¹

1 Introduction↩︎

1.1 Motivations↩︎

Masaki Kashiwara introduced crystal bases for representations of quantum groups, uncovering their remarkable combinatorial properties [1]. He later axiomatized these into combinatorial crystals, which form the underlying structure for representations of Lie groups and quantum enveloping algebras. Independently, George Lusztig developed the canonical basis for quantum enveloping algebras [2], a distinguished basis with positive structure coefficients whose specialization (or tropicalization) gives rise to a crystal basis as its “shadow" structure, with profound applications in representation theory. Around the same time, Peter Littelmann introduced the path model for crystals from another perspective [3].

Let \(C=(c_{i,j})_{i,j\in I}\) be a symmetric Cartan matrix, with \(\Phi\) the associated root system and \(\Lambda\) the weight lattice. A Kashiwara crystal of type \(\Phi\) is a nonempty set \(\mathcal{B}\) equipped with raising and lowering operators \(r_i\) and \(l_i\), string length functions \(\rho_i\) and \(\lambda_i\) for each \(i \in I\), and a weight function \(\mathcal{B}\to \Lambda\) satisfying specific axioms (see [1], [4], [5], or Section 6.2).

Let \(G\) be a simple simply-connected complex algebraic group, and \(U\) be its maximal unipotent subgroup. The coordinate ring \({\mathbb{k}}[U]\) is one of the most studied cluster algebras [6], [7]; as a crystal it is isomorphic to \(\mathcal{B}(\infty)\), one of the most studied crystals. Besides the unipotent groups, many classical cluster algebras admit natural crystal structures via group actions, with well-known examples including \(G\) and its subgroups (along with their strata) [6], and partial flag varieties [8] (including Grassmannian, \(G/U\) [6] and their strata [9]). Crystal structure also feature prominently in the monoidal categorification of cluster algebras [10]. These examples motivate the following questions:

1. How to detect and describe crystal structures of upper cluster algebras “intrinsically" (i.e., directly from the exchange matrix, without prior knowledge of the underlying space)?

2. How to categorify these crystal structures arising from upper cluster algebras (UCAs)?

3. How to algebraically lift these combinatorial crystals to the UCAs themselves?

Additive categorification of crystals is well-established—for instance, Lusztig’s constructions using preprojective algebras and perverse sheaves on quivers to realize canonical bases [2], [11], or Nakajima’s quiver varieties [12]—but existing models are largely limited to \(U\) or \(G/U\). There is prior work describing crystal structures in specific classical cluster algebras, such as [13]–[16].

Our initial aim was to provide a uniform description of crystal structures for all skew-symmetric cluster algebras. Following a 2021 draft, we discovered that any skew-symmetric upper cluster algebra with reachable coefficients possesses a nontrivial crystal structure. Given the additive categorical model provided by Derksen-Weyman-Zelevinsky’s theory of quivers with potentials [17], [18], we anticipate an additive categorification of these crystal structures as well.

1.2 Tropical Crystal Structures↩︎

First, let us review some key notions in the representation theory of quivers with potentials to define a crystal. Let \((\Delta,\mathcal{S})\) be a nondegenerate ice quiver with potential, and \(J\) be its Jacobian algebra [17]. For any \(\delta\in\mathbb{Z}^{\Delta_0}\) we define the presentation space \[\mathop{\mathrm{PHom}}(\delta):=\mathop{\mathrm{Hom}}(P([-\delta]_+),P([\delta]_+)).\] Here \(P(\beta) = \bigoplus_{u\in \Delta_0} \beta(u) P_u\) and \(P_u\) is the indecomposable projective representation of \(J\) corresponding to \(u\). The vector \(\delta\) is called the weight vector or the \(\delta\)-vector of the presentation space. There is also a notion of injective weight vector \({\check{\delta}}\) if working with injective presentations. For generic \(d\in \mathop{\mathrm{PHom}}(\delta)\), the cokernel of \(d\) has a constant dimension vector, denoted by \(\underline{\dim}(\delta)\).

For any presentation \(d:P_-\to P_+\) and any representation \(N\) of \(J\), let \(\mathop{\mathrm{E}}(d,N)\) be the cokernel of the induced map \(\mathop{\mathrm{Hom}}(P_+, N)\to \mathop{\mathrm{Hom}}(P_-, N)\). We denote by \({\rm e}(\delta, N)\) the generic (minimal) value of \(\dim\mathop{\mathrm{E}}(d,N)\) for \(d\in \mathop{\mathrm{PHom}}(\delta)\). For two decorated representations \(\mathcal{M}\) and \(\mathcal{N}\) we define \({\rm e}(\mathcal{M},\mathcal{N}) = {\rm e}(d_{\mathcal{M}}, N)\) where \(d_{\mathcal{M}}\) is the presentation corresponding to \(\mathcal{M}\) (the correspondence will be reviewed in Section 2.1). \(\mathcal{M}\) is called \(\mathop{\mathrm{E}}\)-rigid if \({\rm e}(\mathcal{M},\mathcal{M})=0\).

For a frozen vertex \(i\) of \(\Delta\), there is an associated boundary representation \(E_i\) (detailed construction is given in Section 5). This representation fits into an exact sequence \(0\to {E}_i^\mu \to E_i \to S_i \to 0\) where \(\mathcal{E}_i^\mu\) is \(\mu\)-supported and \(S_i\) is the simple representation supported on \(i\). The frozen vertex \(i\) is called rigid (resp. reachable) if \(\mathcal{E}_i^\mu\) is \(\mathop{\mathrm{E}}\)-rigid (resp. reachable). We denote the projective and injective weights of \(E_i\) by \({\epsilon}_i\) and \({\check{{\epsilon}}}_i\) respectively.

Recall the skew-symmetric matrix \(B(\Delta)\) associated to \(\Delta\): \[B(\Delta)(u,v) = |\text{arrows u\to v}| - |\text{arrows v\to u}|.\] If we delete the rows of \(B(\Delta)\) corresponding to the frozen vertices, the resulting matrix is denoted by \(B_\Delta\). Let \(I\) be a subset of frozen vertices of \(\Delta\).

An important class of operations for a quiver with potential is the mutation operations introduced by Derksen-Weyman-Zelevinsky [17]. The mutation \(\mu_u\) at a vertex \(u\) of \(\Delta\) sends \((\Delta,\mathcal{S})\) to another QP \((\Delta',\mathcal{S}')=\mu_u(\Delta,\mathcal{S})\) and a decorated representation \(\mathcal{M}\) of \((\Delta,\mathcal{S})\) to \(\mathcal{M}'=\mu_u(\mathcal{M})\) of \((\Delta',\mathcal{S}')\). If \(\mathcal{M}\) is general of weight \(\delta\), then \(\delta\) undergoes a tropical transformation 10 to \(\delta'=\mu_u(\delta)\). In view of cluster algebras, it is natural to ask the crystal structure on \(\mathop{\mathrm{trop}}(\Delta,\mathcal{S})\) compatible with the mutations in the following sense.

If the weight function \(\operatorname{wt}\) is compatible with mutations, i.e., \(\operatorname{wt}(\delta) =\operatorname{wt}'(\delta')\), then it annihilates the row space of \(B_\Delta\). Due to this additional restriction, we cannot always expect the weight function to take value in the weight lattice \(\Lambda\). We shall replace \(\Lambda\) by its \(\mathbb{Q}\)-span \(\Lambda_{\mathbb{Q}} := \bigoplus_{i\in I} \mathbb{Q}\varpi_i\), where \(\varpi_i\)’s are the fundamental weights in \(\Lambda\). In this article, all weights in \(\Lambda\) will be written in coordinates in the basis of fundamental weights.

Recall that a crystal \(\mathcal{B}\) is called seminormal if \[\rho_i(x) = \max\{k\in\mathbb{Z}_{\geq 0} \mid r_i^k(x)\neq 0\}\;\text{ and }\;\lambda_i(x) = \max\{k\in\mathbb{Z}_{\geq 0} \mid l_i^k(x)\neq 0\}.\] If just the first condition is assumed, we say \(\mathcal{B}\) is upper seminormal.

Now we state our first main result.

Here, \(\mathop{\mathrm{rank}}(\delta,{\epsilon})\) denotes the general rank from \(\delta\) to \({\epsilon}\) introduced in [20] (see Definition 13), and the Auslander-Reiten transform \(\tau\) makes sense for \(\delta\)-vectors by Theorem 11. If \(\delta\) or \({\epsilon}\) is reachable, then we have an algorithm to compute \(\mathop{\mathrm{rank}}(\delta, {\epsilon})\) based on mutations (see Theorem 16).

Actually Theorem 1 has a dual version if we work with the set \(\check{\mathcal{B}}\) of \(\mu\)-supported \({\check{\delta}}\)-vectors and the dual boundary representations \(E_i^\star\). Due to the bijection between \(\check{\mathcal{B}}\) and \(\mathcal{B}\) (Theorem 11), the crystal cluster structure can be transferred back to \(\mathcal{B}\) from \(\check{\mathcal{B}}\). This is what we call the dual crystal cluster structure.

However, not every \(\mathop{\mathrm{trop}}(\Delta, \mathcal{S})\) admits a seminormal crystal structure. To upgrade to seminormal ones, we need some additional assumptions on the coefficient pattern of \(\Delta\).

For a \(\tau\)-exact pair \((i,{\bar{\imath}})\), there is a natural way to associate an integral weight function \(\operatorname{wt}_i\): \[\operatorname{wt}_i(\delta) = \delta(\underline{\dim}E_i - \underline{\dim}(\tau^{-1} E_i)).\]

The crystal cluster structure can be transferred via the cluster automorphisms as well. The induced crystal cluster structure is related to the original one if the cluster automorphism is direct. If the cluster automorphism is opposite, the induced structure is naturally related to the dual structure. In the classical case of \({\mathbb{k}}[U]\), the dual structure is related to the original one by the Kashiwara involution [5], [21], which is also an opposite cluster automorphism. This motivates us to define the generalized Kashiwara maps (Definition 29) whose induced crystal cluster structure is related to the dual structure (Corollary 10).

1.3 Algebraic Lifts of Tropical Crystals↩︎

Next we briefly recall the skew-symmetric upper cluster algebras. We fix the base ring \(\mathbb{k}=\mathbb{Z}\). For each vertex \(u\) of \(\Delta\) we attach a variable \(x_u\). Those corresponding to the frozen vertices are called frozen variables. Let \(\mathcal{L}_{\boldsymbol{x}}\) be the subalgebra of the Laurent polynomial algebra \({\mathbb{k}}[\boldsymbol{x}^{\pm 1}]\), which is polynomial in the frozen variables: \[\mathcal{L}_{\boldsymbol{x}}:={\mathbb{k}}[\boldsymbol{x}_u^{\pm 1}, \boldsymbol{x}_{v}], \quad u\in\Delta_0^\mu\;\text{ and }\;v\in\Delta_0^{\operatorname{fr}}.\] For each mutable vertex \(u\), Fomin-Zelevinsky’s mutation operation turns a seed \((\Delta, \boldsymbol{x})\) into a new seed \(\mu_u(\Delta, \boldsymbol{x})\). All such reachable seeds will be denoted by \(\mathfrak{T}\). In particular, we get a new Laurent polynomial algebra \(\mathcal{L}_{\boldsymbol{x}'}\). The upper cluster algebra with seed \((\Delta,\boldsymbol{x})\) is \[\overline{\mathcal{C}}(\Delta,\boldsymbol{x}):=\bigcap_{(\Delta',\boldsymbol{x}') \in \mathfrak{T}}\mathcal{L}_{\boldsymbol{x}'}.\] From now on we will write \(\overline{\mathcal{C}}(\Delta)\) for \(\overline{\mathcal{C}}(\Delta,\boldsymbol{x})\). The representation category of \(J\) is related to \(\overline{\mathcal{C}}(\Delta)\) via the generic character \(C_{\operatorname{gen}}\) [18], [22]. Under the full rank and the reachable assumption of \(B_\Delta\), the set \(\mathop{\mathrm{trop}}(\Delta,\mathcal{S})\) is sent bijectively to a basis of \(\overline{\mathcal{C}}(\Delta)\), called the generic basis [22]–[25]. More generally, in the algebraic lifting results below, whenever we speak about the generic basis of \(\overline{\mathcal{C}}(\Delta)\), we work under the realization hypothesis that the generic characters \(C_{\mathop{\mathrm{gen}}}(\delta)\) for \(\delta\in\mathop{\mathrm{trop}}(\Delta,\mathcal{S})\) form a basis of the upper cluster algebra. Without this hypothesis, the same constructions should be read as statements about the linear span of these generic characters (see Remark 38).

We are also interested in the algebraic lifts of the above combinatorial crystals. By an algebraic lift of the (weak) upper seminormal crystal \(\mathcal{B}\) to \(\overline{\mathcal{C}}(\Delta)\), we mean the following.

1. Each \(r_i^{(\star)}\) should be lifted to a \({\mathbb{k}}\)-derivation \(R_i^{(\star)}\) of \(\overline{\mathcal{C}}(\Delta)\).

2. A basis \(\rm B\) of \(\overline{\mathcal{C}}(\Delta)\) indexed by \(\mathcal{B}\) such that for each \(i\in I\) we have that \[\label{eq:introRi} {R}_i^{(\star)}({\rm B}(\delta)) = \rho_i^{(\star)}(\delta) {\rm B}(r_i^{(\star)}(\delta)) + v \quad \text{ for some v\in \operatorname{span}({\rm B}(\eta): \rho_i^{(\star)}(\eta)<\rho_i^{(\star)}(\delta)-1 ) }.\tag{1}\]

3. There is a positive function \(\tilde{\rho}_i\) such that the tropicalization of \(\tilde{\rho}_i\) gives \(\rho_i\).

Here, we write \(R_i^{(\star)}\) for \(R_i\) or \(R_i^{\star}\) respectively, and similarly for \(\rho_i^{(\star)}\) and \(r_i^{(\star)}\). If \(\mathcal{B}\) is seminormal, then we ask that \(l_i\) can be lifted to a \({\mathbb{k}}\)-derivation \(L_i\) of \(\overline{\mathcal{C}}(\Delta)\) as well. Note that what (2) requires is essentially Berenstein-Kazhdan’s biperfect basis. The part (3) is already done in [26] (see Theorem 12). Although no group action is involved in the definition of the upper cluster algebra, we will see that each \(r_i\) can always be lifted to a \({\mathbb{k}}\)-derivation \(R_i\) of \(\overline{\mathcal{C}}(\Delta)\) (see Section 9.1 for the detail).

