(i=i_{p_l}\). We claim that \[\label{eq:subsetl-cat} K(\widetilde{\mathscr{C}}_{w,v^l})=U_q(\overline{v}^{\,l},\overline{w}) \qquad\text{for all } l\in [\ell(v)].\tag{23}\] We prove this by induction on \(l\).
When \(l=0\), the statement is clear. Assume now that 23 holds for \(l-1\), and we prove it for \(l\).
Let \(\mathbf{s}(\overline{v}^{\,l-1},\overline{w})\) be the seed of \(U_q(\overline{v}^{\,l-1},\overline{w})\). By the definition of the map \(\Phi^{l-1}\) and Theorem 19, every cluster variable \(X_k\) of this seed is a factor of a determinantial module in \(\mathscr{T}_{l-1}\).
Now apply the mutation sequence \(\widetilde{\mu}_l\) to \(\mathbf{s}(\overline{v}^{\,l-1},\overline{w})\). After deleting the cluster variable \[X_{(i,n_i-\alpha(i,l)+1)}\] and freezing the variables adjacent to it, we obtain the seed \(\mathbf{s}(\overline{v}^{\,l},\overline{w})\). On the categorical side, the corresponding family is \(\mathscr{T}_{l}\). By equation 21 , the simple module corresponding to \(X_{(i,n_i-\alpha(i,l)+1)}\) does not belong to \(\mathscr{C}_{w,v^l}\). Denote this simple module by \(Y_l\).
We now show that \[\bigl(( [Y_l]-1 )K(\mathscr{C}_{w,v^{l-1}})\bigr)\cap K(\mathscr{C}_{w,v^l})=0.\] Take a nonzero element \(f\in K(\mathscr{C}_{w,v^{l-1}})\), and write \[f=\sum_{a=1}^m c_a[M_a],\] where each \(M_a\) is a simple module and \(c_a\neq 0\). Then \[([Y_l]-1)f=\sum_{a=1}^m c_a\bigl([Y_l\circ M_a]-[M_a]\bigr).\]
Since the sum is finite and each \(Y_l\circ M_a\) is simple, we may choose \(d\in [m]\) such that \[Y_l\circ M_d \not\cong M_b \qquad\text{for all } b\in [m].\] Moreover, since \(Y_l\circ M_d \cong M_d\circ Y_l\), we have \[Y_l\circ M_d \notin \mathscr{C}_{w,v^l}.\]
On the other hand, every element of \(K(\mathscr{C}_{w,v^l})\) is a \(\mathbb{Z}[q^{\pm 1/2}]\)-linear combination of classes of simple modules lying in \(\mathscr{C}_{w,v^l}\), and the classes of simple modules form a basis of \(K(\mathscr{C}_w)\). Since \(c_d\neq 0\), the class \([Y_l\circ M_d]\) appears in \(([Y_l]-1)f\) with nonzero coefficient and cannot be cancelled by any other term in the above expansion. Therefore \[([Y_l]-1)f\notin K(\mathscr{C}_{w,v^l}).\] Hence, for every nonzero \(f\in K(\mathscr{C}_{w,v^{l-1}})\), one has \[([Y_l]-1)f\notin K(\mathscr{C}_{w,v^l}).\] Hence we obtain an injection \[\label{eq:subsetKw-cat} K(\mathscr{C}_{w,v^l})\hookrightarrow K(\mathscr{C}_{w,v^{l-1}})/([Y_l]-1).\tag{24}\]
Next observe that \[\label{eq:isocwc-cat} \left(K(\mathscr{C}_{w,v^{l-1}})/([Y_l]-1)\right)_{\mathrm{loc}} \stackrel{(1)}{\cong} K(\widetilde{\mathscr{C}}_{w,v^{l-1}})/([Y_l]-1) \stackrel{(2)}{=} U_q(\overline{v}^{\,l-1},\overline{w})/([Y_l]-1),\tag{25}\] where \(\mathrm{loc}\) denotes localization with respect to the set \[S_{l-1}:= \left\{ M\binom{w\varpi_j}{v^{l-1}\varpi_j} \;\middle|\; j\in I \right\}.\] Here, (1) follows from the isomorphism \[K(\mathscr{C}_{w,v^{l-1}})_{\mathrm{loc}} \cong K(\widetilde{\mathscr{C}}_{w,v^{l-1}})\] together with the fact that \(Y_l\in S_{l-1}\), while (2) is exactly the induction hypothesis.
Combining 24 and 25 , we obtain \[K(\widetilde{\mathscr{C}}_{w,v^l}) \subset \left(U_q(\overline{v}^{\,l-1},\overline{w})/([Y_l]-1)\right)_{\mathrm{loc}},\] where the localization is taken at \[M\binom{w\varpi_{i_{p_l}}}{v^l\varpi_{i_{p_l}}}.\] By the definition of the seed \(\mathbf{s}(\overline{v}^{\,l},\overline{w})\), the right-hand side is precisely \[U_q(\overline{v}^{\,l},\overline{w}).\] Therefore, \[K(\widetilde{\mathscr{C}}_{w,v^l})\subset U_q(\overline{v}^{\,l},\overline{w}).\]
On the other hand, Theorems 21 and 22 give \[\mathcal{A}_q(\overline{v}^{\,l},\overline{w})\subset K(\widetilde{\mathscr{C}}_{w,v^l}).\] Combining this with Lemma 8, we obtain \[K(\widetilde{\mathscr{C}}_{w,v^l})=U_q(\overline{v}^{\,l},\overline{w}),\] which proves 23 . Taking \(l=\ell(v)\), we obtain \[K(\widetilde{\mathscr{C}}_{w,v})=U_q(\overline{v},\overline{w}) =\mathcal{A}_q(\overline{v},\overline{w}),\] where the last equality follows from Lemma 8. The final statement now follows from Theorem 21. ◻