1 Introduction↩︎

Cluster algebras were introduced by Fomin and Zelevinsky in studying total positivity and dual canonical bases of quantum groups ([1]). When an algebra has a cluster algebra structure, this structure provides a way to organize distinguished generators into groups called seeds, together with procedures called mutations to connect seeds. Since their introduction, cluster algebras have played an important role in several areas of mathematics including representation theory, combinatorics, geometry, and categorification.

Let \(\mathsf{C}\) be a Cartan matrix of finite \(ADE\) type with index set \(I\), and let \(\mathop{\mathrm{Br}}^+\) be the positive monoid of the (generalized) braid group \(\mathop{\mathrm{Br}}\) associated with \(\mathsf{C}\). There exist two interesting cluster algebras arising from \(\mathop{\mathrm{Br}}^+,\) namely the bosonic extensions and the coordinate rings of braid varieties.

The bosonic extension \(\widehat{\mathcal{A}}\) is the associative algebra over \(\mathbb{Q}(q^{1/2})\) generated by \(f_{i,p}\) (\(i\in I\) and \(p\in\mathbb{Z}\)) with quantum Serre relations and \(q\)-boson relations determined by \(\mathsf{C}\). This algebra \(\widehat{\mathcal{A}}\) is decomposed into an infinite tensor product of the negative half \(\mathcal{U}_q^-(\mathfrak{g})\) of the quantum group \(\mathcal{U}_q(\mathfrak{g})\) associated with \(\mathsf{C}\) as a \(\mathbb{Q}(q^{1/2})\)-vector space, i.e., \(\widehat{\mathcal{A}}\simeq \otimes_{\mathbb{Z}\mspace{1mu}} \mathcal{U}_q^-(\mathfrak{g})\). In this sense, \(\widehat{\mathcal{A}}\) can be understood as an affinization of \(\mathcal{U}_q^-(\mathfrak{g})\). Bosonic extensions are motivated by the ring presentations of quantum Grothendieck rings of Hernandez–Leclerc categories over quantum affine algebras. The Hernandez–Leclerc category is a distinguished monoidal subcategory of the category of finite-dimensional integrable modules over a quantum affine algebra. Hernandez and Leclerc investigated such monoidal subcategories from the perspective of cluster algebras, and studied the quantum Grothendieck rings of the categories and their ring presentations, which motivate the definition of bosonic extensions ([2]–[4]).

Braid symmetries on \(\widehat{\mathcal{A}}\), studied in [5]–[7], define the subalgebra \(\widehat{\mathcal{A}}(\mathtt{b})\) of \(\widehat{\mathcal{A}}\) generated by the PBW vectors \(\mathsf{p}_{{\boldsymbol{i}},k}\) associated with an expression sequence \({\boldsymbol{i}}\) of a positive braid \(\mathtt{b}\in \mathop{\mathrm{Br}}^+\) ([7], [8]). The algebras \(\widehat{\mathcal{A}}(\mathtt{b})\) generalize the quantum unipotent coordinate rings \(A_q(\mathfrak n(w))\), a quantum analogue of a coordinate ring of unipotent subgroups, where positive braid elements \(\mathtt{b}\) play the role of Weyl group elements \(w\). The algebra \(\widehat{\mathcal{A}}(\mathtt{b})\) has a (quantum) cluster algebra structure whose initial seeds are described in terms of \(i\)-boxes ([9]). For the purposes of this paper, we consider the specialization \(\mathbb{A}(\mathtt{b})\) of \(\widehat{\mathcal{A}}(\mathtt{b})\) at \(q^{1/2}=1\). We choose an expression sequence \({\boldsymbol{i}}=(i_1,i_2, \ldots,i_r)\) of \(\mathtt{b}\). An \(i\)-box associated with \({\boldsymbol{i}}\) is an interval \([a,b]\subset[1,r]\) with \(i_a=i_b\), which is a combinatorial object used to describe a seed of \(\mathbb{A}(\mathtt{b})\). For each admissible chain \(\mathfrak{C}=(\mathfrak{c}_t=[a_t,b_t])_{1\leqslant t\leqslant r}\) of \(i\)-boxes, one can define distinguished elements \(\mathsf{D}_{{\boldsymbol{i}}}[a,b]\in\mathbb{A}(\mathtt{b})\) using the \(T\)-systems and PBW vectors, which gives a seed \[\Sigma(\mathfrak{C})=(\mathsf{D}(\mathfrak{C}),Q(\mathfrak{C}))\] for \(\mathbb{A}(\mathtt{b})\), where \(\mathsf{D}(\mathfrak{C})=\{\mathsf{D}_{{\boldsymbol{i}}}[a_t,b_t]\mid 1\leqslant t\leqslant r\}\) and \(Q(\mathfrak{C})\) is the quiver determined by \(\mathfrak{C}\) (see [9] and see also Section 4.2).

Braid varieties are algebraic varieties defined from expression sequences of \(\mathtt{b}\) for the simple algebraic group \(\mathsf{G}\) associated with the Cartan matrix \(\mathsf{C}\) ([10]–[12]). They generalize important families such as open Richardson varieties and double Bott-Samelson varieties. Let us fix an expression sequence \({\boldsymbol{i}}= (i_1,i_2,\ldots,i_r)\) of \(\mathtt{b}\), and let \(\delta(\mathtt{b})\) be the element in the Weyl group \(\mathsf{W}\) obtained from \(\mathtt{b}\) by taking the Demazure product (see 12 ). We take a Borel subgroup \(\mathsf{B}\) of \(\mathsf{G}\). The braid variety \(X({\boldsymbol{i}})\) consists of flag configurations \((\mathsf{B}_0,\mathsf{B}_1, \ldots,\mathsf{B}_r)\) in the flag variety \(\mathfrak{Fl}= \mathsf{G}/\mathsf{B}\) such that \(\mathsf{B}_0=\mathsf{B}\), \(\mathsf{B}_r=\delta(\mathtt{b})\mathsf{B}\) and the pair \((\mathsf{B}_{t-1},\mathsf{B}_t)\) is in relative position \(s_{i_t}\) for \(1\leqslant t\leqslant r\), where \(s_i\) is the simple reflection of the Weyl group \(\mathsf{W}\). Although a different choice of expression \({\boldsymbol{i}}\) yields an isomorphic variety, the choice of expression is important: it determines ambient affine coordinates for the braid variety, which in turn give a set of defining equations for \(X({\boldsymbol{i}})\).

Let \(w_\circ\) be the longest element of \(\mathsf{W}\), and let \(\Delta\in\mathop{\mathrm{Br}}^+\) denote the half twist, i.e., the minimal lift of \(w_\circ\) under the projection \(\mathop{\mathrm{Br}}^+ \twoheadrightarrow \mathsf{W}\). Then the braid variety \(X({\underline{\Delta}}{\boldsymbol{i}})\) is isomorphic to the half decorated double Bott-Samelson variety for an expression sequence \({\underline{\Delta}}\) of \(\Delta\), where \({\underline{\Delta}}{\boldsymbol{i}}\) is the concatenation of \({\underline{\Delta}}\) and \({\boldsymbol{i}}\) (see [10] and see also Section 6.2). In this paper, we mainly focus on these braid varieties \(X({\underline{\Delta}}{\boldsymbol{i}})\). Note that the cluster algebra structure on the coordinate ring \(\mathbb{C}[X({\underline{\Delta}}({\boldsymbol{i}}))]\) of the double Bott-Samelson variety is extensively studied in [13], and it provides a starting point for constructing cluster structures on more general braid varieties in [10].

The coordinate ring \(\mathbb{C}[X({\boldsymbol{i}})]\) of a braid variety carries a cluster algebra structure from Demazure weaves ([10]). A Demazure weave for \({\boldsymbol{i}}\) is a planar graph, written \(\mathfrak{W}:{\boldsymbol{i}}\to\delta({\boldsymbol{i}})\), whose edges are labeled by the index set \(I\) and whose horizontal slices record weave moves from \({\boldsymbol{i}}\) to the Demazure product \(\delta({\boldsymbol{i}})=\delta(\mathtt{b})\). The weave \(\mathfrak{W}\) provides a seed of \(\mathbb{C}[X({\boldsymbol{i}})]\) \[\Sigma(\mathfrak{W})=\bigl(\mathcal{A}(\mathfrak{W}),Q(\mathfrak{W})\bigr),\] where \(\mathcal{A}(\mathfrak{W})\) is the set of the regular functions in \(\mathbb{C}[X({\boldsymbol{i}})]\) defined by \(\mathfrak{W}\), and \(Q(\mathfrak{W})\) is the quiver whose vertices are indexed by trivalent vertices of \(\mathfrak{W}\) and whose arrows are determined by Lusztig’s cycles ([10]). We remark that cluster structures on braid varieties were also constructed using \(3\)-dimensional plabic graphs and Deodhar geometry ([11], [12]). The construction using Deodhar geometry was compared with the construction from Demazure weaves in [14].

In this paper, we compare two cluster algebras \(\mathbb{A}(\mathtt{b})\) and \(\mathbb{C}[X({\underline{\Delta}}{\boldsymbol{i}})]\) by investigating their seeds arising from \(i\)-boxes and the associated Demazure weaves, where \({\boldsymbol{i}}\) is an expression sequence of \(\mathtt{b}\). Define \(\mathbb{A}_\mathbb{C}(\mathtt{b}) :=\mathbb{C}\otimes_\mathbb{Z}\mspace{1mu}\mathbb{A}(\mathtt{b})\), and let \(\widetilde{\mathbb{A}}_\mathbb{C}(\mathtt{b})\) be the localization of \(\mathbb{A}_\mathbb{C}(\mathtt{b})\) by inverting the frozen variables of \(\mathbb{A}(\mathtt{b})\) (see Section 4.2). For each admissible chain \(\mathfrak{C}\) of \(i\)-boxes associated with \({\boldsymbol{i}}\), we construct the corresponding Demazure weave \(\mathfrak{W}_{{\underline{\Delta}}}(\mathfrak{C})\) using the combinatorics of \(i\)-boxes. Theorem 5 says that there exists an algebra isomorphism \[\varphi_{{\boldsymbol{i}}}: \widetilde{\mathbb{A}}_\mathbb{C}(\mathtt{b})\xrightarrow{\sim}\mathbb{C}[X({\underline{\Delta}}{\boldsymbol{i}})],\] which sends the seed \(\Sigma(\mathfrak{C})\) to the seed \(\Sigma(\mathfrak{W}_{{\underline{\Delta}}}(\mathfrak{C}))\) in the sense that \[\mathcal{A}(\mathfrak{W}_{{\underline{\Delta}}}(\mathfrak{C})) = \varphi_{{\boldsymbol{i}}}(\mathsf{D}(\mathfrak{C})) \quad\text{{{and}}}\quad Q(\mathfrak{W}_{{\underline{\Delta}}}(\mathfrak{C})) = Q(\mathfrak{C})^{\mathrm{op}},\] where \(Q^{\mathrm{op}}\) is the opposite quiver of \(Q\). Moreover, this morphism \(\varphi_{{\boldsymbol{i}}}\) sends the PBW vectors \(\overline{\mathsf{p}}_{{\boldsymbol{i}},k}\) to the coordinate functions \(z_k\) of \(\mathbb{C}[X({\underline{\Delta}}{\boldsymbol{i}})]\) corresponding to the letters of \({\boldsymbol{i}}\) (see 20 for the coordinates \(z_k\)). We remark that the existence of an algebra isomorphism between \(\widetilde{\mathbb{A}}_\mathbb{C}(\mathtt{b})\) and \(\mathbb{C}[X({\underline{\Delta}}{\boldsymbol{i}})]\) compatible with cluster algebra structures was previously shown by Qin in the context of the cluster algebras associated with signed words (see [15] and [16]). The main contribution of this paper is to show that the isomorphism \(\varphi_{{\boldsymbol{i}}}\) is compatible with our weave construction \(\mathfrak{W}_{{\underline{\Delta}}}(\mathfrak{C})\) arising from admissible chains \(\mathfrak{C}\) of \(i\)-boxes, which provides a natural explanation why the PBW vectors of \(\widetilde{\mathbb{A}}_\mathbb{C}(\mathtt{b})\) correspond to the coordinates of \(\mathbb{C}[X({\underline{\Delta}}{\boldsymbol{i}})]\).