Let \(\mathfrak{d}_{I}\) be the Lie subalgebra of \(\operatorname{Der}_{\mathbb{k}}(\overline{\mathcal{C}}(\Delta))\) generated by the derivations \(R_i\) and \(R_i^\star\) for \(i\in I\), and let \(U(\mathfrak{d}_{I})\) be the enveloping algebra of \(\mathfrak{d}_{I}\). Then \(\overline{\mathcal{C}}(\Delta)\) is a \(U(\mathfrak{d}_I)\)-module algebra. We set \(c_{i,j}^\star=-{\rm e}(E_j^\star, E_i)\).

We remark that it is in general not a \(U(\mathfrak{n})\times U(\mathfrak{n})\)-module. We show that this is the case if and only if \(c_{i,j}^\star=0\) (Corollary 12). We also have the following result corresponding to Theorem 2.

We mention an immediate corollary. Recall that a normal crystal is a disjoint union of crystals, each of which is isomorphic to the one underlying some integrable highest-weight representation of a fixed Kac-Moody Lie algebra \(\mathfrak{g}\).

Let \(W(\mathfrak{g})\) be the Weyl group of \(\mathfrak{g}\). Due to Kashiwara, there is a Weyl group action on any normal crystal [4]. We conjecture that this action on \(\mathop{\mathrm{trop}}(\Delta,\mathcal{S})\) can be lifted to \(\overline{\mathcal{C}}(\Delta)\) as well.

Now we come to the part (2) of the algebraic lift. A basis indexed by \(\mathcal{B}\) satisfying 1 is called BK-biperfect. We reserve the term biperfect basis for something stronger.

We conjecture that known interesting bases, including theta bases [27] and triangular bases [28] are all BK-biperfect.

1.4 Biperfect Bases↩︎

In general, BK-biperfect bases are far from unique. Knowing one BK-biperfect basis (e.g., the generic basis), we are able to describe all BK-biperfect bases based on a result of Baumann [29]. This part of work is motivated by the recent work of Baumann-Kamnitzer-Knutson [30] and Qin [24]. In [29] he introduced an order \(\preceq_{\mathop{\mathrm{str}}}\) called the string order (see Definition 40). Below we write \(\eta \llcurly_{\rho} \delta\) if \(\rho_i(\eta)<\rho_i(\delta)\) and \(\rho_i^\star(\eta)<\rho_i^\star(\delta)\) for each \(i\in I\).

Using this theorem, it is easy to construct an example where some cluster monomials are not in a BK-biperfect basis. But such an example for \({\mathbb{k}}[U]\) is not trivial (see Example 2). So BK-biperfect bases are not really perfect from a cluster algebra perspective.

When \(\overline{\mathcal{C}}(\Delta)={\mathbb{k}}[U]\), Theorem 7 answers a question by J. Kamnitzer [31]. In the same article, he also asks for a refinement of the notion of BK-biperfect bases to incorporate all cluster monomials.

A linear basis of \(\overline{\mathcal{C}}(\Delta)\) indexed by \(\mathop{\mathrm{trop}}(\Delta,\mathcal{S})\) is a rather weak notion in the cluster algebra setting. For one thing, additional orders from the cluster structure do not play a role here. Let \(t\) be a seed \((\Delta,\mathcal{S})\). We recall the dominance order \(\prec_t\) on the lattice \(\mathbb{Z}^{\Delta_0}\) such that \(\delta' \prec_t \delta\) if and only if \(\delta' = \delta + \gamma B_{\Delta}\) for some \(\mu\)-supported dimension vector \(\gamma\).

Recall from [28] that an element \(z\in \overline{\mathcal{C}}(\Delta)\) is called pointed at \(\delta\in \mathop{\mathrm{trop}}(\Delta,\mathcal{S})_t\) if it is of the form \(\boldsymbol{x}_t^{-\delta} F(\boldsymbol{y}_t)\). This is equivalent to say that \(\delta\) is maximal among the monomial degrees of \(z\) with respect to the order \(\prec_t\). Based on this notion, F. Qin introduced the good bases [24], in which each basis element is compatibly pointed at every seed \(t\in \mathfrak{T}\). A good property for them is that they contain all cluster monomials.

In our definition of biperfect bases, we will require that the basis elements be pointed at \(\mathop{\mathrm{trop}}(\Delta,\mathcal{S})\) instead of just being indexed by \(\mathop{\mathrm{trop}}(\Delta,\mathcal{S})\).

It follows from this definition that biperfect bases are good bases. Combining Theorem 7 and Qin’s description of good bases (Theorem 51), we are able to describe all biperfect bases for a fixed UCA (Corollary 17).

1.5 Other Results↩︎

We also mention some side results. The maximal version of the crystal operators \(r_i^{\max}\) and \({\check{r}}_i^{\max}\) can be lifted to the module category as well. It turns out that \(r_i^{\max}\) and \({\check{r}}_i^{\max}\) are quite close to an adjoint pair (see Lemma 30). From there we deduce the following result, which provides an efficient way to calculate some Kashiwara’s data.

Along the way, we further develop the representation theory of quivers of potentials in the following aspects. First, we introduce the extension of QPs as a generalization of a construction in [23]. Second, due to the bijection of the decorated representations and presentations in the homotopy category \(K^b(\operatorname{proj}\text{-}J)\), it is natural to expect that the mutation operation can be defined directly on presentations. We find this indeed can be done based on the extension construction (Definition 11 and Lemma 8). Third, the (dual) boundary representations were introduced in [19] to describe the \(\mu\)-supported \(\delta\)-vector cone of an upper cluster algebra. It was originally defined by injective presentations satisfying certain “boundary" condition (see Proposition 20.(1) and (2)). However, it is unclear whether the original definition would depend on the frozen pattern. In this article, we shall give an intrinsic construction in Section 5.1, and generalize a result in [19] (Theorem 23). Another key ingredient, the general rank of QP representations, is treated in [20].

Some important topics about crystals, such as tensor products, are not touched in the present paper. We will treat them in a follow-up paper.

1.6 Organization↩︎

In Section 2 we first briefly review the theory of quivers with potentials following [17], then we introduce the extension construction and define the mutation of presentations. In Section 3 we briefly review the theory of general presentations and tropical \(F\)-polynomials following [26], [32]. In Section 4 we review the raising and lowering operators \(r_{\epsilon}\) and \(l_{\epsilon}\) introduced in [20] in the special case when \({\epsilon}\) is rigid. Then we further specialize to a class of \({\epsilon}\), called minimally exceptional, which is one of the crucial properties holding for rigid boundary representations. In Section 5 we give an intrinsic construction of boundary representations in Definition 16, and prove various properties for them, including Proposition 20, Corollary 6, and Theorem 23. Then we show some invariance properties for the set of \(\mu\)-supported \(\delta\)-vectors in Lemmas 15 and 16. Finally we introduce the Cartan type and weight functions associated to a set of frozen vertices in Definitions 20 and 21.

In Section 6 we start to describe the crystal structure on the \(\mu\)-supported \(\delta\)-vectors for any ice quiver with potential. We first study the properties of raising and lowering operators associated to boundary representations (notably Lemma 26). After all these preparations, we prove the first two main results Theorem 32 on the (weak) upper seminormal crystal cluster structure and Theorem 34 on the seminormal crystal cluster structure. Then we briefly mention the interaction with the cluster automorphisms. We define the generalized Kashiwara map associated to an opposite cluster automorphism in Definition 29 and relate it to the dual structure in Corollary 10. In Section 7 we study the functors corresponding to the maximal version of the crystal operators, and prove certain adjoint properties for them in Lemma 30 and Corollary 11. We illustrate them in the calculation of the Kashiwara’s data.

In Section 8 we briefly review upper cluster algebras and their generic bases following [6], [22], [26], and recall another tropical pairing. In Section 9 we show that the crystal structure can be algebraically lifted to the upper cluster algebras. This includes two parts – the lift of operators to derivations and the lift of \(\mathcal{B}\) to a BK-biperfect basis. Theorem 40 concerns the explicit structure of the derivations, and Theorem 46 shows that generic bases are BK-biperfect. In Section 10 we briefly review the string order, then give a description of all BK-biperfect bases in Theorem 48 based on [29]. Then we review the good bases [24] and define biperfect bases. In Section 11 we apply these results to various examples. The classical examples include unipotent groups, base affine spaces, Grassmannians, and certain unipotent subgroups of Kac-Moody type. We illustrate in Propositions 60 and 61 how to construct examples with interesting normal crystal structure.

1.7 Notations and Conventions↩︎

The space \(\mathop{\mathrm{E}}\) and the tropical \(F\)-polynomial play important roles throughout. Since \(e\) and \(f\) are naturally assigned to them, we decided to switch the traditional \(e\) and \(f\) for raising and lowering operators to \(r\) and \(l\), and the corresponding function \(\varphi\) and \({\epsilon}\) to \(\rho\) and \(\lambda\).

By a quiver \(\Delta\) we mean a quadruple \(\Delta = (\Delta_0,\Delta_1, t, h)\) where \(\Delta_0\) is a finite set of vertices, \(\Delta_1\) is a finite set of arrows, and \(t\) and \(h\) are the tail and head functions \(\Delta_1 \to \Delta_0\). The sets of mutable and frozen vertices are denoted by \(\Delta_0^\mu\) and \(\Delta_0^{\operatorname{fr}}\) respectively.

All modules are right modules, and all vectors are row vectors. For direct sum of \(n\) copies of \(M\), we write \(nM\) instead of the traditional \(M^{\oplus n}\). We write \(\hom,\mathop{\mathrm{ext}}\) and \({\rm e}\) for \(\dim\mathop{\mathrm{Hom}}, \dim\mathop{\mathrm{Ext}}\), and \(\dim \mathop{\mathrm{E}}\). The superscript \(*\) is the trivial dual for vector spaces. Unadorned \(\mathop{\mathrm{Hom}}\) and \(\mathop{\mathrm{E}}\) are understood over the Jacobian algebra of an appropriate quiver with potential.

In the literature of cluster algebras, the final-seed mutation was first introduced in [33]. But in this article, all mutations except 48 are the initial-seed mutations (see Section 8.1 for the meaning). The mutation defined for quivers with potentials in [17] is to model the initial-seed mutation. A typical example is Lemma 31. Traditionally, one specifies the initial-seed using superscripts. Since no final-seed mutation is involved, we do not strictly follow this tradition.

\[\begin{align} & B(\Delta) && \text{the full skew-symmetric matrix of \Delta}\\ & B_\Delta && \text{the submatrix of B(\Delta) with rows indexed by \Delta_0^\mu}\\ & \mathop{\mathrm{rep}}J && \text{the category of finite-dimensional representations of J} &\\ & S_u && \text{the simple representation supported on the vertex u} &\\ & P_u,\;I_u && \text{the projective cover and the injective envelope of S_u} &\\ & \underline{\dim}M && \text{the dimension vector of M} & \\ & \hat{\tau}\mathcal{M} && \text{the representation obtained by forgetting the decorated part of \tau \mathcal{M}} & \\ & \mathop{\mathrm{trop}}(\Delta,\mathcal{S}) && \text{the set of \mu-supported \delta-vectors of (\Delta,\mathcal{S})} \\ & E_i,\;E_i^\star && \text{the boundary and dual boundary representations attached to i\in\Delta_0^{\operatorname{fr}}}\\ & \mathcal{E}_i^\mu && \text{a general representation of (\Delta,\mathcal{S})_\mu of weight -b_i} \\ & {\epsilon}_i,\;{\check{{\epsilon}}}_i && \text{the projective and injective weight vectors of E_i} \\ & r_i= r_{{\epsilon}_i},\;l_i= l_{{\epsilon}_i} && \text{the raising and lowering operators attached to i\in\Delta_0^{\operatorname{fr}}} \\ & {\check{r}}_i^\star = {\check{r}}_{{\epsilon}_i^\star},\;{\check{l}}_i^\star = {\check{l}}_{{\epsilon}_i^\star} && \text{the dual raising and lowering operators attached to i\in\Delta_0^{\operatorname{fr}}}\\ & \rho_i, \lambda_i;\;\rho_i^\star, \lambda_i^\star && \text{the string length function for r_i,l_i and r_i^\star,l_i^\star}\\ & \operatorname{wt}_i && \text{the i-th coordinate for the weight function \operatorname{wt}} \\ & \overline{\mathcal{C}}(\Delta) && \text{the upper cluster algebra with the seed \Delta} \\ & C_{\operatorname{gen}} && \text{the generic cluster character}\\ & R_i,L_i;\;R_i^\star,L_i^\star && \text{the {\mathbb{k}}-derivation of \overline{\mathcal{C}}(\Delta) lifting r_i,l_i and r_i^\star,l_i^\star} \\ & \prec_{\mathfrak{T}}, \prec_{\mathop{\mathrm{str}}} && \text{the dominance order, the string order} \end{align}\]

2 Representation Theory of Quivers with Potentials↩︎

2.1 Decorated Representations and Presentations↩︎

Let \(\Delta\) be a finite quiver with no loops. For such a quiver, we associate a skew-symmetric matrix \(B(\Delta)\) given by \[B(\Delta)(u,v) = |\text{arrows u\to v}| - |\text{arrows v\to u}|.\] Following [17], we define a potential \(\mathcal{S}\) on a quiver \(\Delta\) as a (possibly infinite) linear combination of oriented cycles in \(\Delta\). More precisely, a potential is an element of the trace space \(\mathop{\mathrm{Tr}}(\widehat{k\Delta}):=\widehat{k\Delta}/[\widehat{k\Delta},\widehat{k\Delta}]\), where \(\widehat{k\Delta}\) is the completion of the path algebra \(k\Delta\) and \([\widehat{k\Delta},\widehat{k\Delta}]\) is the closure of the commutator subspace of \(\widehat{k\Delta}\). The pair \((\Delta,\mathcal{S})\) is a quiver with potential, or QP for short. For each arrow \(a\in \Delta_1\), the cyclic derivative \(\partial_a\) on \(\widehat{k\Delta}\) is defined to be the linear extension of \[\partial_a(a_1\cdots a_d)=\sum_{k=1}^{d}a^*(a_k)a_{k+1}\cdots a_da_1\cdots a_{k-1}.\] For each potential \(\mathcal{S}\), its Jacobian ideal \(\partial \mathcal{S}\) is the closed (two-sided) ideal in \(\widehat{k\Delta}\) generated by all \(\partial_a \mathcal{S}\). The Jacobian algebra \(J(\Delta,\mathcal{S})\) is \(\widehat{k\Delta}/\partial \mathcal{S}\). A QP is Jacobi-finite if its Jacobian algebra is finite-dimensional.