The two cluster algebra structures compared here arise from different constructions: one using bosonic extensions and \(i\)-boxes, which is deeply connected to monoidal categorification, and the other using braid varieties and Demazure weaves arising in geometry. The morphism \(\varphi_{{\boldsymbol{i}}}\) in Theorem 5 provides a bridge between them and allows us to compute explicitly the cluster variables of \(\mathfrak{W}_{{\underline{\Delta}}}(\mathfrak{C})\) and the quiver \(Q(\mathfrak{W}_{{\underline{\Delta}}}(\mathfrak{C}))\) using the combinatorics of \(i\)-boxes. Although the weaves \(\mathfrak{W}_{{\underline{\Delta}}}(\mathfrak{C})\) form a special family of weaves, both \(\mathcal{A}(\mathfrak{W}_{{\underline{\Delta}}}(\mathfrak{C}))\) and \(Q(\mathfrak{W}_{{\underline{\Delta}}}(\mathfrak{C}))\) can be computed easily using \(i\)-boxes, as illustrated through examples in the paper.

The proof of Theorem 5 begins by constructing the weave \(\mathfrak{W}_{{\underline{\Delta}}}(\mathfrak{C})\) from an admissible chain \(\mathfrak{C}\). First, we associate a double string \(\mathbf{s}_{{\underline{\Delta}}}(\mathfrak{C})\) with \(\mathfrak{C}\), and obtain the corresponding double inductive weave \(\mathfrak{W}^B_{{\underline{\Delta}}}(\mathfrak{C})\) following [10]. The weave \(\mathfrak{W}_{{\underline{\Delta}}}(\mathfrak{C})\) is then obtained by vertically concatenating the weave \(\mathfrak{W}^B_{{\underline{\Delta}}}(\mathfrak{C})\) with another weave \(\mathfrak{W}^{\rm T}_{{\underline{\Delta}}}(\mathfrak{C})\) consisting of braid moves chosen so as to make the top equal to \({\underline{\Delta}}{\boldsymbol{i}}\) (see Section 7.1 for precise construction). The argument first treats the distinguished admissible chain \(\mathfrak{C}_{\mathcal{R}}\) in 4 ; for this chain, the weave \(\mathfrak{W}^B_{{\underline{\Delta}}}(\mathfrak{C}_{\mathcal{R}})\) coincides with the right inductive weave \(\overrightarrow{\mathfrak{w}}({\underline{\Delta}}{\boldsymbol{i}})\). We thus identify \(Q(\overrightarrow{\mathfrak{w}}({\underline{\Delta}}{\boldsymbol{i}}))\) with \(Q(\mathfrak{C}_{\mathcal{R}})^{\mathrm{op}}\) (see Lemma 8), which yields the isomorphism \(\varphi_{{\boldsymbol{i}}}\) defined by sending the seed \(\Sigma(\mathfrak{C}_{\mathcal{R}})\) of \(\widetilde{\mathbb{A}}_\mathbb{C}(\mathtt{b})\) to the seed \(\Sigma(\mathfrak{W}_{{\underline{\Delta}}}(\mathfrak{C}_{\mathcal{R}}))\) of \(\mathbb{C}[X({\underline{\Delta}}{\boldsymbol{i}})]\). Every admissible chain can be reached from \(\mathfrak{C}_{\mathcal{R}}\) by box moves (see Section 3.2). The box moves can be described in terms of mutations and permutations (Proposition 1), and Theorem 4 gives the same mutations or permutations for the seeds constructed from the weaves \(\mathfrak{W}_{{\underline{\Delta}}}(\mathfrak{C})\). A key ingredient is the description in [10] of mutations and permutations of inductive weaves arising from the corresponding moves of double strings. This explains why the isomorphism \(\varphi_{{\boldsymbol{i}}}\) induces the correspondence between \(\mathfrak{C}\) and \(\mathfrak{W}_{{\underline{\Delta}}}(\mathfrak{C})\) at the level of seeds, and between the PBW vectors \(\overline{\mathsf{p}}_{{\boldsymbol{i}},k}\) and the coordinates \(z_k\) at the level of rings.

We further investigate the isomorphism \(\varphi_{{\boldsymbol{i}}}\) from the perspective of the monoidal categorification through the representation-theoretic approach using affine determinantial modules in [9]. We remark that the monoidal categorification was also studied and shown by Qin via a different method based on a cluster-algebraic approach (see [15] and [16]). Let \(\mathscr{C}_{\mathfrak{g}}^{0}\) be the Hernandez–Leclerc category over a quantum affine algebra and let \({\mathscr{C}_\mathfrak{g}}^\mathbb{D}(\mathtt{b})\) be the distinguished subcategory of \(\mathscr{C}_{\mathfrak{g}}^{0}\) categorifying \(\mathbb{A}(\mathtt{b})\), i.e., \[K( {\mathscr{C}_\mathfrak{g}}^\mathbb{D}(\mathtt{b})) \simeq \mathbb{A}(\mathtt{b})\] (see Section 9 for precise definitions). Under this categorification, for each expression sequence \({\boldsymbol{i}}\) of \(\mathtt{b}\), the PBW vectors \(\overline{\mathsf{p}}_{{\boldsymbol{i}},t}\) and \(\mathsf{D}_{{\boldsymbol{i}}}[a,b]\) correspond to the affine cuspidal modules \(C_t^{\boldsymbol{i}}\) and the affine determinantial modules \(M^{ {\boldsymbol{i}}}[a,b]\), respectively. Proposition 11 tells us that the modules \(C_t^{\boldsymbol{i}}\) correspond to the coordinates \(z_t\) and the modules \(M^{ {\boldsymbol{i}}}[a,b]\) correspond to cluster variables arising from \(\mathfrak{W}_{{\underline{\Delta}}}(\mathfrak{C})\) via the isomorphism \(\varphi_{{\boldsymbol{i}}}\). In this sense, we categorify the coordinates \(z_t\) of \(X({\underline{\Delta}}{\boldsymbol{i}})\) as the affine cuspidal modules \(C_t^{\boldsymbol{i}}\) in the category \({\mathscr{C}_\mathfrak{g}}^\mathbb{D}(\mathtt{b})\).

As another direction of investigation, we study connections with signed words. Using the connection between signed words and admissible chains of \(i\)-boxes studied in [17], to each admissible chain \(\mathfrak{C}\), one assigns a signed word \(\underline{\mathsf{h}}(\mathfrak{C})\) with \(Q(\underline{\mathsf{h}}(\mathfrak{C}))=Q(\mathfrak{C})\). Combining this with Theorem 5, we obtain \[Q(\underline{\mathsf{h}}(\mathfrak{C}))=Q(\mathfrak{W}_{{\underline{\Delta}}}(\mathfrak{C}))^{\rm op}\] (see Corollary 1).

The paper is organized as follows. Section 2 recalls necessary background on quantum groups and cluster algebras. Section 3 and Section 4 review the combinatorics of \(i\)-boxes, bosonic extensions and their cluster algebra structure. Section 5 and Section 6 discuss Demazure weaves, braid varieties and their cluster algebra structures. Section 7 constructs the weave \(\mathfrak{W}_{{\underline{\Delta}}}(\mathfrak{C})\) for an admissible chain \(\mathfrak{C}\) and proves the existence of the isomorphism \(\varphi_{{\boldsymbol{i}}}\) with the desired properties. Section 8 compares the quivers obtained from \(i\)-boxes, signed words, and Demazure weaves. Section 9 investigates the isomorphism \(\varphi_{{\boldsymbol{i}}}\) from the perspective of monoidal categorification using Hernandez–Leclerc categories.

Acknowledgments. The authors would like to thank Fan Qin for pointing out relevant references concerning the isomorphism \(\varphi_{{\boldsymbol{i}}}\) and for fruitful discussions. E.P.would like to thank the Department of Mathematics at the University of Connecticut for its hospitality and excellent research environment during his visit.

Convention. Throughout this paper, we use the following convention.

1. For \(a,b \in \mathbb{Z} \sqcup \{\pm\infty\}\) with \(a \leqslant b\), we define the interval \([a,b]\) by \[[a,b] := \{ t \in \mathbb{Z} \mid a \leqslant t \leqslant b \}.\] If \(a>b\), we set \([a,b]=\emptyset\). We sometimes write \([a]\) for \([a,a]\) if no confusion arises.

2. For a set \(S\), \(|S|\) denotes the cardinality of \(S\).

3. Let \({\boldsymbol{i}}=(i_1, i_2, \ldots, i_r)\) and \({\boldsymbol{j}}=( j_1, j_2, \ldots,j_s)\) be sequences with entries in an index set \(I\). The concatenation \({\boldsymbol{i}}* {\boldsymbol{j}}\) of \({\boldsymbol{i}}\) and \({\boldsymbol{j}}\) is defined as \[{\boldsymbol{i}}* {\boldsymbol{j}}:= (i_1,\ldots,i_r,j_1,\ldots,j_s).\] For simplicity, we write \({\boldsymbol{i}}{\boldsymbol{j}}\) for \({\boldsymbol{i}}* {\boldsymbol{j}}\) if no confusion arises.

2 Preliminaries↩︎

2.1 Quantum groups↩︎

In this subsection, we recall the notion of quantum groups ([18]).

Let \((\mathsf{C}, \mathsf{P}, \Pi, \mathsf{P}^\vee, \Pi^\vee)\) be a Cartan datum consisting of a symmetrizable generalized Cartan matrix \(\mathsf{C}= (c_{i,j})_{i,j\in I}\), a weight lattice \(\mathsf{P}\), a coweight lattice \(\mathsf{P}^\vee\), the set of simple roots \(\Pi=\{{\mspace{1mu}\alpha}_i\}_{i \in I}\), and the set of simple coroots \(\Pi^\vee=\{h_i \}_{i \in I}\). They satisfy the pairing condition \(\langle h_i, {\mspace{1mu}\alpha}_j \rangle = c_{i,j}\) for \(i,j\in I\), and there exists a \(\mathbb{Q}\)-valued symmetric bilinear form \((\cdot,\cdot)\) on \(\mathsf{P}\) such that \[\langle h_i, \lambda\rangle = \frac{2({\mspace{1mu}\alpha}_i, \lambda)}{({\mspace{1mu}\alpha}_i, {\mspace{1mu}\alpha}_i)}\] for all \(i \in I\) and \(\lambda\in \mathsf{P}\). In this paper, we choose a bilinear form \((\cdot,\cdot)\) such that \(({\mspace{1mu}\alpha}_i, {\mspace{1mu}\alpha}_i)=2\) whenever \(\mathsf{C}\) is symmetric.

Let \(q\) be an indeterminate, and let \(\mathcal{U}_q(\mathsf{g})\) be the quantum group associated with \(\mathsf{C}\), which is the associative algebra over \(\mathbb{Q}(q)\) generated by \(e_i, f_i\) (\(i\in I\)) and \(q^h\) (\(h\in \mathsf{P}^\vee\)) satisfying certain condition (see [18] for details). We denote by \(\mathcal{U}_q^-(\mathsf{g}) = \langle f_i \mid i\in I \rangle\) the negative half of \(\mathcal{U}_q(\mathsf{g})\).

Let \(\mathsf{W}\) be the Weyl group associated with the Cartan matrix \(\mathsf{C}\), which is generated by \(s_i\) \((i \in I)\) given as \[s_i\lambda= \lambda-\langle h_i,\lambda\rangle{\mspace{1mu}\alpha}_i \quad \text{for \lambda\in \mathsf{P}}.\] We denote by \(\mathop{\mathrm{Br}}\) the \((\)generalized\()\) braid group associated with \(\mathsf{C}\). The braid group \(\mathop{\mathrm{Br}}\) has a group presentation: the generating set \(\{ \sigma_i^{\pm1} \mid i \in I\}\) and its defining relations \[\begin{align} \sigma_i \sigma_i^{-1} = \sigma_i^{-1}\sigma_i = {\rm id} \quad\text{{{and}}}\quad \underbrace{\sigma_i\sigma_j\sigma_i\sigma_j \cdots}_{m(i,j) \text{ times}} = \underbrace{\sigma_j\sigma_i\sigma_j\sigma_i \cdots}_{m(i,j) \text{ times}} \quad \text{ for i \ne j \in I}, \end{align}\] where \[\begin{align} m(i,j) \mathbin{:=} \begin{cases} c_{i,j} c_{j, i}+2 & \text{ if } c_{i,j} c_{j, i} \leqslant 2,\\ 6 & \text{ if } c_{i,j} c_{j, i} = 3, \\ \infty & \text{ otherwise.} \end{cases} \end{align}\] We set \(\mathop{\mathrm{Br}}^+\) to be the submonoid of \(\mathop{\mathrm{Br}}\) generated by \(\sigma_i\) for all \(i\in I\). There is a natural group epimorphism \(\mathop{\mathrm{Br}}\twoheadrightarrow \mathsf{W}\) sending \(\sigma_{i}\) to \(s_i\) for \(i\in I\). For an element \(\mathtt{b}\in \mathop{\mathrm{Br}}^+\) (resp.\(w \in \mathsf{W}\)), we denote by \(\ell(\mathtt{b})\) (resp.\(\ell(w)\)) the length of \(\mathtt{b}\) (resp.\(w\)). We denote the canonical lift of \(w \in \mathsf{W}\) to the positive braid monoid \(\mathop{\mathrm{Br}}^+\) by \(\mathtt{b}(w)\). When \(\mathsf{W}\) is of finite type, we define \[\begin{align} \label{Eq:32D32w0} \Delta := \mathtt{b}(w_\circ) \in \mathop{\mathrm{Br}}^+, \end{align}\tag{1}\] where \(w_\circ\) is the longest element in \(\mathsf{W}\). We call \(\Delta\) the half twist. For \(i\in I\), we define \(i^*\) to be the index defined as follows: \[\begin{align} \label{Eq:istar} {\mspace{1mu}\alpha}_{i^*} = - w_\circ ({\mspace{1mu}\alpha}_i). \end{align}\tag{2}\] Note that the correspondence \(i \mapsto i^*\) gives an involution on the index set \(I\).

We now define expression sequences of \(\mathtt{b}\in \mathop{\mathrm{Br}}^+\) and \(w \in \mathsf{W}\) as follows.

1. A sequence \({\boldsymbol{i}}=(i_1,i_2,\ldots,i_r) \in I^{r}\) is said to be an expression (or sequence) of \(\mathtt{b}\) if \(\mathtt{b}= \sigma_{i_1}\sigma_{i_2}\cdots \sigma_{i_r}\).

2. A sequence \({\boldsymbol{i}}=(i_1,\ldots,i_{r}) \in I^{r}\) is said to be a reduced expression (or reduced sequence) of \(w\) if \(w = s_{i_1}\cdots s_{i_{r}}\) and \(\ell(w) = r\).

We denote by \(\mathrm{Seq}(\mathtt{b})\) (resp.\(\mathrm{Seq}(w)\)) the set of all expressions of \(\mathtt{b}\) (resp.\(w\)). We often write \({\underline{\Delta}}\) for an expression of \(\Delta\) in \(\mathrm{Seq}(\Delta)\).

2.2 Cluster algebras↩︎

In this subsection, we recall the definition of a cluster algebra ([1]).

Let \(K\) be a finite index set with the decomposition \(K = K^{\mathrm{ex}} \sqcup K^{\mathrm{fr}}\). We call \(K^{\mathrm{ex}}\) the set of exchangeable indices and \(K^{\mathrm{fr}}\) the set of frozen indices. Let \(\mathfrak{F}\) be a field of characteristic 0.

A seed is a pair \(\Sigma= (\mathbf{x}, B)\) consisting of

• a set \(\mathbf{x} = \{ x_k \mid k \in K \}\) algebraically independent in \(\mathfrak{F}\),

• an integer matrix \[B = (b_{i,j})_{(i,j) \in K \times K^{\mathrm{ex}}}\] whose principal part, namely the submatrix indexed by \(K^{\mathrm{ex}} \times K^{\mathrm{ex}}\), is skew-symmetric.

The exchange matrix \(B\) determines a quiver \(Q\) with no loops or 2-cycles. The vertex set of \(Q\) is \(K\), and the arrows of \(Q\) are determined by \(B\) as follows:

• if \(b_{i,j} > 0\), then there are \(b_{i,j}\) arrows from \(i\) to \(j\),

• if \(b_{i,j} < 0\), then there are \(-b_{i,j}\) arrows from \(j\) to \(i\),

• if \(b_{i,j} = 0\), then there are no arrows between \(i\) and \(j\).

In this setting, vertices in \(K^{\mathrm{fr}}\) are called frozen vertices, and vertices in \(K^{\mathrm{ex}}\) are called mutable vertices. We depict \(K^{\mathrm{fr}}\) by rectangular nodes and \(K^{\mathrm{ex}}\) by circular nodes. We write \(B(\Sigma)\) and \(Q(\Sigma)\) for the above \(B\) and \(Q\) respectively, and we sometimes write such a seed by \(\Sigma = (\mathbf{x}, Q)\) instead of \(\Sigma = (\mathbf{x}, B)\).

A labeled quiver \(Q\) is a quiver together with an assignment \(l : Q_0 \to \mathcal{L}\), where \(Q_0\) is the set of vertices of \(Q\) and \(\mathcal{L}\) is the set of labels. Unless otherwise specified for \(\mathcal{L}\), we understand \(\mathcal{L}= \{ 1,2, \ldots, m \}\), where \(m = |Q_0|\). For a (labeled) quiver \(Q\), we denote by \(Q^{\rm op}\) the opposite quiver of \(Q\), which is the quiver defined by keeping the same set of vertices (and the same labeling) of \(Q\) and reversing all arrows of \(Q\). Note that, if \(B\) is the exchange matrix corresponding to \(Q\), then \(-B\) is the exchange matrix corresponding to \(Q^{\rm op}\).

For any exchangeable index \(k \in K^{\mathrm{ex}}\), the mutation in direction \(k\), denoted by \(\mu_k\), transforms a seed \(\Sigma= (\mathbf{x}, B)\) into a new seed \(\Sigma' = \mu_k(\Sigma) := (\mathbf{x}', B')\). The entries of the mutated exchange matrix \(B' = (b'_{i,j})\) are given by \[b'_{i,j} = \begin{cases} - b_{i,j} & \text{if } i = k \text{ or } j = k, \\[2pt] b_{i,j} + \dfrac{|b_{i,k}| b_{k,j} + b_{i,k} |b_{k,j}|}{2} & \text{otherwise}, \end{cases}\] and the cluster variables \(\mathbf{x}' = \mu_k(\mathbf{x}) = \{ x'_j \mid j \in K \}\) are updated according to the exchange relation: \[x'_j = \begin{cases} \displaystyle{x_k^{-1} \left( \prod_{\substack{i \in K \\ b_{i,k}>0}} x_i^{b_{i,k}} + \prod_{\substack{i \in K \\ b_{i,k}<0}} x_i^{-b_{i,k}} \right)} & \text{if } j=k, \\ x_j & \text{if } j \neq k. \end{cases}\] Two seeds are mutation equivalent if they are connected by a finite sequence of mutations.

We fix an initial seed \(\Sigma= (\mathbf{x}, B)\) (equivalently, \(\Sigma = (\mathbf{x}, Q)\)). Let \(\mathbb{P} = \mathbb{C}[x_j^{\pm 1} \mid j \in K^{\mathrm{fr}}]\) be the ring of Laurent polynomials in the frozen variables, and let \(\overline{\mathbb{P}} = \mathbb{C}[x_j \mid j \in K^{\mathrm{fr}}]\) be the ring of polynomials in the frozen variables.

For an algebra \(A\) in \(\mathfrak{F}\), we say that \(A\) has a cluster algebra structure if there exists a family \(F\) of seeds \(\Sigma= (\mathbf{x}, B)\) of \(A\), i.e., \(\mathbf{x} \subset A\), such that

1. for any seed \(\Sigma\in F\), the cluster algebra \(\mathscr{A}(\Sigma)\) (resp.\({\mathscr{A}_\bullet}(\Sigma)\)) coincides with \(A\),

2. any mutation of a seed in \(F\) is in \(F\),

3. any pair of seeds \(\Sigma\) and \(\Sigma'\) in \(F\) are connected by a finite sequence of mutations.

3 Combinatorics of \(i\)-boxes↩︎

In this section, we review the notion of \(i\)-boxes following [9], [20], [21]. An \(i\)-box is a combinatorial object that yields both of the seeds for the bosonic extensions and the monoidal seeds for the Hernandez–Leclerc categories.

Let \(\mathsf{C}\) be a Cartan matrix of finite \(ADE\) type with an index set \(I\), and let \(\mathop{\mathrm{Br}}\) be the braid group associated with \(\mathsf{C}\). Throughout this section, we choose and fix an expression sequence \[\begin{align} {\boldsymbol{i}}= (i_1, i_2, \ldots, i_r) \in \mathrm{Seq}(\mathtt{b}) \end{align}\] for \(\mathtt{b}\in \mathop{\mathrm{Br}}^+\).

3.1 Chains of \(i\)-boxes↩︎

For an index \(1 \leqslant k \leqslant r\) and \(j \in I\), we define the nearest indices at which \(j\) occurs to the right and to the left of \(k\) by \[\begin{align} k_{\boldsymbol{i}}(j)^+ &:= \min \bigl(\{ t \in [1,r] \mid t\geqslant k,\; i_t=j \} \sqcup \{ +\infty \} \bigr), \\ k_{\boldsymbol{i}}(j)^- &:= \max \bigl(\{t \in [1,r] \mid t\leqslant k,\; i_t=j \} \sqcup \{-\infty\} \bigr). \end{align}\] In addition, we define the strictly next and strictly previous indices at which \(i_k\) occurs by \[\begin{align} k_{\boldsymbol{i}}^+ &:= \min \bigl(\{ t \in [1,r]\mid t> k,\; i_t= i_k \} \sqcup \{+\infty\} \bigr), \\ k_{\boldsymbol{i}}^- &:= \max \bigl(\{t \in [1,r] \mid t< k,\; i_t=i_k \} \sqcup \{-\infty\} \bigr). \end{align}\] We sometimes omit the subscript \({\boldsymbol{i}}\) when the expression is clear from the context.