By abuse of language, we also say that \(\mathcal{M}\) is a representation of \((\Delta,\mathcal{S})\). When appropriate, we will view an ordinary representation \(M\) as the decorated representation \((M,0)\).

Following [32] we call a homomorphism between two projective representations a projective presentation (or presentation in short). As a full subcategory of the category of complexes in \(\mathop{\mathrm{rep}}J\), the category of projective presentations is Krull-Schmidt as well. Sometimes it is convenient to view a presentation \(P_-\to P_+\) as elements in the homotopy category \(K^b(\operatorname{proj}\text{-}J)\) of bounded complexes of projective representations of \(J\). Our convention is that \(P_-\) sits in degree \(-1\) and \(P_+\) sits in degree \(0\). We denote this subcategory of presentations by \(K^{[-1,0]}(J)\).

We denote by \(P_u\) (resp. \(I_u\)) the indecomposable projective (resp. injective) representation of \(J\) corresponding to the vertex \(u\) of \(\Delta\). For \(\beta \in \mathbb{Z}_{\geq 0}^{\Delta_0}\) we write \(P(\beta)\) for \(\bigoplus_{u\in \Delta_0} \beta(u)P_u\).

The \(\delta\)-vector is just the corresponding element in the Grothendieck group of \(K^b(\operatorname{proj}\text{-}J)\).

Let \(\nu\) be the Nakayama functor \(\mathop{\mathrm{Hom}}(-,J)^*\). There is a map still denoted by \(\nu\) sending a projective presentation to an injective one \[P_-\to P_+\;\mapsto\;\nu(P_-) \to \nu(P_+).\] Note that if there is no direct summand of the form \(P_i\to 0\), then \(\ker(\nu d) = \tau\mathop{\mathrm{coker}}(d)\) where \(\tau\) is the classical Auslander-Reiten translation.

Let \(\mathcal{R}ep(J)\) be the set of decorated representations of \(J\) up to isomorphism. There is a bijection between two additive categories \(\mathcal{R}ep(J)\) and \(K^{[-1,0]}(\operatorname{proj}\text{-}J)\) mapping any representation \(M\) to its minimal presentation in \(\mathop{\mathrm{rep}}J\), and the simple representation \(S_u^-\) of \(k^{\Delta_0}\) to \(P_u\to 0\). We also denote \(P_u\to 0\) by \(P_u[1]\). Now we can naturally extend the classical AR-translation to decorated representations: \[\xymatrix{\mathcal{M} \ar[r]\ar@{<->}[d] & \tau \mathcal{M} \ar@{<->}[d] \\ d_{\mathcal{M}} \ar[r] & \nu(d_{\mathcal{M}})}\] Note that this definition agrees with the one in [32].

Suppose that \(\mathcal{M}\) corresponds to a projective presentation \(d_{\mathcal{M}}: P(\beta_-)\to P(\beta_+)\). Consider the resolution of the simple module \(S_u\) \[\begin{align} \tag{2} \cdots \to \bigoplus_{h(a)=u} P_{t(a)}\xrightarrow{_a(\partial_{[ab]})_b} \bigoplus_{t(b)=u} P_{h(b)} \xrightarrow{_b(b)} P_u \to S_u\to 0,\\ \tag{3} 0\to S_u \to I_u \xrightarrow{(a)_a} \bigoplus_{h(a)=u} I_{t(a)} \xrightarrow{_a(\partial_{[ab]})_b} \bigoplus_{t(b)=u} I_{h(b)} \to \cdots. \end{align}\] Applying \(\mathop{\mathrm{Hom}}(M,-)\) and \(\mathop{\mathrm{Hom}}(-,M)\) to 3 and 2 , we get that \[\begin{align} \tag{4} \beta_-(u)&=\dim(\ker \alpha_u/\mathop{\mathrm{im}}\gamma_u)+\dim M^-(u), \text{ and } \beta_+(u)=\dim \mathop{\mathrm{coker}}\alpha_u, \\ \tag{5} {\check{\beta}}_-(u)&=\dim(\ker \gamma_u/\mathop{\mathrm{im}}\beta_u)+\dim M^-(u), \text{ and } {\check{\beta}}_+(u)=\dim \ker \beta_u. \end{align}\] Here, the maps \(\alpha_u\), \(\beta_u\), and \(\gamma_u\) are depicted in the following diagram as in [17]. \[\label{eq:abc} \vcenter{\xymatrix@C=5ex{ & M(u) \ar[dr]^{\beta_u} \\ \bigoplus_{h(a)=u} M(t(a)) \ar[ur]^{\alpha_u} && \bigoplus_{t(b)=u} M(h(b)) \ar[ll]^{\gamma_u} \\ }}\tag{6}\]

The \(\delta\)-vector \(\delta_{\mathcal{M}}\) of \(\mathcal{M}\) is by definition the \(\delta\)-vector of \(d_{\mathcal{M}}\). If working with the injective presentations, we can define the \({\check{\delta}}\)-vector \({\check{\delta}}_{\mathcal{M}}\) of \(\mathcal{M}\). It follows from 4 and 5 that \(\delta_{\mathcal{M}}\) and \({\check{\delta}}_{\mathcal{M}}\) are related by \[\label{eq:delta2dual} {\check{\delta}}_{\mathcal{M}} = \delta_{\mathcal{M}} + (\underline{\dim}M) B(\Delta).\tag{7}\]

Note that according to this definition, we have that \(\mathop{\mathrm{Hom}}(\mathcal{M},\mathcal{N}) = \mathop{\mathrm{Hom}}(M,N)\).³ We also set \(\mathop{\mathrm{E}}(d_{\mathcal{M}},d_{\mathcal{N}}) = \mathop{\mathrm{E}}(\mathcal{M},\mathcal{N})\) and \({\check{\mathop{\mathrm{E}}}}({\check{d}}_{\mathcal{M}},{\check{d}}_{\mathcal{N}}) = {\check{\mathop{\mathrm{E}}}}(\mathcal{M},\mathcal{N})\). We refer readers to [32] for an interpretation of \(\mathop{\mathrm{E}}(\mathcal{M},\mathcal{N})\) in terms of the presentations \(d_{\mathcal{M}}\) and \(d_{\mathcal{N}}\). We call \(\mathcal{M}\) or \(d_{\mathcal{M}}\) rigid if \(\mathop{\mathrm{E}}(\mathcal{M},\mathcal{M})=0\).

2.2 Mutation of Quivers with Potentials↩︎

In [17] and [18], the mutation of quivers with potentials is introduced to model the cluster algebras. The mutation \(\mu_u\) of a QP \((\Delta,\mathcal{S})\) at a vertex \(u\) is defined as follows. The first step is to define the following new QP \(\widetilde{\mu}_u(\Delta,\mathcal{S})=(\widetilde{\Delta},\widetilde{\mathcal{S}})\). We put \(\widetilde{\Delta}_0=\Delta_0\) and \(\widetilde{\Delta}_1\) is the union of three different kinds

1. all arrows of \(\Delta\) not incident to \(u\),

2. a composite arrow \([ab]\) from \(t(a)\) to \(h(b)\) for each \(a,b\) with \(h(a)=t(b)=u\),

3. an opposite arrow \(a^\star\) (resp. \(b^\star\)) for each incoming arrow \(a\) (resp. outgoing arrow \(b\)) at \(u\).

The new potential on \(\widetilde{\Delta}\) is given by \[\widetilde{\mathcal{S}}:=[\mathcal{S}]+\sum_{h(a)=t(b)=u}b^\star a^\star[ab],\] where \([\mathcal{S}]\) is obtained by substituting \([ab]\) for each word \(ab\) occurring in \(\mathcal{S}\). Finally we define \((\Delta',\mathcal{S}')=\mu_u(\Delta,\mathcal{S})\) as the reduced part ([17]) of \((\widetilde{\Delta},\widetilde{\mathcal{S}})\). For this last step, we refer readers to [17] for details. A sequence of vertices is called admissible for \((\Delta,\mathcal{S})\) if its mutation along this sequence is defined. If all sequences are admissible for \((\Delta,\mathcal{S})\) then we call \((\Delta,\mathcal{S})\) nondegenerate.

Now we start to define the mutation of decorated representations of \(J:=J(\Delta,\mathcal{S})\). Recall the triangle of linear maps 6 with \(\alpha_u\gamma_u=0\) and \(\gamma_u\beta_u =0\). We first define a decorated representation \(\widetilde{\mathcal{M}}=(\widetilde{M},\widetilde{M}^-)\) of \(\widetilde{\mu}_u(\Delta,\mathcal{S})\). We set \[\begin{align} &\widetilde{M}(v)=M(v),\quad \widetilde{M}^-(v)=M^-(v)\quad (v\neq u); \\ &\widetilde{M}(u)=\frac{\ker \gamma_u}{\mathop{\mathrm{im}}\beta_u}\oplus \mathop{\mathrm{im}}\gamma_u \oplus \frac{\ker \alpha_u}{\mathop{\mathrm{im}}\gamma_u} \oplus M^-(u),\quad \widetilde{M}^-(u)=\frac{\ker \beta_u}{\ker \beta_u\cap \mathop{\mathrm{im}}\alpha_u}. \end{align}\] We then set \(\widetilde{M}(a)=M(a)\) for all arrows not incident to \(u\), and \(\widetilde{M}([ab])=M(ab)\). It is defined in [17] a choice of linear maps \(\widetilde{M}(a^\star)\) and \(\widetilde{M}(b^\star)\) making \(\widetilde{M}\) a representation of \((\widetilde{\Delta},\widetilde{\mathcal{S}})\). We refer readers to [18] for details. Finally, we define \(\mathcal{M}'=\mu_u(\mathcal{M})\) to be the reduced part ([17]) of \(\widetilde{\mathcal{M}}\).

Let us recall several formula relating the \(\delta\)-vector of \(\mathcal{M}\) and its mutation \(\mu_{u}(\mathcal{M})\). We will use the notation \([b]_+\) for \(\max(b,0)\).

We remark that the mutated \(\delta\)-vector \(\delta'\) is not completely determined by \(\delta\) (we need \(\beta_-\) and \(\beta_+\)). But see also Remark 9.

We say a decorated representation \(\mathcal{M}\) of \((\Delta,\mathcal{S})\) negative reachable or just reachable if there is a sequence of mutations \(\mu_{\boldsymbol{u}}\) such that \(\mu_{\boldsymbol{u}}(\mathcal{M})\) is negative, i.e., \(\mu_{\boldsymbol{u}}(\mathcal{M})\) has only the decorated part. Similarly we say \(\mathcal{M}\) positive reachable if there is a sequence of mutations \(\mu_{\boldsymbol{u}}\) such that \(\mu_{\boldsymbol{u}}(\mathcal{M})\) is a projective representation. More generally, we say \(\mathcal{M}\) extended reachable if there is an extended sequence of mutations \(\mu_{\boldsymbol{u}}\) such that \(\mu_{\boldsymbol{u}}(\mathcal{M})\) is negative.

2.3 Extension of Quivers with Potentials↩︎

Let \((\Delta,\mathcal{S})\) be a quiver with potential, and \(\mathcal{V}\) be a decorated representation of \((\Delta,\mathcal{S})\). By the extension of \((\Delta,\mathcal{S})\) by \(\mathcal{V}\), we mean the following construction.

We start with \((\Delta,\mathcal{S})\) and a new vertex \(v\). Take the projective presentation \(d_{\mathcal{V}}: P(\beta_-)\xrightarrow{} P(\beta_+)\) corresponding to \(\mathcal{V}\). We assume that \(P(\beta_-)\) and \(P(\beta_+)\) share no common summands. Note that this is always the case if \(d_{\mathcal{V}}\) is in general position [34]. Then we draw \(\beta_+(w)\) arrows from \(v\) to \(w\) and \(\beta_-(u)\) arrows from \(u\) to \(v\). We view the map \(d_{\mathcal{V}}\) as a matrix with entries a linear combination of paths. For each entry of \(c:=d_{\mathcal{V}}(u,w): P_u \to P_w\), we add the potential \(a b c\) to the original potential \(\mathcal{S}\) where \(a\) is the added arrow corresponding to \(P_u\) and \(b\) is the added arrow corresponding to \(P_w\). We denote the resulting quiver with potential by \((\Delta[\mathcal{V}], \mathcal{S}[\mathcal{V}])\) or, for short \((\Delta,\mathcal{S})[\mathcal{V}]\) or \((\Delta,\mathcal{S})[d_{\mathcal{V}}]\), and abbreviate its Jacobian algebra to \(J[\mathcal{V}]\). If we restrict \((\Delta,\mathcal{S})[\mathcal{V}]\) on \(\Delta\) in the sense of [17], then we get the original QP \((\Delta,\mathcal{S})\) back.

There is an obvious dual construction \((\Delta,\mathcal{S})[\mathcal{V}^*]\) from the injective presentation \({\check{d}}_{\mathcal{V}}: I_+\to I_-\) corresponding to \(\mathcal{V}\). It is easy to see that \((\Delta,\mathcal{S})[\mathcal{V}] = (\Delta,\mathcal{S})[\tau \mathcal{V}^*]\). For any decorated representation \(\mathcal{M}\) of \((\Delta,\mathcal{S})\), we denote by \(\mathcal{M}[0]\) the extension of \(\mathcal{M}\) by zeros to \(\Delta[\mathcal{V}]\). It is easy to see from the definition of the new potential \(\mathcal{S}[\mathcal{V}]\) that \(\mathcal{M}[0]\) is in fact a representation of \((\Delta,\mathcal{S})[\mathcal{V}]\). When writing a vector in \(\mathbb{Z}^{\Delta[\mathcal{V}]}\), our convention is to let the new vertex \(v\) correspond to the last coordinate. We will find it convenient to introduce the notation \(\hat{\tau}\mathcal{V}\) to denote the representation obtained from \(\tau \mathcal{V}\) by forgetting the decorated part.

::: {#L:V[0] .lemma} Lemma 5. Consider \(\mathcal{M}[0]\) as a representation of \((\Delta,\mathcal{S})[\mathcal{V}]\). We have that

1. The \({\check{\delta}}\)-vector of \(\mathcal{M}[0]\) is equal to \(({\check{\delta}}_{\mathcal{M}}, -\hom(V,M))\).

2. The \(\delta\)-vector of \(\mathcal{M}[0]\) is equal to \((\delta_{\mathcal{M}}, -{\rm e}(\mathcal{V},M))\).

3. Let \(P_{v,v}\) be the space spanned by all paths from \(v\) to \(v\) without passing \(v\) in the middle. Then \(P_{v,v}\cong k\oplus \mathop{\mathrm{E}}(\mathcal{V}, \mathcal{V})\). :::

::: {#C:V[0] .corollary} Corollary 2. If \(\mathcal{V}\) is rigid, then

1. \(\delta_{\mathcal{V}[0]}=(\delta_{\mathcal{V}}, 0)\) and \({\check{\delta}}_{\mathcal{V}[0]}=({\check{\delta}}_{\mathcal{V}}, -{\check{\delta}}_{\mathcal{V}}(\underline{\dim}V))\).

2. We have exact sequences \(0\to V\to P_v\to S_v\to 0\) and \(0\to S_v\to I_v\to {\hat{\tau}\mathcal{V}}\to 0\). :::

Warning: Recall that \((\tau V)^* \cong \mathop{\mathrm{Hom}}(V,A)\), so we have a natural evaluation map \(\mathop{\mathrm{Hom}}(V,A)\times V \to A\). One might think that if \(V\) is rigid, then the Jacobian algebra is given by the matrix algebra \(\left(\begin{smallmatrix}A & (\tau V)^* \\ V & k\end{smallmatrix}\right)\). But this is not true in general because the construction may introduce new paths between two vertices of \(\Delta\).

2.4 Mutation of Presentations↩︎

Conversely, given a quiver with potentials \((\Delta,\mathcal{S})\) and a vertex \(v\in \Delta_0\), let \((\Delta,\mathcal{S})_{\hat{v}}\) be the restriction of \((\Delta,\mathcal{S})\) to the full subquiver of \(\Delta_0\setminus\{v\}\).

Note that \(v\) is simple in \((\Delta,\mathcal{S})[\mathcal{V}]\). If \(v\) is simple in \((\Delta,\mathcal{S})\), then we can obtain a presentation \(d_v\) of \((\Delta,\mathcal{S})_{\hat{v}}\) as follows. Let \(P_-\) (resp. \(P_+\)) be the direct sum of \(P_u\) (resp. \(P_w\)) for each arrow \(u\to v\) (resp. \(v\to w\)). We define \[\label{eq:dv} d_v: \bigoplus_{a:u\to v} P_u \xrightarrow{\partial_{[ab]}[\mathcal{S}]} \bigoplus_{b:v\to w} P_w.\tag{9}\] Clearly we have that \((\Delta,\mathcal{S})_{\hat{v}}[d_v] = (\Delta,\mathcal{S})\). We also define \({\check{d}}_v = \nu(d_v)\).

To make sense of Definition 11, we observe that the restriction of \(\mu_u((\Delta,\mathcal{S})[\mathcal{V}])\) to \(\mu_u(\Delta)\) is \(\mu_u(\Delta,\mathcal{S})\) and the following lemma.

The following lemma was proved in [32] as an easy consequence of a result of K. Bongartz [35].

In particular, we obtain the following corollary.

3 General Presentations and Tropical \(F\)-polynomials↩︎

3.1 General Presentations↩︎

We shall start our discussion by reviewing some results in [32]. We will consider a more general setting where the algebra \(A\) is any basic finite-dimensional \(k\)-algebra, which can be presented as \(k\Delta / I\).

Any \(\delta\in \mathbb{Z}^{\Delta_0}\) can be written as \(\delta = \delta_+ - \delta_-\) where \(\delta_+=\max(\delta,0)\) and \(\delta_- = \max(-\delta,0)\). Here the maximum is taken coordinate-wise. We put \[\mathop{\mathrm{PHom}}_A(\delta):=\mathop{\mathrm{Hom}}_A(P(\delta_-),P(\delta_+)).\] We say that a general presentation in \(\mathop{\mathrm{PHom}}_A(\delta)\) has property \(\heartsuit\) if there is some open (and thus dense) subset \(U\) of \(\mathop{\mathrm{PHom}}_A(\delta)\) such that all presentations in \(U\) have property \(\heartsuit\). For example, a general presentation \(d\) in \(\mathop{\mathrm{PHom}}_A(\delta)\) has the following properties:

1. \(\mathop{\mathrm{Hom}}(d,N)\) has constant dimension for a fixed \(N\in \mathop{\mathrm{rep}}A\).

2. \(\mathop{\mathrm{Gr}}_\gamma(\mathop{\mathrm{coker}}(d))\) has constant topological Euler characteristic.

Note that (1) implies that \(\mathop{\mathrm{E}}(d,N)\) has constant dimension on \(U\) as well. We denote these two generic values by \(\hom(\delta,N)\) and \({\rm e}(\delta,N)\). If we apply (1) to \(N=A^*\), then \(\mathop{\mathrm{coker}}(d)\) has a constant dimension vector, which will be denoted by \(\underline{\dim}(\delta)\).

The presentation space \(\mathop{\mathrm{PHom}}_A(\delta)\) comes with a natural group action by \[\mathop{\mathrm{Aut}}_A(\delta):=\mathop{\mathrm{Aut}}_A(P(\delta_-))\times \mathop{\mathrm{Aut}}_A(P(\delta_+)).\] A rigid presentation in \(\mathop{\mathrm{PHom}}_A(\delta)\) has a dense \(\mathop{\mathrm{Aut}}_A(\delta)\)-orbit [32]. In particular, a rigid presentation is always general.

If we freeze a vertex \(v\), then we are not allowed to mutate at \(v\). A quiver with frozen vertices is called an ice quiver. The vertices of \(\Delta\) split into two disjoint sets \(\Delta_0 = \Delta_0^{\mu} \sqcup \Delta_0^{\operatorname{fr}}\). An arrow between frozen vertices is called a frozen arrow. The \(B\)-matrix \(B_{\Delta}\) of an ice quiver \(\Delta\) is obtained from the original \(B\)-matrix \(B(\Delta)\) by removing the rows corresponding to the frozen vertices. Note that the information on frozen arrows is lost in \(B_{\Delta}\).

Due to the relation \(\delta_{\tau^{-1}\mathcal{M}} = -{\check{\delta}}_{\mathcal{M}}\) and 7 , we have that for a general presentation \(d\) of weight \(\delta\), the \(\delta\)-vector of \(\tau^{-1} d\) is constant. We denote this constant vector by \(\tau^{-1}\delta\).

3.2 Tropical \(F\)-polynomials↩︎

Motivated by the \(F\)-polynomial of \(M\) defined in [18], we introduced its tropical version in [26]. We will review the \(F\)-polynomial of \(M\) in Section 8.2.

Clearly \(f_M\) and \({\check{f}}_M\) are related by \(f_M(\delta)-{\check{f}}_M(-\delta)= \delta(\underline{\dim}M)\). Moreover, it follows from 8 that for any presentation \(d\) of weight \(\delta\), \[\begin{align} \tag{11} \delta(\underline{\dim}M) &= \hom(d,M) - {\rm e}(d,M);\\ \tag{12} \check{\delta} (\underline{\dim}M) &= \hom(M,{\check{d}}) - {\check{{\rm e}}}(M,{\check{d}}). \end{align}\]

If \(M\) is general of weight \({\check{{\epsilon}}}\) then we will write \(f_{\check{{\epsilon}}}(\delta)\) for \(f_M(\delta)\), and similarly for \(\check{f}_M\).

This is a stronger version of a conjecture in [26], where we only conjecture the reciprocity \(\check{f}_{\delta}({\check{{\epsilon}}}) = {f}_{\check{{\epsilon}}}(\delta)\).

4 The Lowering and Raising Operators↩︎

4.1 Lowering and Raising Operators for Rigid \({\epsilon}\)↩︎

Schofield introduced the general rank for quiver representations in his theory of general representations [38]. The following lemma is a straightforward generalization of [38].

Below the algebra \(A\) will always be the Jacobian algebra of some quiver with potential. Let \(\alpha\) be the maximal rank vector of \(\mathop{\mathrm{Hom}}(P_-, P_+)\), and \(U\) be the open subset of \(\mathop{\mathrm{Hom}}(P_-, P_+)\) attaining the maximal rank \(\alpha\). Then the cokernel of homomorphisms in \(U\) lies in a single component of \(\mathop{\mathrm{rep}}_{\underline{\dim}(\delta)}(A)\) [20], [22], [32], and we call this component the principal component of \(\delta\), denoted by \(\mathop{\mathrm{PC}}(\delta)\).

For quivers with potentials, we have that \(\mathop{\mathrm{PC}}(\delta)=\mathop{\mathrm{PC}}({\check{\delta}})\) by [20], so \(\mathop{\mathrm{rank}}(\delta,{\epsilon}) = \mathop{\mathrm{rank}}({\check{\delta}},{\epsilon})\).

The following theorem is a direct consequence of [20].

We remark that if \({\epsilon}\) is not rigid, the compositions may not be identities. In fact, we gave a necessary and sufficient condition in [20] for such a composition being the identity. The following result is also proved there.

4.2 Minimally Exceptional Representations↩︎

A dimension vector \(\gamma\) is called a quotient (resp. sub-)dimension vector of \(E\) if \(\gamma\) is the dimension vector of some quotient (sub-) representation of \(E\).

We need to point out that \({\epsilon}\) is minimally exceptional is not equivalent to the corresponding \({\check{{\epsilon}}}\) being minimally exceptional. Recall that \({\epsilon}\) is called Schur if \(\hom(E,E)=1\) for some \(E\) of weight \({\epsilon}\).

The lemma shows in particular \(E\) is indecomposable and in fact exceptional, that is, both Schur and \(\mathop{\mathrm{E}}\)-rigid. As any rigid \({\epsilon}\) satisfies that \({\epsilon}({\check{\gamma}})>0\) hence the name minimal. The following corollary follows directly from Theorem 17 and Lemma 11.

5 Boundary Representations↩︎

5.1 Boundary Representations↩︎

The (dual) boundary representations were introduced in [19] to describe the \(\mu\)-supported \({\sf g}\)-vector cone of an upper cluster algebra. It was originally defined by injective presentations satisfying certain “boundary" condition (see Proposition 20.(1) and (2)). However, it is unclear whether the original definition would depend on the frozen pattern. Here, we are going to give an intrinsic construction.

Let \((\Delta,\mathcal{S})\) be an ice quiver with potential, and \(\Delta^\mu\) be the mutable part of \(\Delta\). We write \((\Delta,\mathcal{S})_\mu\) for the restriction of \((\Delta,\mathcal{S})\) to \(\Delta^\mu\). We denote by \(\Delta^\mu[i]\) the full subquiver of the mutable vertices together with (a frozen vertex) \(i\). We shall denote \((\Delta,\mathcal{S})\) restricting to \(\Delta^{\mu}[i]\) and its Jacobian algebra by \((\Delta,\mathcal{S})_{\mu[i]}\) and \(J_{\mu[i]}\) respectively. Note that we have a chain of extensions \(\Delta^\mu \subset \Delta^\mu[i] \subset \Delta\). Let \(P_{[i]}\) and \(I_{[i]}\) be respectively the indecomposable projective and injective representation of \(J_{\mu[i]}\) corresponding to \(i\). This distinguishes them from the representations \(P_i\) and \(I_i\) of \((\Delta,\mathcal{S})\).

It follows from the definition that the dual boundary representation \(E_i^\star\) is dual of the boundary representation of the opposite quiver with potential. We also remark that in general \(E_i\) is not the projective representation of \((\Delta,\mathcal{S})\).