An admissible chain \(\mathfrak{C}=(\mathfrak{c}_t)_{1\leqslant t\leqslant r}\) is uniquely determined by its initial envelope \(\widetilde{\mathfrak{c}}_1=[c,c]\) and its sequence of horizontal moves \(\mathfrak{H}=(\mathcal{H}_1,\ldots,\mathcal{H}_{r-1})\in\{\mathcal{L},\mathcal{R}\}^{r-1}\), where \(c\) equals the number of occurrences of \(\mathcal{L}\) in \(\mathfrak{H}\) plus one. We often shortly write \(\mathfrak{H}\) for the pair \((c, \mathfrak{H})\) when no confusion arises. For \(t\geqslant 2\), the envelope \(\widetilde{\mathfrak{c}}_t\) is obtained from \(\widetilde{\mathfrak{c}}_{t-1}=[\widetilde{a}_{t-1},\widetilde{b}_{t-1}]\) by \[\widetilde{\mathfrak{c}}_t= \begin{cases} [\widetilde{a}_{t-1}-1,\widetilde{b}_{t-1}] & if \mathcal{H}_{t-1}=\mathcal{L} (\emph{left move}),\\ [\widetilde{a}_{t-1},\widetilde{b}_{t-1}+1] & if \mathcal{H}_{t-1}=\mathcal{R} (\emph{right move}). \end{cases}\] We call \(\mathfrak{H}\) the LR sequence of \(\mathfrak{C}\). The unique element \(z_t\in\widetilde{\mathfrak{c}}_t\setminus\widetilde{\mathfrak{c}}_{t-1}\) is called the effective end of \(\mathfrak{c}_t\); equivalently, \(z_t=\widetilde{a}_t\) if \(\mathcal{H}_{t-1}=\mathcal{L}\) and \(z_t=\widetilde{b}_t\) if \(\mathcal{H}_{t-1}=\mathcal{R}\). We often denote \(z_t\) by \(z_{\mathfrak{c}_t}\) when we wish to emphasize the dependence on the \(i\)-box \(\mathfrak{c}_t\) rather than on the position \(t\) in the chain. Note that the \(i\)-box \(\mathfrak{c}_t\) is given by \[\mathfrak{c}_t = \begin{cases} [\widetilde{a}_t,\widetilde{b}_t\} & \text{if } \mathcal{H}_{t-1}=\mathcal{L},\\ \{\widetilde{a}_t,\widetilde{b}_t] & \text{if } \mathcal{H}_{t-1}=\mathcal{R}. \end{cases}\]

3.2 Box moves↩︎

Let \(\mathfrak{C}=(\mathfrak{c}_t)_{1\leqslant t\leqslant r}\) be the admissible chain of \(i\)-boxes associated with \({\boldsymbol{i}}\) determined by a pair \((c,\mathfrak{H})\) of \(c\in [1,r]\) and an LR sequence \(\mathfrak{H}\).

For \(1\leq k < r\), we say that \(\mathfrak{c}_k\) is movable if \(k=1\), or if \(k\geqslant 2\) and \(\mathcal{H}_{k-1}\neq\mathcal{H}_k\). For a movable \(\mathfrak{c}_k\) in \(\mathfrak{C}\), the box move of \(\mathfrak{C}\) at \(k\) is the admissible chain \(\mathcal{B}_k(\mathfrak{C})\) which is determined by the pair \((c',\mathfrak{H}')\) such that \[{\rm (a)}~~ c'=\begin{cases} c-1 & if k=1 and \mathcal{H}_1=\mathcal{L},\\ c+1 & if k=1 and \mathcal{H}_1=\mathcal{R},\\ c & if k>1, \end{cases} \quad {\rm (b)}~~ \mathcal{H}'_s=\begin{cases} \mathcal{R}& if s\in \{ k-1,k \} and \mathcal{H}_s=\mathcal{L},\\ \mathcal{L}& if s\in \{ k-1,k \} and \mathcal{H}_s=\mathcal{R},\\ \mathcal{H}_s & if s\not\in \{ k-1,k \}. \end{cases}\] That is, \(\mathcal{B}_k(\mathfrak{C})\) is obtained from \(\mathfrak{C}\) by moving \(\widetilde{\mathfrak{c}}_k\) by \(1\) to the right or to the left inside \(\widetilde{\mathfrak{c}}_{k+1}\).

We now construct an exchange matrix associated with the admissible chain \(\mathfrak{C}=(\mathfrak{c}_t)_{1\leqslant t\leqslant r}\). We define the index sets as follows: \[\begin{align} \label{eq:32frakC32index} \mathsf{K}(\mathfrak{C}) &:= \{1,2,\ldots,r\}, \nonumber\\ \mathsf{K}(\mathfrak{C})^{\mathrm{fr}} &:= \{\, t \in \mathsf{K}(\mathfrak{C}) \mid \mathfrak{c}_t = [1(i)^+,\, r(i)^-] \text{ for some } i \in I \,\},\\ \mathsf{K}(\mathfrak{C})^{\mathrm{ex}} &:= \mathsf{K}(\mathfrak{C}) \setminus \mathsf{K}(\mathfrak{C})^{\mathrm{fr}}. \nonumber \end{align}\tag{3}\] Following [20], we associate to an admissible chain \(\mathfrak{C}\) an exchange matrix \(B(\mathfrak{C})\) and the corresponding quiver \(Q(\mathfrak{C})\).

As a special case, we consider the admissible chain of \(i\)-boxes associated with \({\boldsymbol{i}}\) \[\begin{align} \label{Eq:32C32calR} \mathfrak{C}_{\mathcal{R}} := (\mathfrak{c}_t)_{1\leqslant t\leqslant r} \end{align}\tag{4}\] determined by the LR sequence \((\mathcal{R},\mathcal{R},\ldots,\mathcal{R})\). Then \(\mathfrak{c}_t = \{ 1, t ]\) for \(t\in [1,r]\) and the corresponding sequence of colors coincides with \({\boldsymbol{i}}\), i.e., \[(\mathrm{color}(\mathfrak{c}_t))_{1\leqslant t\leqslant r} = {\boldsymbol{i}}.\] In this case, the exchange matrix is given by \[\begin{align} B(\mathfrak{C}_{\mathcal{R}}) =(b_{s,t})_{(s,t)\in \mathsf{K}(\mathfrak{C}_{\mathcal{R}})\times \mathsf{K}(\mathfrak{C}_{\mathcal{R}})^{\mathrm{ex}} } \end{align}\] where the entries \(b_{s,t}\) are given as follows: \[\begin{align} \label{eq:32exchange32matrix32for32RRR} b_{s,t} := \begin{cases} 1 & \text{if either } s=t^+ \text{ or } \bigl( s < t < s^+ < t^+ \text{ and } ({\mspace{1mu}\alpha}_{i_s},{\mspace{1mu}\alpha}_{i_t})=-1 \bigr),\\[2pt] -1 & \text{if either } t=s^+ \text{ or } \bigl( t < s < t^+ < s^+ \text{ and } ({\mspace{1mu}\alpha}_{i_s},{\mspace{1mu}\alpha}_{i_t})=-1 \bigr),\\[2pt] 0 & \text{otherwise.} \end{cases} \end{align}\tag{5}\] Note that the quiver \(Q(\mathfrak{C}_{\mathcal{R}})\) is the GLS quiver associated with \({\boldsymbol{i}}\) (see [21]).

4 Bosonic extensions for quantum groups↩︎

In this section, we recall the bosonic extension for quantum groups ([3], [7], [8], [22], [23]) and its cluster algebra structure arising from \(i\)-boxes.

4.1 Bosonic extensions↩︎

Let \(\mathsf{C}= (c_{i,j})_{i,j\in I}\) be a Cartan matrix of finite \(ADE\) type and let \(\mathcal{U}_q(\mathfrak{g})\) be the corresponding quantum group. We denote by \(\mathop{\mathrm{Br}}= \langle \sigma_i \mid i \in I \rangle\) the braid group of \(\mathfrak{g}\).

For integers \(m \geqslant n \geqslant 0\), we define the \(q\)-integers, factorials, and binomial coefficients by \[[n]_q := \frac{q^n-q^{-n}}{q-q^{-1}}, \qquad [n]_q! := \prod_{t=1}^n [t]_q, \qquad \begin{bmatrix} m\\ n\end{bmatrix}_q := \frac{[m]_q!}{[n]_q!\,[m-n]_q!}.\] The bosonic extension \(\widehat{\mathcal{A}}\) of \(\mathcal{U}_q^-(\mathfrak{g})\) is the \({\mathbb{Q}(q^{1/2})}\)-algebra generated by \(\{ f_{i,p} \}_{(i,p) \in I \times \mathbb{Z}\mspace{1mu}}\) subject to the following relations: for any \(i,j \in I\) and \(m,p \in \mathbb{Z}\mspace{1mu}\),

1. \(\displaystyle \sum^{1-c_{i,j} }_{t=0} (-1)^t \begin{bmatrix} 1-c_{i,j} \\ t \end{bmatrix}_{q} {f_{i,p}}^{1-c_{i,j} -t} f_{j,p} f_{i,p}^{t} = 0\) if \(i \ne j\),

2. \(f_{i,m} f_{j,p} = q^{(-1)^{p-m+1}({\mspace{1mu}\alpha}_i,{\mspace{1mu}\alpha}_j)} f_{j,p} f_{i,m} + \delta_{i,j}\delta_{p,m+1}(1- q^{2})\) if \(m < p\).

We define the subalgebras \(\widehat{\mathcal{A}}_{\leqslant k}\) (resp.\(\widehat{\mathcal{A}}_{\geqslant k}\)) as the \(\mathbb{Q}(q^{1/2})\)-subalgebras of \(\widehat{\mathcal{A}}\) generated by \(\{f_{i, m} \mid i \in I, m \leqslant k\}\) (resp.\(\{f_{i,m} \mid i \in I, m \geqslant k\}\)), and set \(\widehat{\mathcal{A}}_{< k} := \widehat{\mathcal{A}}_{\leqslant k-1}\) and \(\widehat{\mathcal{A}}_{> k} := \widehat{\mathcal{A}}_{\geqslant k+1}\).

For each \(i \in I\), we denote by \(\mathrm{T}_i\in\mathrm{Aut}_{\mathbb{Q}(q^{1/2})}(\widehat{\mathcal{A}})\) the \({\mathbb{Q}(q^{1/2})}\)-algebra automorphism defined by \[\begin{align} \mathrm{T}_i(f_{j,m}) = \begin{cases} f_{j, m+\delta_{i,j}} & \text{if } ({\mspace{1mu}\alpha}_i,{\mspace{1mu}\alpha}_j) \ne -1, \\ \dfrac{ q^{1/2} f_{j,m}f_{i,m} - q^{-1/2}f_{i,m}f_{j,m} }{q - q^{-1}} & \text{if } ({\mspace{1mu}\alpha}_i,{\mspace{1mu}\alpha}_j) = -1. \end{cases} \end{align}\] As the family \(\{\mathrm{T}_i\}_{i\in I}\) satisfies the braid relations ([5]–[7]), one can write \[\mathrm{T}_\mathtt{b}:=\mathrm{T}_{i_1}^{\epsilon_1}\cdots \mathrm{T}_{i_r}^{\epsilon_r}\] for any \(\mathtt{b}\in\mathop{\mathrm{Br}}\) written as \(\mathtt{b}=\sigma_{i_1}^{\epsilon_1}\cdots \sigma_{i_r}^{\epsilon_r}\) with \(\epsilon_t\in\{\pm1\}\).

Let \(\mathtt{b}\in\mathop{\mathrm{Br}}^+\) and define \[\widehat{\mathcal{A}}(\mathtt{b}):=\widehat{\mathcal{A}}_{\geqslant 0}\cap \mathrm{T}_\mathtt{b}(\widehat{\mathcal{A}}_{<0}).\] Let us take \({\boldsymbol{i}}=(i_1,i_2,\dots,i_r)\in\mathrm{Seq}(\mathtt{b})\), and define \[\mathsf{p}_{{\boldsymbol{i}}, t} := q^{1/2} \mathrm{T}_{i_1}\cdots \mathrm{T}_{i_{t-1}}(f_{i_t,0}) \quad \text{ for } 1\leqslant t \leqslant r.\] It was shown in [7], [8] that \(\widehat{\mathcal{A}}(\mathtt{b})\) is generated by \(\mathsf{p}_{{\boldsymbol{i}}, t}\) for \(t\in [1,r]\) as an algebra. The generators \(\mathsf{p}_{{\boldsymbol{i}}, t}\) are called PBW vectors of \(\widehat{\mathcal{A}}(\mathtt{b})\) with respect to \({\boldsymbol{i}}\) (see [7], [8] for details). We define \(\widehat{\mathcal{A}}_{ \mathbb{Z}\mspace{1mu}[q^{\pm 1/2}] }(\mathtt{b})\) to be the \({ \mathbb{Z}\mspace{1mu}[q^{\pm 1/2}] }\)-vector space generated by the ordered products \(\mathsf{p}_{{\boldsymbol{i}}, r}^{a_r} \mathsf{p}_{{\boldsymbol{i}}, r-1}^{a_{r-1}} \cdots \mathsf{p}_{{\boldsymbol{i}}, 1}^{a_1}\) for all \((a_1, \ldots, a_r) \in \mathbb{Z}\mspace{1mu}_{\geqslant 0}^{r}\). Since \(\widehat{\mathcal{A}}_{ \mathbb{Z}\mspace{1mu}[q^{\pm 1/2}] }(\mathtt{b})\) forms a ring (see [23]), we define the specialization of \(\widehat{\mathcal{A}}(\mathtt{b})\) at \(q^{1/2}=1\) by \[\mathbb{A}(\mathtt{b}) := \widehat{\mathcal{A}}_{ \mathbb{Z}\mspace{1mu}[q^{\pm 1/2}] }(\mathtt{b}) \big/ (q^{1/2}-1)\widehat{\mathcal{A}}_{ \mathbb{Z}\mspace{1mu}[q^{\pm 1/2}] }(\mathtt{b}),\] and set \(\mathbb{A}_\mathbb{C}(\mathtt{b}) := \mathbb{C}\otimes_{\mathbb{Z}\mspace{1mu}} \mathbb{A}(\mathtt{b})\). We set \[\begin{align} \label{Eq:32PBW32vector} \overline{\mathsf{p}}_{{\boldsymbol{i}}, t} := \mathsf{p}_{{\boldsymbol{i}}, t}|_{q^{1/2}=1} \in \mathbb{A}(\mathtt{b}) \qquad \text{for t \in [1,r]}. \end{align}\tag{8}\] It was shown in [7], [8] that \(\mathbb{A}(\mathtt{b})\) is the polynomial ring with the generators \(\overline{\mathsf{p}}_{{\boldsymbol{i}}, t}\) for \(t \in [1,r]\).

4.2 Cluster structure of \(\mathbb{A}(\mathtt{b})\)↩︎

Recall the \(i\)-boxes and related notations in Section 3.

Let \(\mathtt{b}\in \mathop{\mathrm{Br}}^+\) and take \({\boldsymbol{i}}=(i_1,i_2,\ldots,i_r) \in \mathrm{Seq}(\mathtt{b})\). For any \(i\)-box \([a,b]\) of \({\boldsymbol{i}}\), we define the element \(\mathsf{D}_{{\boldsymbol{i}}}[a,b] \in \mathbb{A}(\mathtt{b})\) inductively as follows: \[\label{Def:32minors} \begin{align} & \mathsf{D}_{{\boldsymbol{i}}}[k] := \overline{\mathsf{p}}_{{\boldsymbol{i}}, k} \qquad \text{ for any k\in [1, r]}, \\ & \mathsf{D}_{{\boldsymbol{i}}}[a,b] \mathsf{D}_{ {\boldsymbol{i}}}[a^+,b^-] = \mathsf{D}_{{\boldsymbol{i}}}[a^+,b] \mathsf{D}_{ {\boldsymbol{i}}}[a,b^-] - \prod_{ \substack{ j \in I \\ ({\mspace{1mu}\alpha}_{i_a},{\mspace{1mu}\alpha}_j)=-1}} \mathsf{D}_{{\boldsymbol{i}}}[a(j)^+,b(j)^-]. \end{align}\tag{9}\] We use the convention that \(\mathsf{D}_{{\boldsymbol{i}}}[\emptyset]=1\). Note that the second relation in 9 comes from the \(T\)-systems among affine determinantial modules in the Hernandez–Leclerc category (see Theorem 6 and Proposition 11), which is a generalization of \(T\)-systems among Kirillov–Reshetikhin modules. For simplicity, we write \(\mathsf{D}[a,b]\) and \(\overline{\mathsf{p}}_{ k}\) for \(\mathsf{D}_{{\boldsymbol{i}}}[a,b]\) and \(\overline{\mathsf{p}}_{{\boldsymbol{i}}, k}\), respectively, if no confusion arises.

Let \(\mathfrak{C}=(\mathfrak{c}_t)_{1\leqslant t\leqslant r}\) be an admissible chain of \(i\)-boxes associated with \({\boldsymbol{i}}\). We now consider the index sets \(\mathsf{K}(\mathfrak{C})\), \(\mathsf{K}(\mathfrak{C})^{\mathrm{fr}}\) and \(\mathsf{K}(\mathfrak{C})^{\mathrm{ex}}\) given in 3 , and define \[\begin{align} \label{eq:32frakC32cluster} \mathsf{D}(\mathfrak{C}) &:= \{\, \mathsf{D}(\mathfrak{c}_t) \mid t \in \mathsf{K}(\mathfrak{C}) \,\} \subset \mathbb{A}(\mathtt{b}). \end{align}\tag{10}\]

The box moves \(\mathcal{B}_k\) of \(\mathfrak{C}\) can be understood as mutations (see [21] and also [17]).

Thanks to Theorem 1, we have an isomorphism of \(\mathbb{C}\)-algebras \[{\mathscr{A}_\bullet}(\Sigma(\mathfrak{C})) \simeq \mathbb{A}_\mathbb{C}(\mathtt{b}),\] where \({\mathscr{A}_\bullet}(\Sigma(\mathfrak{C}))\) is the partially compactified cluster algebra over \(\mathbb{C}\) in Definition 1.

For any \({\boldsymbol{i}}= (i_1, \ldots, i_r) \in \mathrm{Seq}(\mathtt{b})\), we set \[\begin{align} \mathcal{P}({\boldsymbol{i}}) := \mathbb{C}[x_{1}, x_{2}, \ldots ,x_{r}] \end{align}\] to be the polynomial ring in \(r\) variables \(x_{k}\) (\(k\in [1,r]\)). Then the PBW vectors of \(\mathbb{A}(\mathtt{b})\) associated with \({\boldsymbol{i}}\) give an isomorphism \[\begin{align} \psi_{{\boldsymbol{i}}} : \mathcal{P}({\boldsymbol{i}}) \buildrel \sim \over \longrightarrow \mathbb{A}_\mathbb{C}(\mathtt{b}), \qquad \text{ x_{k} \mapsto \overline{\mathsf{p}}_{{\boldsymbol{i}}, k} for anyk\in [1,r].} \end{align}\] We denote by \(\widetilde{\mathbb{A}}(\mathtt{b})\) (resp.\(\widetilde{\mathbb{A}}_\mathbb{C}(\mathtt{b})\)) the localization of \(\mathbb{A}(\mathtt{b})\) (resp.\(\mathbb{A}_\mathbb{C}(\mathtt{b})\)) at the frozen variables \[\begin{align} \mathrm{F}_{t} := \mathsf{D}(\mathfrak{c}_t) \qquad \text{for all t\in \mathsf{K}(\mathfrak{C})^{\mathrm{fr}}}, \end{align}\] and set \(\widetilde{\mathcal{P}}({\boldsymbol{i}})\) to be the localized ring of \(\mathcal{P}({\boldsymbol{i}})\) by the elements \(\psi_{\boldsymbol{i}}^{-1}(\mathrm{F}_{t})\) for all \(t\in \mathsf{K}(\mathfrak{C})^{\mathrm{fr}}\). Using the same notation, we have \[\begin{align} \label{Eq:32iso32PA} \psi_{{\boldsymbol{i}}} : \widetilde{\mathcal{P}}( {\boldsymbol{i}}) \buildrel \sim \over \longrightarrow \widetilde{\mathbb{A}}_\mathbb{C}(\mathtt{b}), \qquad \text{ x_{ k } \mapsto \overline{\mathsf{p}}_{{\boldsymbol{i}}, k} for anyk\in [1, r].} \end{align}\tag{11}\] This induces an isomorphism of algebras \[\mathscr{A}(\Sigma(\mathfrak{C})) \simeq \widetilde{\mathbb{A}}_\mathbb{C}(\mathtt{b}) \simeq \widetilde{\mathcal{P}}( {\boldsymbol{i}}),\] where \(\mathscr{A}(\Sigma(\mathfrak{C}))\) denotes the cluster algebra over \(\mathbb{C}\) in Definition 1.

5 Double inductive weaves↩︎

The Demazure weaves are combinatorial objects which provide initial seeds for the coordinate ring of a braid variety. In this section, we recall the notation for double inductive weaves following [10] for our purposes.

Recall the half twist \(\Delta = \mathtt{b}(w_\circ)\) given in 1 . The Demazure product \(\delta: \mathop{\mathrm{Br}}^+ \to \mathsf{W}\) is the map defined inductively by \(\delta(\sigma_i) = s_i\) \((i\in I)\) and, for any \(\mathtt{b}\in \mathop{\mathrm{Br}}^+\), \[\begin{align} \label{Eq:32Dp} \delta(\mathtt{b}\sigma_i) = \begin{cases} \delta(\mathtt{b}) s_i & \text{if } \ell(\delta(\mathtt{b})s_i) > \ell(\delta(\mathtt{b})), \\ \delta(\mathtt{b}) & \text{if } \ell(\delta(\mathtt{b})s_i) < \ell(\delta(\mathtt{b})). \end{cases} \end{align}\tag{12}\]

5.1 Demazure weaves↩︎

A Demazure weave \(\mathfrak{W}\subset \mathbb{R}^2\) is a planar graph where each edge is labeled by a simple root index \(i \in I\). The vertices of the graph fall into specific types as depicted in Figure 1: 6-valent vertices, 4-valent vertices, and 3-valent vertices. We also call the three diagrams in Figure 1 weave moves. By convention, we assume that all weaves are oriented downwards, and use the notation \(\mathfrak{W}: {\boldsymbol{i}}\to {\boldsymbol{j}}\) to specify that a weave \(\mathfrak{W}\) starts from the top with \({\boldsymbol{i}}\) and ends at the bottom with \({\boldsymbol{j}}\).

Let \({\boldsymbol{i}}\in \mathrm{Seq}(\mathtt{b})\) for \(\mathtt{b}\in \mathop{\mathrm{Br}}^+\) and let \({\boldsymbol{j}}\in \mathrm{Seq}(\delta(\mathtt{b}))\). As horizontal slices of a weave \(\mathfrak{W}: {\boldsymbol{i}}\to {\boldsymbol{j}}\) can be viewed as expression sequences of elements in \(\mathop{\mathrm{Br}}^+\), one can understand that a weave \(\mathfrak{W}\) represents a history of the weave moves applied to \({\boldsymbol{i}}\) at the top. Note that the weave moves are compatible with the Demazure product (see 12 ). When we would not emphasize the explicit description of \({\boldsymbol{j}}\), by abuse of notation, we may replace \({\boldsymbol{j}}\) with \(\delta({\boldsymbol{i}}) := \delta(\mathtt{b})\) in the weave, i.e., \(\mathfrak{W}: {\boldsymbol{i}}\to \delta({\boldsymbol{i}})\). Throughout this paper, a weave \(\mathfrak{W}: {\boldsymbol{i}}\to \delta({\boldsymbol{i}})\) will be referred to as a Demazure weave on \({\boldsymbol{i}}\).

Given a Demazure weave \(\mathfrak{W}\), we associate to it a quiver \(Q(\mathfrak{W})\). The vertex set of \(Q(\mathfrak{W})\) is identified with the set \(\mathfrak{W}_3\) of trivalent vertices in \(\mathfrak{W}\), and the arrows, as well as the set \(\mathcal{F}\subset \mathfrak{W}_3\) of frozen vertices, are determined by the local intersections of Lusztig cycles at \(\mathfrak{W}_3\) (see [10] for details).

An indexed weave is a weave with a bijection from the set \(\mathfrak{W}_3\) of trivalent vertices to \(\{ 1,2,\ldots,m \}\), where \(m:= |\mathfrak{W}_3|\) is the total number of trivalent vertices. If \(\mathfrak{W}\) is indexed, then the quiver \(Q(\mathfrak{W})\) can be understood as a labeled quiver whose vertices are in bijection with \(\{ 1,2,\ldots,m \}\).