Throughout we assume that \((\Delta,\mathcal{S})\) is nondegenerate and each frozen vertex is simple in \((\Delta,\mathcal{S})_{\mu[i]}\). Recall the presentation \(d_i\) defined in 9 . Note that \(d_i\) depends on the ambient quiver of the vertex \(i\). Here, we specify that \(d_i\) is defined inside the quiver \(\Delta^\mu[i]\) so that \(d_i\) is a presentation of \((\Delta,\mathcal{S})_\mu\). By the proof of Lemma 12 we may assume that \(d_i\) is a general presentation of weight \(-b_i\), where \(b_i\) is the \(i\)-th column of the matrix \(B_{\Delta}\). Let \(\mathcal{E}_i^\mu\in \mathop{\mathrm{rep}}(J_{\mu})\) be the decorated representation corresponding to \(d_i\). If \(\mathcal{E}_i^\mu\) is rigid, then by Corollary 2.(2) there are exact sequences \[\begin{align} \tag{21} 0 \to E_i^\mu \to &E_i \to S_i \to 0. \shortintertext{Dually, we can work with the general decorated representation E_i^{\star\mu} of injective weight b_i:} \tag{22} 0 \to S_i \to &E_i^\star \to E_i^{\star\mu} \to 0, \end{align}\] Note that \(\tau_\mu E_i^\mu = E_i^{\star\mu}\) by Theorem 11, where \(\tau_\mu\) is the AR-translation restricted on \(J_\mu\).

We will use the following easy observation (see also [39]) more than once. Recall the map \(\alpha_u,\beta_u\) and \(\gamma_u\) in 6 .

By the frozen dimension of a representation \(M\), we mean the total dimension of its restriction on the frozen part of \(\Delta\). From now on, we set \({\epsilon}_i\) and \({\check{{\epsilon}}}_i\) to be \(\delta\)-vector and \({\check{\delta}}\)-vector of \(E_i\) respectively.

With a little effort, we can show that \({\epsilon}_i(j) = {\check{{\epsilon}}}_j^\star(i)\). We will only prove this for a special case in Lemma 18.

5.2 The \(\mu\)-supported \(\delta\)-vectors↩︎

In this subsection, we do not require the frozen vertices are rigid.

Recall that an arrow between frozen vertices is called a frozen arrow.

The following theorem was proved in [19] (see also [26]) for “non-intrinsically” defined boundary representations.

5.3 Extensions without Frozen Arrows↩︎

Suppose that there is no frozen arrows and \(i\) is a frozen vertex. Then \(d_i\) can be viewed as a presentation of \((\Delta,\mathcal{S})_\mu\). In this subsection, all frozen vertices are assumed to be rigid.

Throughout this article we will denote by \(e_i\) the standard unit vector supported at \(i\)-th coordinate. We will not specify the ambient space of \(e_i\) if it is clear from the context. For example, the \(e_i\) in the following lemma is in \(\mathbb{Z}^{\Delta_0^{\operatorname{fr}}}\).

5.4 Cartan Type and Weight Functions↩︎

Let \(I\) be a set of rigid frozen vertices of \(\Delta_0\).

Note that by Lemma 1 this is also equal to \(\require{upgreek} 2\updelta_{i,j} - {\check{{\rm e}}}_{J_\mu}(\tau_\mu\mathcal{E}_i^\mu, \tau_\mu\mathcal{E}_j^\mu) - {\check{{\rm e}}}_{J_\mu}(\tau_\mu\mathcal{E}_j^\mu, \tau_\mu\mathcal{E}_i^\mu)\).

If \({\sf L}=\mathbb{Q}^I\), then we will drop \(\mathbb{Q}^I\) and simply call it a compatible grading. If \({\sf L}=\mathbb{Z}^I\), then we will call it an integral compatible grading. For a compatible grading \(\operatorname{wt}\), one can define its mutation at \(u\) \[\mu_u(\operatorname{wt})(v) = \begin{cases}\sum_{u\to w}\operatorname{wt}(w) - \operatorname{wt}(u) & \text{if v=u} \\ \operatorname{wt}(v) & \text{if v\neq u.} \end{cases}\]

Now we shall give a natural construction of weight functions. Let \(E=E_i\) be a boundary representation. We set \(\operatorname{wt}_{\epsilon}: \mathbb{Z}^{\Delta_0}\to\mathbb{Z}\) to be the linear functional given by \[\label{eq:wti} \delta \mapsto \delta(\underline{\dim}E - \underline{\dim}(\tau^{-1} E)).\tag{29}\]

The representation \(E_i^\mu\) and the boundary representation \(E_i\) do not depend on other frozen vertices. However, \(\tau^{-1} E_i\) will depend. So being \(\tau\)-exact for \((i,{\bar{\imath}})\) depends on other frozen vertices as seen in the following lemma.

Let \(n=|I|\) and \(r=\mathop{\mathrm{rank}}(C_I)\). By a realization of \(C_I\), we mean a complex vector space \(\mathfrak{h}\) of dimension \(2n-r\) together with a basis \(\{h_i\}_{i\in I \sqcup K}\) and a basis \(\{\alpha_i\}_{i\in I\sqcup K}\) of \(\mathfrak{h}^*\) such that \(\alpha_i(h_j) = c_{i,j}\) for \(i,j\in I\) and \(\alpha_i(h_j)\in\mathbb{Z}\) for \(i,j\in I\sqcup K\). Let \(\Lambda\) be the corresponding weight lattice with fundamental weights \(\{\varpi_i\}_{i\in I}\).

6 The Tropical Crystal Structures↩︎

6.1 The Lowering and Raising Operators attached to Boundary Representations↩︎

From now on, all the ice quivers are assumed to have no frozen arrows. Now we let \(E_i\) (resp. \(E_i^\star\)) be the (resp. dual) boundary representations attached to a rigid frozen vertex \(i\).

Later we will define an upper semi-normal crystal structure on the set of \(\mu\)-supported \(\delta\)-vectors. The function \({\rm e}(-, E_i)\) will play a role of string length function for the operator \(r_i\).

6.2 The Upper Seminormal Crystal with Weaker Weights↩︎

Let \(C_I=(c_{i,j})_{i,j\in I}\) be a generalized Cartan matrix, which will be assumed to be symmetric in this paper. Let \(\Phi=(\mathfrak{h}; h_i,\alpha_i)\) be a realization of \(C\), and \(\Lambda\) be the corresponding weight lattice.

Fix a quiver with potential \((\Delta,\mathcal{S})\). According to the language from cluster algebras, we will call any mutation of \((\Delta,\mathcal{S})\) a seed. Let \(\mathfrak{T}\) be the index set for all seeds. By abuse of language, we also call an element \(t\in \mathfrak{T}\) a seed. A seed \(t'\) obtained from \(t\) by a sequence \(\mu_{\boldsymbol{u}}\) of mutations is denoted by \(\mu_{\boldsymbol{u}}:t\to t'\).

As we have seen in Lemma 20, to have a weight function compatible with mutations, it is necessary to require that the weight function is compatible with the \(B\)-matrix, that is, a compatible grading. Due to this restriction, we cannot always expect that the weight function is integral. So we will consider a slightly weaker version of the weight function. Namely, we allow the range of \(\operatorname{wt}\) to be \(\Lambda_{\mathbb{Q}} = \bigoplus_{i\in I}\mathbb{Q}\varpi_i\). If \(\operatorname{wt}(x) = \sum_i \operatorname{wt}_i(x) \varpi_i\), then this is equivalent to saying that \((\operatorname{wt}_i)_{i\in I}: \mathbb{Z}^{\Delta_0}\to \mathbb{Q}^I\). We will call a crystal with such a weight function, a crystal with weaker weights. The following lemma is immediate.

Fix a subset \(I\) of frozen vertices, and let \(C_I\) be the Cartan type of \(I\) (Definition 20). Throughout we assume that \(\operatorname{span}({\check{{\epsilon}}}_i)_{i\in I} \cap \operatorname{span}(B) = \{0\}\) so at least one \(\mathbb{Q}^I\)-valued compatible weight function adapted to \(I\) exists. This assumption is generically satisfied.

The prototypical examples of this type in the cluster theory are the coordinate ring of the maximal unipotent subgroups of a simple simply-connected complex algebraic groups. These classical examples will be briefly reviewed in Section 11.1.

Using the dual boundary representations, we can similarly define a crystal structure on \(\check{\mathcal{B}}=\check{\mathop{\mathrm{trop}}}(\Delta,\mathcal{S})\), the set of \(\mu\)-supported \({\check{\delta}}\)-vectors of \((\Delta,\mathcal{S})\). Suppose that there is a compatible grading \((\check{\operatorname{wt}}_i)_{i\in I}\) dually adapted to \(I\). Then we define \[\begin{align} {\check{r}}_i^\star({\check{\delta}})&={\check{r}}_{{\epsilon}_i^\star}({\check{\delta}}) & \check{\rho}_i^\star({\check{\delta}}) &= {\check{{\rm e}}}(E_i^\star, {\check{\delta}}) \\ {\check{l}}_i^\star({\check{\delta}})&={\check{l}}_{{\epsilon}_i^\star}({\check{\delta}}) & \check{\lambda}_i^\star({\check{\delta}}) &= \check{\operatorname{wt}}_i({\check{\delta}}) + \check{\rho}_i^\star({\check{\delta}}). \end{align}\] As remarked below Definition 20, this crystal has the same Cartan type as \(\mathop{\mathrm{trop}}(\Delta,\mathcal{S})\) but in general the weight function \((\check{\operatorname{wt}}_i)_{i\in I}\) can be different from \(({\operatorname{wt}}_i)_{i\in I}\).

6.3 The Seminormal Crystal Structure↩︎

Next we discuss the nicest situation from a crystal perspective.

The prototypical examples of this type in the cluster algebra theory is the base affine spaces and the affine coordinate ring of the Grassmannians. We will briefly review them in Section 11.3 and 11.4.

6.4 Intertwining with Cluster Automorphisms↩︎

By a permutation of \(\Delta_0\), we mean a permutation of \(\Delta_0\) that restricts to the set of frozen vertices. If \(\pi\) is a permutation on \(\Delta_0\), then we get another ice QP \(\pi(\Delta,\mathcal{S})\) by relabelling the vertices. Each representation \(M\) of \((\Delta,\mathcal{S})\) is naturally a representation of \(\pi(\Delta,\mathcal{S})\), and we denote this induced functor still by \(\pi\).

Let \(\sigma=(\mu_{\boldsymbol{u}}, \pi)\) be a cluster automorphism. We have a bijection \[\sigma: \mathop{\mathrm{trop}}(\Delta, \mathcal{S}) \to \mathop{\mathrm{trop}}(\pi\mu_{\boldsymbol{u}}(\Delta, \mathcal{S}))\;\text{ given by }\; \delta \mapsto \pi\mu_{\boldsymbol{u}}(\delta).\] Since the essential part of the crystal structure in Theorem 34 is \(r_i\), \(\rho_i\), and \(\operatorname{wt}_i\), we will ignore \(l_i\) and \(\lambda_i\) in the notation for this structure. The crystal cluster structure \((r_i^\sigma;\rho_i^\sigma;\operatorname{wt}_i^\sigma)\) induced by \(\sigma\) is the crystal cluster structure given by \[\label{eq:crystalsigma} r_i^\sigma(\delta) = r_{\pi(i)}(\delta),\;\rho_i^\sigma(\delta) = {\rm e}(\delta, E_{\pi(i)}),\;\text{ and }\;\operatorname{wt}_i^\sigma=\pi\mu_{\boldsymbol{u}}(\operatorname{wt}_i)\tag{35}\] for \(\delta \in \mathop{\mathrm{trop}}(\pi\mu_{\boldsymbol{u}}(\Delta,\mathcal{S}))\). Note that if \(\sigma\) is direct, then by Lemma 16 \(\mathop{\mathrm{trop}}(\pi\mu_{\boldsymbol{u}}(\Delta, \mathcal{S})) = \mathop{\mathrm{trop}}(\Delta, \mathcal{S})\). In this case, the crystal operators and the string length functions on the right-hand sides are the ordinary ones on \((\Delta,\mathcal{S})\).

Now we shall consider the case when \(\sigma\) is opposite. Recall that the boundary representation \(E_i\) of \((\Delta,\mathcal{S})^{\mathop{\mathrm{op}}}\) is isomorphic to the dual boundary representation of \((\Delta,\mathcal{S})\), and the \(\delta\)-vectors of \((\Delta,\mathcal{S})^{\mathop{\mathrm{op}}}\) is naturally a \({\check{\delta}}\)-vectors of \((\Delta,\mathcal{S})\). Hence, 35 becomes \[\label{eq:sigmaopp} r_i^\sigma(\delta) = {\check{r}}_{\pi(i)}^\star(\delta),\;\rho_i^\sigma(\delta) = {\check{{\rm e}}}(E_{\pi(i)}^\star, \delta) = \check{\rho}_{\pi(i)}^\star(\delta),\;\text{ and }\;\operatorname{wt}_i^\sigma=\pi\mu_{\boldsymbol{u}}(\operatorname{wt}_i)\tag{36}\] where \(\delta \in \mathop{\mathrm{trop}}(\Delta,\mathcal{S})^{\mathop{\mathrm{op}}}\) but \(\delta\)’s on the right-hand sides are viewed as \({\check{\delta}}\)-vectors of \((\Delta,\mathcal{S})\).

By Theorem 11 the map \(\delta\mapsto {\check{\delta}}\) is a bijection from the set \({\mathop{\mathrm{trop}}}(\Delta, \mathcal{S})\) to the set \(\check{\mathop{\mathrm{trop}}}(\Delta, \mathcal{S})\), so we can transfer this crystal structure \(({\check{r}}_i^\star,{\check{l}}_i^\star; \check{\rho}_i^\star,\check{\lambda}_i^\star; \check{\operatorname{wt}}_i^\star)\) from \(\check{\mathcal{B}}\) to \(\mathcal{B}\), denoted by \((r_i^\star, l_i^\star;\;\rho_i^\star, \lambda_i^\star;\;\operatorname{wt}_i^\star)\). In fact, they can be explicitly written down (see Remark 15). We call this the dual crystal structure of \(\mathcal{B}\), denoted by \(\mathcal{B}^\star\). Note that \(\mathcal{B}\) and \(\mathcal{B}^\star\) have the same underlying set.