Two weaves \(\mathfrak{W}_1, \mathfrak{W}_2: {\boldsymbol{i}}\to {\boldsymbol{j}}\) are said to be equivalent if they are related by a sequence of the following local moves:

1. isotopies of strands corresponding to braid relations in \(\mathop{\mathrm{Br}}^+\),

2. local moves involving \(i j i j \to j i j j \to j i j\) and \(i j i j \to i i j i \to i j i \to j i j\) for adjacent \(i\) and \(j\),

3. local moves involving \(i k i \to i i k \to i k \to k i\) and \(i k i \to k i i \to k i\) for non-adjacent \(i\) and \(k\).

Note that these local moves involve at most one trivalent vertex. Hence, by transporting the indexing, these moves also define an equivalence relation on the set of indexed weaves.

Equivalent (indexed) weaves induce the same (labeled) quivers and cluster variables (see, e.g., [10]). By abuse of notation, we will often denote by \(\mathfrak{W}\) the equivalence class of an (indexed) weave.

5.2 Double inductive weaves↩︎

A double string is a sequence of pairs \[\mathbf{s}=(i_1 \mathbf{X}_1, i_2 \mathbf{X}_2, \ldots, i_m \mathbf{X}_m),\] where \(i_t\in I\) and \(\mathbf{X}_t\in \{\mathbf{L}, \mathbf{R}\}\). Each double string \(\mathbf{s}\) determines a sequence \({\boldsymbol{i}}_0, {\boldsymbol{i}}_1, \ldots, {\boldsymbol{i}}_m\) recursively, where \(i_t \mathbf{X}_t\) specifies whether \(i_t\) is added on the left or the right of \({\boldsymbol{i}}_{t-1}\) according to \(\mathbf{X}_t\). More precisely, set \({\boldsymbol{i}}_0 := \emptyset\) and for \(t=1,\ldots,m\), define inductively \[{\boldsymbol{i}}_t := \begin{cases} (i_t)* {\boldsymbol{i}}_{t-1} & \text{if } \mathbf{X}_t=\mathbf{L},\\ {\boldsymbol{i}}_{t-1}* (i_t) & \text{if } \mathbf{X}_t=\mathbf{R}, \end{cases}\] where \({\boldsymbol{j}}_1*{\boldsymbol{j}}_2\) denotes the concatenation of \({\boldsymbol{j}}_1\) and \({\boldsymbol{j}}_2\). We denote by \({\boldsymbol{i}}(\mathbf{s})\) the final sequence \({\boldsymbol{i}}_m\), which is called the \({\boldsymbol{i}}\)-sequence of \(\mathbf{s}\). If adding the index \(i_t\) increases the length of the Demazure product, i.e., \(\ell(\delta({\boldsymbol{i}}_t))=\ell(\delta({\boldsymbol{i}}_{t-1}))+1\), we put the symbol “\(+\)” as a superscript on \(i_t\mathbf{X}_t\) for more clarity.

For any double string \(\mathbf{s}\), we define the corresponding double inductive weave \[\mathfrak{W}(\mathbf{s}): {\boldsymbol{i}}(\mathbf{s}) \to \delta({\boldsymbol{i}}(\mathbf{s}))\] as follows. We start with the empty weave. At the \(t\)-th stage, we add one strand to the current weave according to \(i_t\mathbf{X}_t\) and apply a sequence of weave moves to the bottom of the diagram:

1. Length-additive step (\(i_t \mathbf{X}_t^+\)): If \(\ell(\delta({\boldsymbol{i}}_t)) = \ell(\delta({\boldsymbol{i}}_{t-1})) + 1\), then we add a strand of color \(i_t\) on the side specified by \(\mathbf{X}_t\). This extends the sequence without introducing new trivalent vertices.

2. Non-additive step (\(i_t \mathbf{X}_t\)): If \(\ell(\delta({\boldsymbol{i}}_t)) = \ell(\delta({\boldsymbol{i}}_{t-1}))\), then the Demazure product already has a descent at \(i_t\) on the side \(\mathbf{X}_t\). We add a strand of color \(i_t\) to the side of the current weave specified by \(\mathbf{X}_t\), and merge the new strand with the current weave to make a 3-valent vertex on the side \(\mathbf{X}_t\) using 6-valent and 4-valent moves if necessary.

Since the trivalent vertices occur only in the non-additive steps, we order the set \(\mathfrak{W}_3 (\mathbf{s})\) of trivalent vertices from top to bottom according to the non-additive steps. This gives an indexing of the vertex set \(\mathfrak{W}_3 (\mathbf{s})\) by \(\{1,2,\ldots,m\}\), where \(m:=|\mathfrak{W}_3(\mathbf{s})|\). In this sense, we consider \(Q(\mathfrak{W}(\mathbf{s}))\) as a labeled quiver. We set \[Q(\mathbf{s}) := Q(\mathfrak{W}(\mathbf{s}))\] to be the quiver associated with the double string \(\mathbf{s}\). Note that the resulting weave \(\mathfrak{W}(\mathbf{s}): {\boldsymbol{i}}(\mathbf{s})\to\delta({\boldsymbol{i}}(\mathbf{s}))\) is well-defined up to weave equivalence.

A double inductive weave is called a right inductive weave if the associated double string is \((i_1\mathbf{R}, i_2\mathbf{R}, \ldots,i_m\mathbf{R}).\) In this case, putting \({\boldsymbol{i}}= (i_1, i_2, \ldots, i_m)\), we write \(\overrightarrow{\mathfrak{w}}({\boldsymbol{i}})\) for this right inductive weave for brevity, and call it the right inductive weave associated with \({\boldsymbol{i}}\).

Let \({\underline{\Delta}}\in \mathrm{Seq}(\Delta)\) be an expression sequence of the half twist \(\Delta\) defined in 1 . By definition, for any \({\boldsymbol{i}}\in \mathrm{Seq}(\mathtt{b})\), we have \(\delta({\underline{\Delta}}{\boldsymbol{i}}) = w_\circ\). Then the quiver of the right inductive weave \(\overrightarrow{\mathfrak{w}}({\underline{\Delta}}{\boldsymbol{i}})\) is described recursively as follows.

6 Braid varieties↩︎

This section reviews the cluster algebra structures on the coordinate rings of braid varieties and the connection to the (half decorated) double Bott-Samelson varieties, following [10]. This cluster algebra structure is combinatorially encoded by Demazure weaves. In particular, we will focus on the cluster algebra structures arising from double inductive weaves associated with double strings.

6.1 Braid varieties↩︎

Let \(\mathsf{G}\) be a simply connected simple algebraic group over \(\mathbb{C}\) of finite \(ADE\) type, and denote by \(\mathfrak{g}\) the corresponding Lie algebra. We fix a Borel subgroup \(\mathsf{B}\), denote by \(\mathsf{B}_-\) the opposite one, and set \(\mathsf{T}:= \mathsf{B}\cap \mathsf{B}_-\). Let \(\mathsf{W}= \langle s_i \mid i\in I\rangle\) be the Weyl group associated with \(\mathfrak{g}\) with the index set \(I\). Elements of the flag variety \(\mathfrak{Fl}:= \mathsf{G}/\mathsf{B}\) are considered as Borel subgroups of \(\mathsf{G}\) and the identity coset of \(\mathfrak{Fl}\) is called the standard flag corresponding to the Borel subgroup \(\mathsf{B}\). There is the Bruhat decomposition \(\mathsf{G}= \bigsqcup_{w \in \mathsf{W}} \mathsf{B}w \mathsf{B}\), where we identify \(\mathsf{W}\) with the quotient \(N_\mathsf{G}(\mathsf{T}) / \mathsf{T}\). We say that a pair of flags \((x\mathsf{B}, y\mathsf{B}) \in \mathfrak{Fl}\times \mathfrak{Fl}\) is in relative position \(w\) if \[x^{-1}y \in \mathsf{B}w \mathsf{B},\] which is denoted by \(x\mathsf{B}\xrightarrow{w} y\mathsf{B}\).

By [10], different choices of expression sequences of \(\mathtt{b}\) yield isomorphic varieties.

Following [10], we fix a pinning of the group \(\mathsf{G}\). This choice determines a homomorphism \(\phi_i: \mathrm{SL}_2(\mathbb{C}) \to \mathsf{G}\) for each \(i \in I\). We define the matrix \(B_i(z)\) for a parameter \(z \in \mathbb{C}\) by \[\begin{align} \label{eq:32B95i40z41} B_i(z) := \phi_i \begin{pmatrix} z & -1 \\ 1 & 0 \end{pmatrix} = x_i(z) \dot{s}_{i} \qquad \text{ for i\in I,} \end{align}\tag{14}\] where \(x_i(z)\) and \(\dot{s}_{i}\) are defined as \[x_i(z) := \phi_i \begin{pmatrix} 1 & z \\ 0 & 1 \end{pmatrix} \quad\text{{{and}}}\quad \dot{s}_{i} := \phi_i \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}.\] Note that \(\dot{s}_{ i } \in N_{\mathsf{G}}(T)\) is a representative of the simple reflection \(s_i \in \mathsf{W}\cong N_{\mathsf{G}}( \mathsf{T})/ \mathsf{T}\). For an expression sequence \({\boldsymbol{i}}= (i_{1}, \dots, i_{r}) \in \mathrm{Seq}(\mathtt{b})\) and a sequence of parameters \(z = (z_1, \dots, z_r) \in \mathbb{C}^r\), the element \(B_{{\boldsymbol{i}}}(z) \in \mathsf{G}\) is defined as the ordered product: \[\begin{align} \label{Eq:32Biz} B_{{\boldsymbol{i}}}(z) := B_{i_{1}}(z_1)B_{i_{2}}(z_2)\cdots B_{i_{r}}(z_r) \in \mathsf{G}. \end{align}\tag{15}\] By [10], the braid variety \(X({\boldsymbol{i}})\) for \({\boldsymbol{i}}= (i_1, \dots, i_r)\) can be realized as an affine subvariety of \(\mathbb{C}^r\) via the isomorphism \[\begin{align} \label{Eq:32Br32coord} X({\boldsymbol{i}}) \cong \left\{ (z_1, \dots, z_r) \in \mathbb{C}^{r} \;\middle|\; \delta(\mathtt{b})^{-1}B_{{\boldsymbol{i}}}(z) \in \mathsf{B}\right\}. \end{align}\tag{16}\] We now understand \(z_j \in \mathbb{C}[X({\boldsymbol{i}})]\) as the \(j\)-th coordinate function restricted from the ambient space \(\mathbb{C}^r\) under the isomorphism 16 . We refer to the tuple \(z = (z_1, \ldots, z_r)\) as the coordinates on the braid variety \(X({\boldsymbol{i}})\).

6.2 Cluster structure↩︎

Recall Demazure weaves and double strings in Section 5.

Let \(\mathfrak{W}: {\boldsymbol{i}}\to \delta({\boldsymbol{i}})\) be a Demazure weave for \({\boldsymbol{i}}= (i_1, i_2, \dots, i_r) \in \mathrm{Seq}(\mathtt{b})\) and assign coordinates \(z_1, \ldots, z_r\) to the top strands of the weave \(\mathfrak{W}\) from left to right. Following the inductive procedure of [10], we associate a regular function \(A_k(\mathfrak{W}) \in \mathbb{C}[X({\boldsymbol{i}})]\) to each \(k \in \mathfrak{W}_3\), which is expressed as a polynomial in \(z_1, \ldots, z_r\) (see [10]). We denote the collection of these functions by \[\begin{align} \label{Eq:32cv} \mathcal{A}(\mathfrak{W}) := \{ A_t(\mathfrak{W}) \}_{t\in [1,m]}, \end{align}\tag{17}\] where \(m = |\mathfrak{W}_3|\). The set \(\mathcal{A}(\mathfrak{W})\) together with the labeled quiver \(Q(\mathfrak{W})\) gives an initial seed for \(\mathbb{C}[X({\boldsymbol{i}})]\).

The double inductive weaves arising from double strings have enjoyable properties. For any two double strings \(\mathbf{s}_1\) and \(\mathbf{s}_2\) with \({\boldsymbol{i}}(\mathbf{s}_1)={\boldsymbol{i}}(\mathbf{s}_2)\), they are connected by the following two types of local moves: \[\label{Eq:32moves32for32strings} \begin{align} (j_1 \mathbf{L}, j_2 \mathbf{X}_2, \dots) &\longleftrightarrow (j_1 \mathbf{R}, j_2 \mathbf{X}_2, \dots), \\ (\dots, i \mathbf{L}, j \mathbf{R}, \dots) &\longleftrightarrow (\dots, j \mathbf{R}, i \mathbf{L}, \dots). \end{align}\tag{18}\] From the proof of [10], we have the following theorem.

We now recall the double Bott-Samelson variety following [10], [13]. For any expression sequence \({\boldsymbol{i}}= (i_1,i_2, \dots, i_r) \in \mathrm{Seq}(\mathtt{b})\), the (half decorated) double Bott-Samelson variety \(\mathop{\mathrm{Conf}}({\boldsymbol{i}})\) is defined as the subvariety of \(\mathbb{C}^r\): \[\mathop{\mathrm{Conf}}({\boldsymbol{i}}) := \{(z_1, \dots, z_{r}) \in \mathbb{C}^r \mid B_{{\boldsymbol{i}}}(z) \in \mathsf{B}_{-} \cdot\mathsf{B}\},\] where \(B_{{\boldsymbol{i}}}(z)\) is given in 15 . The variety \(\mathop{\mathrm{Conf}}({\boldsymbol{i}})\) is smooth and affine ([13]). Similar to the case of braid varieties, different expression sequences \({\boldsymbol{i}}\) for the same braid element \(\mathtt{b}\) yield isomorphic varieties.

We write \((w,z)\) for the coordinates of the braid variety \(X({\underline{\Delta}}{\boldsymbol{i}})\), where \(w = (w_1, \ldots, w_{ \ell({\underline{\Delta}})})\) (resp.\(z = (z_1, \ldots, z_r)\)) corresponds to the letters of \({\underline{\Delta}}\) (resp.\({\boldsymbol{i}}\)), and write \(z\) for the coordinates of the double Bott-Samelson variety \(\mathop{\mathrm{Conf}}({\boldsymbol{i}})\). By [10], there exists an isomorphism between the braid variety and the double Bott-Samelson variety \[\begin{align} \label{Eq:32X61Conf} \zeta_{\boldsymbol{i}}: X({\underline{\Delta}}{\boldsymbol{i}}) \buildrel \sim \over \longrightarrow \mathop{\mathrm{Conf}}({\boldsymbol{i}}) \end{align}\tag{19}\] defined by sending the coordinates \((w,z)\) to the coordinates \(z\).

Recall the right inductive weave \(\overrightarrow{\mathfrak{w}}({\underline{\Delta}}{\boldsymbol{i}})\) in Section 5.2. Under the isomorphism \(\zeta_{\boldsymbol{i}}\) in 19 , the cluster variables \(\mathcal{A}(\overrightarrow{\mathfrak{w}}({\underline{\Delta}}{\boldsymbol{i}}))\) (see 17 ) can be described in terms of generalized principal minors.

Thanks to Proposition 6, for any \({\boldsymbol{i}}= (i_1, i_2, \ldots, i_r) \in \mathrm{Seq}(\mathtt{b})\), we define \[\begin{align} \mathcal{B}({\boldsymbol{i}}) := \mathbb{C}\bigl[z_{ 1},z_{ 2}, \dots,z_{ r}\bigr] \bigl[f_i (z)^{-1} \mid i \in I \bigr]. \end{align}\] Throughout the remainder of the paper, we identify the three algebras \(\mathcal{B}({\boldsymbol{i}})\), \(\mathbb{C}\bigl[\mathop{\mathrm{Conf}}({\boldsymbol{i}})\bigr]\) and \(\mathbb{C}\bigl[X({\underline{\Delta}}{\boldsymbol{i}})\bigr]\) via the isomorphisms \[\begin{align} \label{Eq:32332iso} \mathcal{B}({\boldsymbol{i}}) \cong \mathbb{C}\bigl[\mathop{\mathrm{Conf}}({\boldsymbol{i}})\bigr] \cong \mathbb{C}\bigl[X({\underline{\Delta}}{\boldsymbol{i}})\bigr], \end{align}\tag{20}\] where the first isomorphism is given in Proposition 6 and the second is given by 19 .

7 Comparison of two cluster algebra structures↩︎

In this section, we establish an explicit isomorphism between the localized bosonic extension \(\widetilde{\mathbb{A}}(\mathtt{b})\) and the coordinate ring of the braid variety \(X({\underline{\Delta}}{\boldsymbol{i}})\) for \({\boldsymbol{i}}\in \mathrm{Seq}(\mathtt{b})\), and prove that this isomorphism respects the cluster structures. The comparison of these cluster structures relies on the construction of a Demazure weave associated with an admissible chain of \(i\)-boxes, which is formulated using double strings.

Throughout this section, let \(\mathsf{C}= (c_{i,j})_{i,j\in I}\) be a Cartan matrix of finite \(ADE\) type, and let \(\mathfrak{g}\) be the corresponding simple Lie algebra. We denote by \(\mathop{\mathrm{Br}}\) the generalized Braid group associated with \(\mathsf{C}\).

7.1 Double inductive weaves arising from \(i\)-boxes↩︎

Let \(\mathtt{b}\in \mathop{\mathrm{Br}}^+\) and take an expression \[{\boldsymbol{i}}:=( i_1, i_2, \ldots,i _r)\in\mathrm{Seq}(\mathtt{b}).\] Let \(\mathfrak{C}= (\mathfrak{c}_t)_{t \in [1,r]}\) be the admissible chain of \(i\)-boxes associated with \({\boldsymbol{i}}\) and denote its LR sequence by \(\mathfrak{H}=(\mathcal{H}_1,\ldots,\mathcal{H}_{r-1})\). For simplicity, we set \(\mathcal{H}_0 := \mathcal{R}\). For an expression sequence \({\underline{\Delta}}= (j_1, j_2, \ldots, j_\ell) \in \mathrm{Seq}(\Delta)\), we define \[\begin{align} \label{Eq:32s40C41} \mathbf{s}_{{\underline{\Delta}}}(\mathfrak{C}) := (j_1 \mathbf{R}^+, j_2 \mathbf{R}^+, \ldots, j_\ell \mathbf{R}^+, h_1 \mathbf{X}_1, h_2 \mathbf{X}_2, \ldots, h_r \mathbf{X}_r), \end{align}\tag{21}\] where \[\begin{align} h_t := \begin{cases} \mathrm{color}(\mathfrak{c}_t)^* & \text{if } \mathcal{H}_{t-1} = \mathcal{L}, \\ \mathrm{color}(\mathfrak{c}_t) & \text{if } \mathcal{H}_{t-1} = \mathcal{R}, \end{cases} \quad\text{{{and}}}\quad \mathbf{X}_t := \begin{cases} \mathbf{L}& \text{if } \mathcal{H}_{t-1} = \mathcal{L},\\ \mathbf{R}& \text{if } \mathcal{H}_{t-1} = \mathcal{R}, \end{cases} \end{align}\] (see 2 for \(i^*\)) and set \[\begin{align} {\boldsymbol{i}}^\mathrm{L}(\mathfrak{C}) &:= (h_t)_{\mathbf{X}_t=\mathbf{L}} \quad\text{(taken in decreasing order of t)},\\ {\boldsymbol{i}}^\mathrm{R}(\mathfrak{C}) &:= (h_t)_{\mathbf{X}_t=\mathbf{R}} \quad\text{(taken in increasing order of t)}. \end{align}\] Then the \({\boldsymbol{i}}\)-sequence of the double string \(\mathbf{s}_{{\underline{\Delta}}}(\mathfrak{C})\) is given in terms of \({\boldsymbol{i}}^\mathrm{L}(\mathfrak{C})\) and \({\boldsymbol{i}}^\mathrm{R}(\mathfrak{C})\) as follows: \[{\boldsymbol{i}}_{\underline{\Delta}}(\mathfrak{C}):= {\boldsymbol{i}}(\mathbf{s}_{{\underline{\Delta}}}(\mathfrak{C})) = {\boldsymbol{i}}^\mathrm{L}(\mathfrak{C})\,{\underline{\Delta}}\,{\boldsymbol{i}}^\mathrm{R}(\mathfrak{C}).\] We define \[\begin{align} \label{Eq:32fWT} \mathfrak{W}_{{\underline{\Delta}}}^{\rm T}(\mathfrak{C}) : {\underline{\Delta}}{\boldsymbol{i}}\rightarrow {\boldsymbol{i}}_{\underline{\Delta}}(\mathfrak{C}) \end{align}\tag{22}\] to be the Demazure weave consisting of the braid moves, i.e., 4-valent and 6-valent vertices in Figure 1, and set \(\mathfrak{W}_{{\underline{\Delta}}}^{B}(\mathfrak{C})\) to be the double inductive weave associated with \(\mathbf{s}_{{\underline{\Delta}}}(\mathfrak{C})\) (see Section 5.2), i.e., \[\begin{align} \label{Eq:32fWB} \mathfrak{W}_{{\underline{\Delta}}}^{B}(\mathfrak{C}) := \mathfrak{W}( \mathbf{s}_{{\underline{\Delta}}}(\mathfrak{C})) :{\boldsymbol{i}}_{\underline{\Delta}}(\mathfrak{C}) \rightarrow \delta ( {\boldsymbol{i}}_{\underline{\Delta}}(\mathfrak{C})). \end{align}\tag{23}\] Note that \(\delta ( {\boldsymbol{i}}_{\underline{\Delta}}(\mathfrak{C})) = w_\circ\) in the Weyl group \(\mathsf{W}\). We now define \[\begin{align} \mathfrak{W}_{{\underline{\Delta}}}(\mathfrak{C}) := \mathfrak{W}_{{\underline{\Delta}}}^{B}(\mathfrak{C}) \circ \mathfrak{W}_{{\underline{\Delta}}}^{T}(\mathfrak{C}) : {\underline{\Delta}}{\boldsymbol{i}}\rightarrow \delta ( {\boldsymbol{i}}_{\underline{\Delta}}(\mathfrak{C})), \end{align}\] where \(\mathfrak{W}_2\circ\mathfrak{W}_1\) denotes the vertical concatenation obtained by placing \(\mathfrak{W}_2\) below \(\mathfrak{W}_1\). Although the weave \(\mathfrak{W}_{{\underline{\Delta}}}(\mathfrak{C})\) itself depends on the choice of \(\mathfrak{W}_{{\underline{\Delta}}}^{\rm T}(\mathfrak{C})\), all such \(\mathfrak{W}_{{\underline{\Delta}}}(\mathfrak{C})\) belong to the same equivalence class (see Section 5.1). Thanks to Proposition 7 below, the seeds arising from \(\mathfrak{W}_{{\underline{\Delta}}}(\mathfrak{C})\) do not depend on the choice of the weave \(\mathfrak{W}_{{\underline{\Delta}}}^{\rm T}(\mathfrak{C})\).

We call \(\mathbf{s}_{{\underline{\Delta}}}(\mathfrak{C})\) (resp.\(\mathfrak{W}_{{\underline{\Delta}}}(\mathfrak{C})\)) the double string (resp.the Demazure weave) associated with \(\mathfrak{C}\). We sometimes write \(\mathbf{s}(\mathfrak{C})\) and \(\mathfrak{W}(\mathfrak{C})\) instead of \(\mathbf{s}_{{\underline{\Delta}}}(\mathfrak{C})\) and \(\mathfrak{W}_{{\underline{\Delta}}}(\mathfrak{C})\) respectively when there is no need to emphasize the expression \({\underline{\Delta}}\).

7.2 Box moves and weaves↩︎

In this subsection, we will investigate how box moves on an admissible chain \(\mathfrak{C}\) affect the weave \(\mathfrak{W}_{{\underline{\Delta}}}(\mathfrak{C})\).