When \((\Delta,\mathcal{S})\) corresponds to the cluster algebra \({\mathbb{k}}[U]\), the original Kashiwara involution for \(\mathcal{B}(\infty)\) [4] is the generalized Kashiwara map associated to an opposite cluster automorphism \((\mu_{\boldsymbol{u}},\pi)\) where \(\pi\) is the Lusztig involution (see Section 11.1). In the literature, a Kashiwara involution is often denoted by \(\star\), but unfortunately \(\star\) has different meaning in our notation. So our \(\rho_i^\kappa = \rho_{\pi(i)}^\star\) is the \(\rho_i^\star\) in the literature for \(\mathcal{B}(\infty)\).

7 Kashiwara’s Data↩︎

Readers can skip the whole Section 7 without harm to understand the full story.

7.1 The Right Mutations w.r.t. a Representation↩︎

We recall another identification of the space \(\mathop{\mathrm{E}}(d',d'')\) in [40]. According to [40], \(\mathop{\mathrm{E}}(d',d'') \cong \mathop{\mathrm{Ext}}_{\mathcal{C}}^1(d',d'')\) where \(\mathcal{C}\) is the abelian category of complexes of representations of \((\Delta,\mathcal{S})\). By applying \(\mathop{\mathrm{Hom}}(-,N)\) to an exact sequence of presentations in \(\mathcal{C}\) \[0\to d'' \to d\to d'\to 0\] we get by the snake lemma a long exact sequence \[0\to \mathop{\mathrm{Hom}}(d',N)\to \mathop{\mathrm{Hom}}(d,N) \to \mathop{\mathrm{Hom}}(d'',N) \xrightarrow{\partial} \mathop{\mathrm{E}}(d',N)\to \mathop{\mathrm{E}}(d,N)\to \mathop{\mathrm{E}}(d'',N)\to 0.\]

Let \(\mathcal{E}=\mu_{\boldsymbol{u}}(P_k)\) be a positive-reachable representation of \((\Delta,\mathcal{S})\). For any representation \(\mathcal{M}'\) of \(\mu_{\boldsymbol{u}}(\Delta,\mathcal{S})\), let \(\check{\rho}={\rm e}(\mathcal{M}', P_k)\). There is an exact sequence \(\xi\) of presentations \[0\to \check{\rho}d_{P_k} \to d \to d_{\mathcal{M}'} \to 0\] representing the universal extension in \(\mathop{\mathrm{Ext}}^1_{\mathcal{C}}(d_{\mathcal{M}'}, d_{P_k})\). Namely, the pull-backs of \(\xi\) under the \(i\)-th canonical injections \(d_{P_k} \hookrightarrow \check{\rho}d_{P_k}\) form a basis of \(\mathop{\mathrm{Ext}}^1_{\mathcal{C}}(d_{\mathcal{M}'}, d_{P_k})\). By construction the induced map \(\partial'\) is surjective, and so is the map \(\partial\): \[\xymatrix@R=2ex{ \check{\rho}\mathop{\mathrm{Hom}}_{\mathcal{C}}(d_{P_k}, d_{P_k}) \ar[r]^{\partial'} \ar@{}[d]|{\rotatebox{90}{\scalebox{1.5}[1]{\cong}}} &\mathop{\mathrm{Ext}}^1_{\mathcal{C}}(d_{\mathcal{M}'}, d_{P_k}) \ar@{}[d]|{\rotatebox{90}{\scalebox{1.5}[1]{\cong}}} \\ \check{\rho}\mathop{\mathrm{Hom}}(P_k, P_k) \ar[r]^{\partial} &\mathop{\mathrm{E}}(\mathcal{M}', P_k) }\] Let \(\check{\mathcal{R}}'\) be the decorated representation corresponding to \(d\). We set \(\check{\mathcal{R}}=\mu_{\boldsymbol{u}}(\check{\mathcal{R}}')\).

Similarly, if \(\mathcal{E}=\mu_{\boldsymbol{u}}(P_k[1])\) is negative-reachable and \(\rho={\rm e}(P_k[1],\mathcal{M}')\), then there is an exact sequence of presentations \(0\to d_{\mathcal{M}'} \to d \to \rho P_k[1]\to 0\) such that the induced map \(\rho\mathop{\mathrm{Hom}}(P_k, P_k) \to \mathop{\mathrm{E}}(P_k[1], \mathcal{M}')\) is surjective. Let \(\mathcal{R}'\) be the decorated representation corresponding to \(d\). We set \(\mathcal{R}=\mu_{\boldsymbol{u}}(\mathcal{R}')\).

It is shown in [41] that the definition does not depend on the choices the mutation sequence \(\mu_{\boldsymbol{u}}\) ⁴. From the exact sequence \[\mathop{\mathrm{Hom}}(\check{\rho}P_k,P_k)\twoheadrightarrow{} \mathop{\mathrm{E}}(\mathcal{M}', P_k)\to \mathop{\mathrm{E}}(\check{\mathcal{R}}', P_k) \to \mathop{\mathrm{E}}(\check{\rho}P_k, P_k)=0,\] we see that \(\mathop{\mathrm{E}}(\check{\mathcal{R}}', P_k)=0\). By Lemma 3, \({\rm e}(\check{\mathcal{R}},\mathcal{E})+{\rm e}(\mathcal{E},\check{\mathcal{R}}) = {\rm e}(\check{\mathcal{R}}',P_k)+{\rm e}(P_k,\check{\mathcal{R}}')=0\). Similarly we can show that \({\rm e}(\mathcal{R},\mathcal{E})+{\rm e}(\mathcal{E},\mathcal{R})=0\). Moreover, as shown in [32] that if \(\mathcal{M}'\) is rigid, then so are \(\mathcal{R}'\) and \(\check{\mathcal{R}}'\). Hence, if \(\mathcal{M}\) is rigid, so are \(\sqcup^+_{\mathcal{E}}(\mathcal{M})\) and \(\sqcup^-_{\mathcal{E}}(\mathcal{M})\). We conclude that

It also follows from [20] (as Theorem 17) that the representations \(R\) and \(\check{R}\) fit into respectively the long exact sequences \[\begin{align} \tag{38}\cdots\to \hat{\tau}^{-1}\mathcal{M}\to \hat{\tau}^{-1}\mathcal{R}\to\rho\hat{\tau}^{-1}\mathcal{E}\to &M\to R \to \rho E \to \hat{\tau}\mathcal{M} \to \hat{\tau}\mathcal{R} \to \cdots \\ \tag{39} \cdots \to\hat{\tau}^{-1}\check{\mathcal{R}}\to \hat{\tau}^{-1}\mathcal{N}\to \check{\rho}E\to &\check{R}\to N \to \check{\rho}\hat{\tau}\mathcal{E} \to \hat{\tau}\check{\mathcal{R}}\to \hat{\tau}\mathcal{N} \to \cdots \end{align}\] Moreover, we can assume that \(M\) is general of weight \(\delta\) and \(R\) is general of weight \(r_{\epsilon}^{\max}(\delta)\); \(N\) is general of weight \({\check{\eta}}\) and \(\check{R}\) is general of weight \({\check{r}}_{\epsilon}^{\max}({\check{\eta}})\). We also note that \[\label{eq:rho} \rho = {\rm e}(P_k[1], \mathcal{M}') = {\rm e}(P_k[1], \mathcal{M}') + {\rm e}(\mathcal{M}', P_k[1]) = {\rm e}(\mathcal{E}, \mathcal{M}) + {\rm e}(\mathcal{M}, \mathcal{E}).\tag{40}\] Similarly we have that \(\check{\rho}= {\rm e}(\mathcal{E}, \mathcal{N}) + {\rm e}(\mathcal{N}, \mathcal{E})={\check{{\rm e}}}(\mathcal{E}, \mathcal{N}) + {\check{{\rm e}}}(\mathcal{N}, \mathcal{E})\).

7.2 Adjoint Properties↩︎

We say \({\epsilon}\) is a summand of \(\eta\) if a general presentation of weight \(\eta\) has a summand of weight \({\epsilon}\).

7.3 Calculating Kashiwara’s Data↩︎

For a fixed sequence \(\boldsymbol{i} = (i_1,i_2,\dots, i_n)\) in \(I\), we write \(\boldsymbol{i}_{<k}\) for the sequence \((i_1,i_2,\dots, i_{k-1})\); and write \(\boldsymbol{i}_{k>}\) for \((i_{k-1},\dots,i_2,i_1)\). In a similar fashion, we write \(r_{k>}^{\max}\) for \(r_{{\epsilon}_{i_{k-1}}}^{\max}\cdots r_{{\epsilon}_{i_2}}^{\max} r_{{\epsilon}_{i_1}}^{\max}\) and \({\check{r}}_{< k}^{\max}\) for \({\check{r}}_{{\epsilon}_{i_1}}^{\max} {\check{r}}_{{\epsilon}_{i_2}}^{\max}\cdots {\check{r}}_{{\epsilon}_{i_{k-1}}}^{\max}\).

The adjoint property (Corollary 11) provides us a way to calculate the Kashiwara data in the following style \[{\rm e}(r_{k>}^{\max}(\delta), {\check{{\epsilon}}}_{i_k}) = {\rm e}(\delta, \check{R}).\] However, due to the restriction of Corollary 11, we cannot always expect the representation \(\check{R}\) to exist. We can show using the algorithm in [26] that the right adjoint does not exist in Example 1 below. The best one can hope may be the following: there is a sequence of mutations \(\mu_{\boldsymbol{u}}\) such that \[\label{eq:muadj} {\rm e}(r_{k>}^{\max}(\delta), {\check{{\epsilon}}}_{i_k}) = {\rm e}(\mu_{\boldsymbol{u}}(\delta), \mu_{\boldsymbol{u}}({\check{r}}_{<k}^{\max}({\check{{\epsilon}}}_{i_k}))).\tag{47}\] In the current paper, we are not going to fully develop the machinery to solve this problem. It actually involves some intricate algebraic combinatorics, which will be treated somewhere else.

We do not know if 47 always exists for any \(\boldsymbol{i}\)-Kashiwara data. But we conjecture this (actually something stronger) holds for some classical examples, such as \({\mathbb{k}}[U]\).

8 Upper Cluster Algebras and their Generic Bases↩︎

8.1 Upper Cluster Algebras↩︎

In this subsection, we will briefly review the skew-symmetric upper cluster algebras. Readers can find the definition of general upper cluster algebras in the original paper [6]. In most cases we will assume the base ring \({\mathbb{k}}\) to be \(\mathbb{Z}\).

Two seeds \((\Delta,\boldsymbol{x})\) and \((\Delta',\boldsymbol{x}')\) that can be obtained from each other by a sequence of mutations are called mutation-equivalent, denoted by \((\Delta,\boldsymbol{x})\sim (\Delta',\boldsymbol{x}')\).

Let \(\mathcal{L}(\boldsymbol{x}):={\mathbb{k}}[\boldsymbol{x}_\mu^{\pm 1}, \boldsymbol{x}_{\operatorname{fr}}]\) be the subalgebra of the Laurent polynomial algebra \({\mathbb{k}}[\boldsymbol{x}^{\pm 1}]\), which is polynomial in the frozen variables.

Note that by the Laurent Phenomenon [33] the upper cluster algebra \(\overline{\mathcal{C}}(\Delta,\boldsymbol{x})\) contains the corresponding cluster algebra \(\mathcal{C}(\Delta,\boldsymbol{x})\).

By an initial-seed mutation \(\mu_u\) of \(f\in \overline{\mathcal{C}}(\Delta,\boldsymbol{x})\), we mean that \(f(\boldsymbol{x})\) is viewed as an element in \(\mathcal{L}(\boldsymbol{x}')\), that is, we express \(f(\boldsymbol{x})\) in terms of the adjacent cluster \(\boldsymbol{x}'\). It is sensible to just write \(f(\boldsymbol{x}')\) for this mutation, but we follow the tradition to denote it by \(\mu_u(f)\) or \(\mu_u(f)(\boldsymbol{x}')\). Although \(\overline{\mathcal{C}}(\Delta',\boldsymbol{x}')\) is equal to \(\overline{\mathcal{C}}(\Delta,\boldsymbol{x})\), we will write \(\mu_u(f)\in \overline{\mathcal{C}}(\Delta',\boldsymbol{x}')\) to indicate our choice of the cluster to express \(\mu_u(f)\).

In general, there may be infinitely many seeds mutation equivalent to \((\Delta,\boldsymbol{x})\). So the following theorem is useful to test the membership in an upper cluster algebra.

8.2 Generic Bases↩︎

We set \(y_u = \boldsymbol{x}^{-b_u}\) for each mutable vertex \(u\), where \(b_u\) is the \(u\)-th row of \(B_{\Delta}\). Recall from [18] that the \(F\)-polynomial of a representation \(M\) and its dual \(\check{F}\) are the generating functions \[\begin{align} {F}_M(\boldsymbol{y}) &= \sum_{\gamma} \chi(\operatorname{Gr}_\gamma(M)) \boldsymbol{y}^{-\gamma}, \\ \check{F}_M(\boldsymbol{y}) &= \sum_{\gamma} \chi(\operatorname{Gr}^\gamma(M)) \boldsymbol{y}^\gamma, \end{align}\] where \(\operatorname{Gr}_\gamma(M)\) and \(\operatorname{Gr}^\gamma(M)\) are the projective varieties parametrizing respectively the \(\gamma\)-dimensional subrepresentations and quotient representations of \(M\).

We can choose an open subset \(U\subset \mathop{\mathrm{PHom}}(\delta)\) such that \(\mathop{\mathrm{coker}}(d)\) has a constant \(F\)-polynomial for any \(d\in U\). We denote by \(\mathop{\mathrm{coker}}(\delta)\) the cokernel of a general presentation in \(\mathop{\mathrm{PHom}}(\delta)\), and by \(\ker({\check{\delta}})\) the kernel of a general presentation in \(\mathop{\mathrm{IHom}}({\check{\delta}})\).

It was shown in [18] that \(C_{\mathop{\mathrm{gen}}}(\delta)\) is a cluster variable if \(\delta\) is negative-reachable. We will call \(C_{\mathop{\mathrm{gen}}}(\delta)\) a generalized cluster variable if \(\delta\) is rigid.

If the generic character maps the set of \(\mu\)-supported \(\delta\)-vectors to a basis of \(\overline{\mathcal{C}}(\Delta,\boldsymbol{x})\), then such a basis is called the generic basis of \(\overline{\mathcal{C}}(\Delta,\boldsymbol{x})\). It was proved [23], [24] that the generic bases exist for a large class of upper cluster algebras.

Combining [24] with the recent results in [44] (eg., [24] and [44]), we know that for a full-rank reachable quiver, the corresponding upper cluster algebra has the generic basis. From now on we assume that all upper cluster algebras in our discussion admit generic bases. But we want to make the following remark.

8.3 Tropical \(x\)-polynomials and their Pairing↩︎

Recall the following mutation rule for a vector \(d\in \mathbb{Z}^{\Delta_0}\), which is obtained by tropicalizing the mutation of \(x\)-variables 48 \[\label{eq:muatrop} d'(v) = \begin{cases} -d(u) + \max \left\{ d [b_u]_+, d[-b_u]_+ \right\} & \text{if v=u;} \\ d(v) & \text{otherwise}. \end{cases}\tag{50}\] Any vector in \(\mathbb{Z}^{\Delta_0}\) satisfying the mutation rule 50 is called a tropical point of the \(\mathcal{A}\)-variety of \(\Delta\), or a tropical \(\mathcal{A}\)-point for short.

In this definition, we do not require that \(f(\boldsymbol{x})\) has positive coefficients. The tropicalization is determined by the Newton polytope of \(f\).

The (initial-seed) mutation rule of \(f^{\mathop{\mathrm{trop}}}\) is induced by the mutation rule of \(f\), namely, \(\mu_u(f^{\mathop{\mathrm{trop}}})=\mu_u(f)^{\mathop{\mathrm{trop}}}\). Similar to the pairing between tropical \(F\)-polynomials and \(\delta\)-vectors, we also consider the pairing between tropical \(x\)-polynomials and tropical \(\mathcal{A}\)-points (though its representation-theoretic meaning is unclear). We conclude that

9 Algebraic Lifting of Crystal Structures↩︎

9.1 Lifting of Crystal Operators↩︎

Recall that a derivation \(D\) on an algebra \(A\) is a map \(A\to A\) satisfying

1. \(D(a+b)=Da+Db\)

2. \(D(ab)=aDb+bDa\).

If \(R\) is any subring of \(A\), \(D\) is called an \(R\)-derivation if \(D(R)=0\). We are mainly interested in \(\operatorname{Der}_{\mathbb{k}}(A)\), the \({\mathbb{k}}\)-derivation of an upper cluster algebra \(A\). It follows from the definition that \(D\) is uniquely defined by its image on any set of generators of \(A\) as a \({\mathbb{k}}\)-algebra. Since \({\mathbb{k}}\)-derivations of \(A\) are closed under the Lie bracket, \(\operatorname{Der}_{\mathbb{k}}(A)\) forms a Lie algebra over \({\mathbb{k}}\). We also recall that the algebra \(A\) admits an action of a Lie algebra \(\mathfrak{d}\) by derivations if and only if \(A\) is a \(U(\mathfrak{d})\)-module algebra where \(U(\mathfrak{d})\) is the enveloping algebra of \(\mathfrak{d}\).

Let \(I\) be a subset of reachable frozen vertices of \((\Delta,\mathcal{S})\). So there is a seed \(t\) in which \(E_i\) is simple (\(i\) is a sink). To define a derivation \(R_i\) on the corresponding upper cluster algebra, we first define a derivation \(R_i\) on the Laurent polynomial ring at \(t\). According to the Lemma 35 below, it is enough to define an action on \(\boldsymbol{x}\). Explicitly this action is given by \[\begin{align} \label{eq:Ri} {R}_i (x_k) &=\begin{cases} \prod_{u\to i} x_u & \text{ if k=i}\\ 0 & \text{otherwise}. \end{cases} \end{align}\tag{51}\] Note that \(C_{\mathop{\mathrm{gen}}} (r_i(-e_i)) = C_{\mathop{\mathrm{gen}}}(-e_i+{\check{{\epsilon}}}_i) = \prod_{u\to i} x_u\). Similarly, suppose that \(E_i^\star\) is simple at \(t\) (\(i\) is a source). We define \[\begin{align} \label{eq:Rist} {R}_i^\star (x_k) &=\begin{cases} \prod_{i\to u} x_u & \text{ if k=i}\\ 0 & \text{otherwise}. \end{cases} \end{align}\tag{52}\] Also note that \(\check{C}_{\mathop{\mathrm{gen}}} ({\check{r}}_i^\star(-e_{i})) = \check{C}_{\mathop{\mathrm{gen}}}(-e_{i}+{\epsilon}_{i}) = \prod_{i\to u} x_u\).

We need to verify that the definition of \(R_i\) and \(R_i^\star\) does not depend on the choice of seeds.

Next, we check that the action of \(R_i\) restricts to \(\overline{\mathcal{C}}(\Delta)\). By Theorems 37, it suffices to show for a set of generators \(\{f_k\}\) of \(\overline{\mathcal{C}}(\Delta)\) that each \({R}_i(f_k)\) is Laurent in any adjacent seed \(t'\) provided the matrix \(B_\Delta\) is of full-rank. But it is not easy to determine a finite set of generators of \(\overline{\mathcal{C}}(\Delta)\) in general so we will do this for the generic basis.

We shall write \(\partial_i\) for \(\partial_{x_i}\). We keep the assumption that \(E_i\) is simple at the seed \(t\).

We remark that it is clear by definition that \(R_i (x_v)\) is a polynomial in \(\boldsymbol{x}'\) for \(v\neq u\).

Combining Lemmas 36 – 39 , we see that each \(R_i\) is a well-defined \({\mathbb{k}}\)-derivation on \(\overline{\mathcal{C}}(\Delta)\). The justification for \(R_i^\star\) is similar. Let \(\mathfrak{d}_{I}\) be the Lie subalgebra of \(\operatorname{Der}_{\mathbb{k}}(\overline{\mathcal{C}}(\Delta))\) generated by the derivations \(R_i\) and \(R_i^\star\) for \(i\in I\). Then \(\overline{\mathcal{C}}(\Delta)\) is a \(U(\mathfrak{d}_{I})\)-module algebra.

Suppose that we are in the situation of Theorem 34, in which \((i,{\bar{\imath}})\) is a \(\tau\)-exact pair. We define \(L_i = R_{{\bar{\imath}}}^\star\). More explicitly, we choose a seed \(t\) of a cluster algebra in which \(\tau^{-1} E_i = E_{{\bar{\imath}}}^\star\) is simple. Then \[\begin{align} {L}_i (x_k) &=\begin{cases} \prod_{{\bar{\imath}}\to u} x_u & \text{ if k={\bar{\imath}}}\\ 0 & \text{otherwise}. \end{cases} \end{align}\]

Finally, recall the weight functions \(\operatorname{wt}_i\) given as in 29 . If the Cartan matrix \(C_I\) does not have full rank, then we extend \((\operatorname{wt}_i)_{i\in I}\) to a larger set \((\operatorname{wt}_i)_{i\in I \sqcup K}\) as in Lemma 23. Then we define the action \(H_i\) in a natural way: \[\begin{align} \label{eq:Hi} H_i(x_k) = \operatorname{wt}_i(-e_k) x_k. \end{align}\tag{54}\] By Lemma 20 the definition of \(H_i\) does not depend on the choice of seed as well.

9.2 The Structure Theorem on Derivations↩︎

We set \(c_{i,j}=-{\rm e}(\mathcal{E}_i^\mu,\mathcal{E}_j^\mu)-{\rm e}(\mathcal{E}_j^\mu,\mathcal{E}_i^\mu)\) as before, and introduce some new numbers \[c_{i,j}^\star=-{\rm e}(E_j^\star, E_i) \;\text{ and } \;\check{c}_{i,j}^\star=-{\check{{\rm e}}}(E_i^\star, E_j).\] Unlike \({\rm e}(E_i,E_j)\), in general we do not have \({\rm e}(E_j^\star, E_i) = {\rm e}(\tau_\mu\mathcal{E}_j^\mu,\mathcal{E}_i^\mu)\). Also note that \(c_{i,j}^\star\) and \(\check{c}_{i,j}^\star\) are not symmetric about \(i\) and \(j\) but we have the following relation.

Now we have two \(U(\mathfrak{n})\)-actions on \(\overline{\mathcal{C}}(\Delta)\), one from \(R_i\)’s and the other from \(R_i^*\)’s.

9.3 The Weyl Group Action↩︎

Let \(W(\mathfrak{g})\) be the Weyl group of \(\mathfrak{g}\). It is known [4] that there is a \(W(\mathfrak{g})\) action on any \(\mathfrak{g}\)-crystal: \[\begin{align} \label{eq:si} s_i(x) = \begin{cases} l_i^n(x) & \text{if n=\operatorname{wt}_i(x)\geq 0} \\ r_i^{-n}(x) & \text{if n=\operatorname{wt}_i(x)\leq 0.} \end{cases} \end{align}\tag{71}\] The action is compatible with that on the corresponding weight lattice by reflections: \[\operatorname{wt}(s_i(x)) = s_i(\operatorname{wt}(x)),\] where the reflection \(s_i\) is given by \(s_i(\nu) = \nu -\operatorname{wt}_i(\nu)\alpha_i\).

In general, the Weyl group action is not induced by cluster automorphisms. But we see from Corollary 13 that the Weyl group action commutes with the mutations, and each \(s_i\) can be conjugated to a linear transformation by a sequence of mutations. We remark that by Theorem 16 the action is equivalent to \[s_i (\delta) = (\mu_i\mu_{\boldsymbol{u}})^{-1}(\mu_i\mu_{\boldsymbol{u}}(\delta) - n e_i).\] Here, we allow the mutation at \(i\) though \(i\) is a frozen vertex.

9.4 Generic Bases are BK-biperfect↩︎

In [46] Berenstein and Kazhdan introduced perfect basis for a unipotent crystal. Recall that \(\mathcal{B}=\mathop{\mathrm{trop}}(\Delta,\mathcal{S})\) is an upper normal crystal \((r_i,\rho_i, \operatorname{wt}_i)_{i\in I}\), and we lifted \(r_i\) to a \({\mathbb{k}}\)-derivation \(R_i\) of \(\overline{\mathcal{C}}(\Delta,\boldsymbol{x})\). Let \(\rm{B}\) be a homogeneous basis of \(\overline{\mathcal{C}}(\Delta,\boldsymbol{x})\) indexed by \(\mathcal{B}\). When talking about homogeneous basis, we always refer to the grading by \((\operatorname{wt}_i)_{i\in I}\). Following [46], we say \({\rm B}\) is BK-perfect for the crystal \(\mathcal{B}\) if for each \(i\in I\) we have that \[\label{eq:perfectRi0} {R}_i({{\rm B}}(\delta)) = \rho_i(\delta) {{\rm B}}(r_i(\delta)) + v \quad \text{ for some v\in \operatorname{span}({\rm B}(\eta): \rho_i(\eta)<\rho_i(\delta)-1 ) }.\tag{72}\]

In [30] authors considered the bicrystal structure of \({\mathbb{k}}[U]\), and introduced the BK-biperfect bases, which are BK-perfect with respect to the two crystal structures. Now taking Remark 45 into account we will generalize the BK-biperfect bases for \({\mathbb{k}}[U]\) to the setting of cluster algebras. As in [30], we will add the normalization condition \({\rm B}(0)=1\) to the definition in [46]. Since the two weight functions of \(\mathop{\mathrm{trop}}(\Delta,\mathcal{S})\) in \((r_i,\rho_i,\operatorname{wt}_i)\) and \((r_i^\star,\rho_i^\star,\operatorname{wt}_i^\star)\) are not necessarily the same, we call a basis of \(\overline{\mathcal{C}}(\Delta)\) homogeneous with respect to both \(\operatorname{wt}_i\) and \(\operatorname{wt}_i^\star\) a bihomogeneous basis.

Here, \(R_i^{(\star)}\) means \(R_i\) and \(R_i^\star\) respectively, and similarly for \(\rho_i^{(\star)}\) and \(r_i^{(\star)}\). We will indulge in this notation in this section.

In the situation of Theorem 34, the crystal operator \(l_i\) can be lifted to the \({\mathbb{k}}\)-derivation \(L_i\) of \(\overline{\mathcal{C}}(\Delta)\). In this case, a homogeneous basis of \(\overline{\mathcal{C}}(\Delta)\) indexed by \(\mathcal{B}\) is called BK-perfect if in addition to 72 we have that \[{L}_i({{\rm B}}(\delta)) = \lambda_i(\delta) {{\rm B}}(l_i(\delta)) + v \quad \text{ for some v\in \operatorname{span}({\rm B}(\eta): \lambda_i(\eta)<\lambda_i(\delta)-1 ) }.\]

Below we will show that the generic bases are BK-biperfect.

Recall from Theorem 17 that there is an exact sequence \(\tau^{-1} E_i\to M \to R\to E_i\), and we may assume that \(M\) and \(R\) are general of weight \(\delta\) and \(r_i(\delta)\). In particular, we may assume that all \(\mathop{\mathrm{Gr}}^{*}(M)\) and \(\mathop{\mathrm{Gr}}^{*}(R)\) take generic Euler characteristics. If \(E_i\) is simple and \(r_i(\delta)\neq 0\), then the last map \(R\to E_i\) must be zero as \(R\) is \(\mu\)-supported.