We first consider the weave \(\mathfrak{W}_{{\underline{\Delta}}}^B(\mathfrak{C})\) given in 23 . Recall the admissible chain \(\mathfrak{C}_{\mathcal{R}}\) determined by the LR sequence \((\mathcal{R},\mathcal{R},\ldots,\mathcal{R})\) (see 4 ).

The following proposition shows that the quivers and cluster variables associated with \(\mathfrak{W}_{{\underline{\Delta}}}^B(\mathfrak{C})\) transform by the same mutation and permutation operations as the exchange matrices in Proposition 1.

The following lemma tells us how cluster variables are changed under the weave \(\mathfrak{W}_{{\underline{\Delta}}}^{\rm T}(\mathfrak{C}) : {\underline{\Delta}}{\boldsymbol{i}}\rightarrow {\boldsymbol{i}}_{\underline{\Delta}}(\mathfrak{C})\) given in 22 .

We now have the main theorem of this subsection.

7.3 Cluster algebra isomorphisms↩︎

Recall the bosonic extension \(\widetilde{\mathbb{A}}_\mathbb{C}(\mathtt{b})\) and its seed \(\Sigma(\mathfrak{C})\) given in Section 4.

8 Connection to signed words↩︎

In this section, we review the connection between \(i\)-boxes and signed words studied in [17], and obtain an explicit connection among \(i\)-boxes, weaves and signed words through the isomorphism given in Theorem 5.

Let \(\mathfrak{g}\) be a simple Lie algebra associated with a Cartan matrix \(\mathsf{C}= (c_{i,j})_{i,j\in I}\) of finite \(ADE\) type. A signed word is a sequence of pairs \[\underline{\mathsf{h}}= (\epsilon_1 h_1, \epsilon_2 h_2, \ldots, \epsilon_r h_r),\] where \(h_t \in I\) and \(\epsilon_t \in \{\pm 1\}\) for \(1 \leqslant t \leqslant r\). Given a signed word \(\underline{\mathsf{h}}=(\epsilon_t h_t)_{1\leq t \leq r}\), we define the index sets \[\begin{align} \mathsf{K}(\underline{\mathsf{h}}) &:= \{1,2,\ldots,r \},\\ \mathsf{K}(\underline{\mathsf{h}})^{\rm fr} &:= \{ j\in \mathsf{K}(\underline{\mathsf{h}}) \mid j_\mathbf{h}^+ = +\infty \}, \\ \mathsf{K}(\underline{\mathsf{h}})^{\rm ex} &:= \mathsf{K}(\underline{\mathsf{h}})\setminus \mathsf{K}(\underline{\mathsf{h}})^{\rm fr}, \end{align}\] where \(\mathbf{h}= ( h_1, h_2, \ldots, h_r )\). For a signed word \(\underline{\mathsf{h}}\), the corresponding exchange matrix \(B(\underline{\mathsf{h}}) = (b_{j,t})_{j\in \mathsf{K}(\underline{\mathsf{h}}),\;t \in \mathsf{K}(\underline{\mathsf{h}})^{\rm ex}}\) is defined by \[b_{j,t} = \begin{cases} \epsilon_{t} & \text{if } t=j^+, \\ -\epsilon_{j} & \text{if } j=t^+, \\ \epsilon_{t}c_{h_j,h_t} & \text{if } \epsilon_{j^+}=\epsilon_{t} \text{ and } j<t<j^+<t^+, \\ \epsilon_{t}c_{h_j,h_t} & \text{if } \epsilon_{t}=-\epsilon_{t^+} \text{ and } j<t<t^+<j^+, \\ -\epsilon_{j}c_{h_j,h_t} & \text{if } \epsilon_{t^+}=\epsilon_{j} \text{ and } t<j<t^+<j^+, \\ -\epsilon_{j}c_{h_j,h_t} & \text{if } \epsilon_{j}=-\epsilon_{j^+} \text{ and } t<j<j^+<t^+, \\ 0 & \text{otherwise}. \end{cases}\] We denote by \(Q(\underline{\mathsf{h}})\) the quiver associated with the exchange matrix \(B(\underline{\mathsf{h}})\).

Let \(\mathtt{b}\in \mathop{\mathrm{Br}}^+\) and let \({\boldsymbol{i}}\in \mathrm{Seq}(\mathtt{b})\). We consider an admissible chain \(\mathfrak{C}=(\mathfrak{c}_t)_{1\leqslant t \leqslant r}\) of \(i\)-boxes associated with \({\boldsymbol{i}}\), and denote by \(\mathfrak{H}=(\mathcal{H}_1,\ldots,\mathcal{H}_{r-1})\) the corresponding LR sequence. The associated signed word \[\underline{\mathsf{h}}(\mathfrak{C}):=(\epsilon_t h_t)_{1\leq t \leq r}\] is defined by \[h_t := \mathrm{color}(\mathfrak{c}_t) \quad\text{{{and}}}\quad \epsilon_t := \begin{cases} +1 & \text{if } t=1 \text{ or } \mathcal{H}_{t-1}=\mathcal{L}, \\ -1 & \text{if } \mathcal{H}_{t-1}=\mathcal{R}. \end{cases}\] It was shown in [17] that the quivers \(Q(\mathfrak{C})\) and \(Q\bigl(\underline{\mathsf{h}}(\mathfrak{C})\bigr)\) coincide for an admissible chain \(\mathfrak{C}\) of \(i\)-boxes associated with \({\boldsymbol{i}}\in \mathrm{Seq}(\mathtt{b})\). Combining this with Theorem 5, we have an explicit connection among \(i\)-boxes, weaves and signed words.

9 Connection to quantum affine algebras↩︎

Let \(U_q'(\mathfrak{g})\) be a quantum affine algebra over an algebraic closure of \(\mathbb{C}(q)\), where \(q\) is an indeterminate. We denote by \({\mathscr{C}_\mathfrak{g}}\) the category of finite-dimensional integrable \(U_q'(\mathfrak{g})\)-modules and by \({\mathscr{C}^0_\mathfrak{g}}\) the Hernandez-Leclerc category, which is a distinguished subcategory of \({\mathscr{C}_\mathfrak{g}}\) (see [2], [3], [21] and [9] for details). To each \(U_q'(\mathfrak{g})\), we associate a simply-laced finite simple Lie algebra \({\mathfrak{g}_{\rm fin}}\) as listed in Table 1 (see [24] and see also [9]). We use the notations \(I_{{\rm fin}}\), \(\mathsf{C}_{{\rm fin}}\), etc., for the Lie algebra \({\mathfrak{g}_{\rm fin}}\).

We take a complete duality datum \(\mathbb{D}=\{L_i^\mathbb{D}\}_{i \in I_{{\rm fin}}} \subset {\mathscr{C}^0_\mathfrak{g}}\), which is the set of root modules satisfying certain categorical conditions determined by \(\mathsf{C}_{{\rm fin}}\) (see [9]). Let \(\widehat{\mathcal{A}}\) be the bosonic extension associated with \(\mathsf{C}_{{\rm fin}}\). It was shown in [9] that there exists a homomorphism \[{\boldsymbol{\Phi}}_\mathbb{D}\colon \widehat{\mathcal{A}}_{\mathbb{Z}\mspace{1mu}[q^{\pm 1/2}]}\to K({\mathscr{C}^0_\mathfrak{g}})\] satisfying \({\boldsymbol{\Phi}}_\mathbb{D}(f_{i,p}) = \bigl[\mathscr{D}^p(L_i^\mathbb{D})\bigr]\) for any \((i,p) \in I_{{\rm fin}}\times \mathbb{Z}\mspace{1mu}\) and \({\boldsymbol{\Phi}}_\mathbb{D}(q^{1/2}) = 1\), where \(\mathscr{D}\) is the right dual functor in \({\mathscr{C}^0_\mathfrak{g}}\). Specializing \(\mathbb{A}:= \widehat{\mathcal{A}}_{\mathbb{Z}\mspace{1mu}[q^{\pm 1/2}]}/ (q^{1/2}-1) \widehat{\mathcal{A}}_{\mathbb{Z}\mspace{1mu}[q^{\pm 1/2}]}\), ¹ we obtain the induced isomorphism \[{^\circ}{\boldsymbol{\Phi}}_\mathbb{D}\colon \mathbb{A}\to K({\mathscr{C}^0_\mathfrak{g}})\] (see [9]).

We denote by \(\mathop{\mathrm{Br}}\) the Braid group associated with \(\mathsf{C}_{{\rm fin}}\). Let \(\mathtt{b}\in \mathop{\mathrm{Br}}^+\) and let \[{\boldsymbol{i}}= (i_1, i_2, \ldots, i_r) \in \mathrm{Seq}(\mathtt{b}).\] For each \(t\in [1,r]\), we consider the affine cuspidal module \(C^{\mathbb{D},{\boldsymbol{i}}}_t\) defined in [9]. For an \(i\)-box \([a,b]\) of \({\boldsymbol{i}}\), we define the affine determinantial module \(M^{\mathbb{D},{\boldsymbol{i}}}[a,b]\) by \[M^{\mathbb{D},{\boldsymbol{i}}}[a,b] := {\operatorname{hd}}\left(C^{\mathbb{D},{\boldsymbol{i}}}_{b} \mathop{\otimes} C^{\mathbb{D},{\boldsymbol{i}}}_{b^-} \mathop{\otimes}\cdots \mathop{\otimes} C^{\mathbb{D},{\boldsymbol{i}}}_{a^+} \mathop{\otimes} C^{\mathbb{D},{\boldsymbol{i}}}_a\right)\] (see [9]). The affine determinantial modules are real simple modules in \({\mathscr{C}^0_\mathfrak{g}}\) and satisfy the \(T\)-system relations. We simply write \(C^{{\boldsymbol{i}}}_{t}\) and \(M^{{\boldsymbol{i}}}[a,b]\) for \(C^{\mathbb{D},{\boldsymbol{i}}}_{t}\) and \(M^{\mathbb{D},{\boldsymbol{i}}}[a,b]\) respectively if no confusion arises.

We define \(\mathscr{C}_\mathfrak{g}^{\mathbb{D}} (\mathtt{b})\) to be the smallest full subcategory of \({\mathscr{C}^0_\mathfrak{g}}\) such that

1. \(\mathscr{C}_\mathfrak{g}^{\mathbb{D}} (\mathtt{b})\) contains all affine cuspidal modules \(\{ C_t^{\mathbb{D},{\boldsymbol{i}}} \}_{1 \leqslant t \leqslant r}\) and the trivial module \(\mathbf{1}\),

2. \(\mathscr{C}_\mathfrak{g}^{\mathbb{D}} (\mathtt{b})\) is stable under taking tensor products, subquotients, and extensions.

The category \(\mathscr{C}_\mathfrak{g}^{\mathbb{D}} (\mathtt{b})\) provides a monoidal categorification of \(\mathbb{A}(\mathtt{b})\) using the affine determinantial modules ([9]).

Combining this with Theorem 5, we realize the cluster structure on \(\mathbb{C}[X(\Delta {\boldsymbol{i}})]\) at the level of the monoidal category \(\mathscr{C}_\mathfrak{g}^{\mathbb{D}}(\mathtt{b})\).

The isomorphism \(\varphi_{{\boldsymbol{i}}}\) given in Theorem 5 together with Proposition 6 yields an explicit formula for the decomposition of \([M^{ {\boldsymbol{i}}}[a,b]]\) in terms of \([C_{t}^{ {\boldsymbol{i}}}]\) in the Grothendieck ring \(K({\mathscr{C}_\mathfrak{g}}^{\mathbb{D}}(\mathtt{b}))\).

References↩︎

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1. \(\mathbb{A}\) and \(\mathbb{A}(\mathtt{b})\) are denoted by \(^{\circ}\mathbb{A}\) and \(^{\circ}\mathbb{A}(\mathtt{b})\) respectively in [9].↩︎