Let \(R_i^{\max}\) be the operator on \(\overline{\mathcal{C}}(\Delta)\) defined by \[R_i^{\max}(C_{\mathop{\mathrm{gen}}}(\delta)) = R_i^{(\rho_i(\delta))}(C_{\mathop{\mathrm{gen}}}(\delta)),\] and similarly we define \({R_i^\star}^{\max}\).

Recall that a normal crystal is a disjoint union of crystals, each of which is isomorphic to the one underlying some integrable highest-weight representation of a fixed Kac-Moody Lie algebra \(\mathfrak{g}\). Such a crystal is also called a \(\mathfrak{g}\)-crystal.

10 Biperfect Bases↩︎

10.1 All BK-biperfect Bases↩︎

We are going to give a description of all BK-biperfect bases for a fixed \(\overline{\mathcal{C}}(\Delta)\). Let us recall the string order introduced in [29]. Given \((b',b'')\) and \((c',c'')\) in \(\mathcal{B}\times \mathcal{B}\), we write \((b',b'')\approx (c',c'')\) if one of the following two conditions holds:

1. There is \(i\in I\) such that \(\rho_i^{(\star)}(b')=\rho_i^{(\star)}(b'')\) and \((c', c'')=(l_i^{(\star)}(b'), l_i^{(\star)}(b''))\).

2. There is \(i\in I\) such that \(\rho_i^{(\star)}(b')=\rho_i^{(\star)}(b'')>0\) and \((c', c'')=(r_i^{(\star)}(b'), r_i^{(\star)}(b''))\).

To simplify the notation, we write \(\eta \llcurly_{\rho} \delta\) if \(\rho_i(\eta)<\rho_i(\delta)\) and \(\rho_i^\star(\eta)<\rho_i^\star(\delta)\) for each \(i\in I\). We also write \(\eta \preceq_{\rho} \delta\) if \(\rho_i(\eta)\leq \rho_i(\delta)\) and \(\rho_i^\star(\eta)\leq \rho_i^\star(\delta)\) for each \(i\in I\). The set \(\{\eta\mid \eta\llcurly_{\rho} \delta\}\) contains lattice points in a (not necessarily bounded) polyhedral set by Theorem 12. We denote this polyhedral set by \({\sf R}(\delta)\). It is clear that \(\llcurly_{\rho}\) implies \(\preceq_{\mathop{\mathrm{str}}}\) and \(\preceq_{\mathop{\mathrm{str}}}\) implies \(\preceq_{\rho}\). The set \(\{\eta\mid \eta\preceq_{\mathop{\mathrm{str}}} \delta\}\) is also given by polyhedral conditions. But it is hard to write down all conditions explicitly.

Let \(\sigma=(\mu_{\boldsymbol{u}}, \pi)\) be a cluster automorphism of \(\Delta\) (Definition 28). Then we have an algebra automorphism \(\mathcal{L}_{\boldsymbol{x}} \to \mathcal{L}_{\mu_{\boldsymbol{u}}(\boldsymbol{x})}\) induced by \(x_{v} \mapsto \mu_{\boldsymbol{u}}(x_{\pi(v)})\). Since \(\pi \mu_{\boldsymbol{u}}(\Delta) = \Delta\) or \(\Delta^{\mathop{\mathrm{op}}}\), this automorphism is compatible with mutations. Hence, it induces an automorphism of \(\overline{\mathcal{C}}(\Delta,\boldsymbol{x})\), still denoted by \(\sigma\). We say \({\rm B}\) is \(\sigma\)-invariant if \({\rm B}(\sigma(\delta)) = \sigma({\rm B}(\delta))\).

One point following from Theorem 48 is that there are excessively many BK-biperfect bases from the point of view of cluster algebras. We can easily construct examples in which some cluster variables are not in a BK-biperfect basis. But such examples for \({\mathbb{k}}[U]\) are not trivial. We found the following example with the help of the software Normaliz [47].

10.2 Good Bases and Biperfect Bases↩︎

A linear basis of \(\overline{\mathcal{C}}(\Delta)\) indexed by \(\mathop{\mathrm{trop}}(\Delta,\mathcal{S})\) is a rather weak notion. For one thing, additional orders from the cluster structure do not play a role here. Let \(t\) be a seed \((\Delta,\mathcal{S})\). We recall the dominance order \(\prec_t\) on the lattice \({\sf M}_t\cong \mathbb{Z}^{\Delta_0}\) such that \(\delta' \prec_t \delta\) if and only if \(\delta' = \delta + \gamma B_{\Delta}\) for some \(\mu\)-supported dimension vector \(\gamma\). We write \(\delta' \prec_{\mathfrak{T}} \delta\) if \(\delta' \prec_{t} \delta\) for all \(t\in \mathfrak{T}\).

Before we introduce the biperfect bases, let us first briefly review the good bases introduced by F. Qin [24]. As in [24] we shall denote the equivalence class of \(\delta\) under the tropical transform 10 by \([\delta]\), and the set of all equivalence classes \([\delta]\) by \({\sf M}\).

In the theorem below (and only in the theorem below), we shall consider the localized upper cluster algebra \(\overline{\mathcal{C}}(\Delta)_{\operatorname{loc}}:= \bigcap_{t\in\mathfrak{T}}{\mathbb{k}}[\boldsymbol{x}_t^{\pm}].\)

Any collection \({\rm B}\) in Theorem 51 is called a good basis of \(\overline{\mathcal{C}}(\Delta)_{\operatorname{loc}}\). If such a basis exists in the non-localized \(\overline{\mathcal{C}}(\Delta)\), we also call it good.

In the cluster algebra setting, it is natural to replace “indexed by \(\mathop{\mathrm{trop}}(\Delta, \mathcal{S})\)" by”pointed at \(\mathop{\mathrm{trop}}(\Delta, \mathcal{S})\)" as in Definition 42.

It follows from the definitions that a biperfect basis is automatically good. Conversely, a good basis which is BK-biperfect at a particular seed \(t\) is biperfect. This follows from Remark 44. We already know that generic bases are good bases. So they are biperfect by Theorem 46.

The following corollary is immediate from Theorems 48 and 51.

Although the \(\eta\)’s in both summations are lattice points in some bounded polyhedral sets, we have no explicit description for either of them.

11 Examples↩︎

As shown in [23], [48], there is a potential \(\mathcal{S}\) such that \((\Delta,\mathcal{S})\) is rigid (and thus nondegenerate) for each of the examples below. Throughout this section, we do not need the explicit forms of these potentials.

11.1 Unipotent Groups↩︎

Let \(Q\) be a Dynkin quiver, and \(\overline{Q}\) be its underlying graph. Let \(G\) be a simple, simply-connected, algebraic group over \(\mathbb{C}\) of type \(\overline{Q}\), and \(U\) be a maximal unipotent subgroup of \(G\). Recall from [6], [7] that \({\mathbb{k}}[U]\) is the upper cluster algebra of \(\Delta_Q\), where \(\Delta_Q\) is the Auslander-Reiten quiver of \(Q\) with translation. It is well known that as a crystal \({\mathbb{k}}[U]\) is isomorphic to \(\mathcal{B}(\infty)\). For any quiver \(Q\), we let \(A_Q\) be its arrow matrix.

So by Theorem 32 \(\mathop{\mathrm{trop}}(\Delta_Q, \mathcal{S})\) has an upper seminormal crystal cluster structure of type \(C_{\overline{Q}}\). In fact, it is isomorphic to \(\mathcal{B}(\infty)\). (1) is already checked in [48]. We also checked there that the weight function \((\operatorname{wt}_i)_{i\in I}\) agrees with the natural grading given by the conjugate action of \(T\subset G\) on \(U\). (2) can be checked by straightforward calculation. (3) is a result of an undergraduate research program (to appear).

11.2 Simple Canonical Models↩︎

We will construct for any acyclic quiver \(Q\) a quiver with potential such that its tropical points carry an upper seminormal crystal structure of type \(C_{\overline{Q}}\).

By definition the \(B\)-matrix of \(\Omega_Q\) is equal to \((B_Q, -E_Q^{\operatorname{T}})\) where \(E_Q=I_{n}-A_Q\) is the Euler matrix of \(Q\) and \(n=|Q_0|\). Let \(I\) be the set of all frozen vertices of \(\Omega_Q\), which is just a copy of \(Q_0\). By construction, the Cartan type of \(I\) is clearly \(C_{\overline{Q}}\). We put the weight function \(\operatorname{wt}_i\) given by the columns of the following matrix: \[\left(\begin{matrix}I_n \\ {E_Q^{\operatorname{T}}}^{-1}B_Q \end{matrix}\right) (I_n + {E_Q^{\operatorname{T}}}^{-1} E_Q).\]

We remark that this construction is a special case of [50]. Here we constructed the quiver of a cluster algebra of \(\mathbb{C}[U^{\omega}]\) for \(\omega = (1,2,\dots,n)\in W(\mathfrak{g})\). Readers can easily generalize this example to construct other crystals of Kac-Moody type.

11.3 Base Affine Spaces↩︎

Let \(G\) be a simple, simply-connected, algebraic group over \(\mathbb{C}\) of type \(\overline{Q}\). Recall from [48] that the algebra of regular functions on the base affine space \(G/U\) is the upper cluster algebra of \(\Delta_Q^\sharp\). The ice quiver \(\Delta_Q^\sharp\) is obtained from \(\Delta_Q\) by adding a set of frozen vertices \({\bar{\imath}}\) for \(i\in Q_0\), which correspond to the shifted projectives in \(\mathop{\mathrm{rep}}(Q)\).

It is known [48] that the set \(\mathop{\mathrm{trop}}(\Delta_Q^\sharp, \mathcal{S})\) is given by lattice points in the polytope \({\sf G}_Q^\sharp\), which can be described by Theorem 23. Also recall that we have a categorical description of the dimension vectors of the (dual) boundary representations. As have seen in Proposition 56 that the Cartan type of \(I\) is \(C_{\overline{Q}}\). The following lemma can be checked, although not trivial, using Lemma 22. For \(i\in \overline{Q}_0\), let \(i \leftrightarrow i^*\) be Lusztig’s involution.

The Weyl group action on \(\mathop{\mathrm{trop}}(\Delta_Q^\sharp, \mathcal{S})_\lambda\) is just the usual Weyl group action on \(\mathcal{B}(\lambda)\). Instead of the cluster coordinates, this was done in [1] in Gelfand-Zeitlin coordinates. We can recover most results in [16] from our description of the crystal structure for \(G/U\).

From the above corollary, we can recover a main result in [51] on the tensor product multiplicity of representations of \(G\). Let \(L(\lambda)\) be the irreducible representation of \(G\) of highest weight \(\lambda\). The tensor product \(L(\mu) \otimes L(\nu)\) decomposes as \(\bigoplus_{\lambda} c^\lambda_{\mu\,\nu}L(\lambda)\).

The tropical function \(\rho_i(\delta)\) in this setting was also considered in [53].

11.4 Grassmannians↩︎

Let \(\mathop{\mathrm{Gr}}\left(\begin{smallmatrix}n \\k\end{smallmatrix}\right)\) be the Grassmannian of \(k\)-planes in \(\mathbb{C}^n\). Recall that the coordinate ring of the affine cone over \(\mathop{\mathrm{Gr}}\left(\begin{smallmatrix}n \\k\end{smallmatrix}\right)\) is the cluster algebra of the following quiver \(\Square:=\Square_{k,l}\), where \(l=n-k\). \[\grass\] We also give another labelling on the frozen vertices: \(0=(0,0),\;i=(i,k),\;l+j=(l,k-j)\) for \(i=1,\dots,l\) and \(j=1,\dots, k\). The following proposition is straightforward to check.

The weight function \(\operatorname{wt}_m = \underline{\dim}E_m - \underline{\dim}\tau^{-1} E_m\) can be read off from (1) and (3). One can check that the weight function agrees with the usual grading from the torus action of \(T\subset \mathop{\mathrm{GL}}_n\). We can also recover most results in [25] from our description of the crystal structure.

There are similar results for isotropic Grassmannians [8]. We will give below an example of an exceptional Grassmannian. In the notation of [8], the group is of type \(E_6\) and the index set \(J\) complementary to the parabolic subgroup is \(\{1\}\).

11.5 The Exact Coefficient Patterns↩︎

The following proposition is clearly follows from Corollary 7 and Lemma 22.

According to Proposition 60, whether the coefficient pattern of \(\Delta\) is exact can be read off from the (rigid) representations \(\mathcal{V}\) of \(\Delta^\mu\).

One can check this is really the cluster tilted algebra for the canonical algebra of type \((a_1,a_2,\cdots,a_r)\) [54].

We note that the cluster tilted algebras of canonical type \((2,3,6)\) as well as the mutable part of \(\Square_{3,9}\) are mutation-equivalent to \(\Delta_6^\mu\). So far we have three different seminormal crystal cluster structures on \(\Delta_6^\mu\): One of Cartan type \(\tilde{A}_8\) from the Grassmannian \(\mathop{\mathrm{Gr}}\left(\begin{smallmatrix}9\\ 3\end{smallmatrix}\right)\), one of type \(\tilde{E}_6\) from an exceptional Grassmannian, and one of type \(\tilde{A}_1\times \tilde{A}_2 \times \tilde{A}_5\). Conjecturally they are the only three maximal exact coefficient patterns on \(\Delta_6^\mu\).

Acknowledgement↩︎

The author would like to thank Pierre Baumann and John Stembridge for some discussions. He would also like to thank Xiaoyue Lin and Yueyang Yan for proof-reading the manuscript.

References↩︎

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1. The author was supported in part by National Natural Science Foundation of China (No. 11971305 and No. 12131015)↩︎

2. The \(\delta\)-vector is the same one defined in [32], but is the negative of the \({\sf g}\)-vector defined in [18].↩︎

3. This definition is slightly different from the one in [18], which involves the decorated part.↩︎

4. The construction in [32], [41] uses triangles in the homotopy category of projective presentations. Here, we work with the abelian category of complexes to slightly simplify the construction.↩︎