1 Introduction↩︎

Introduced by Drinfeld [1] and Jimbo [2] around 1985, the quantum group \(U_q(\mathfrak{g})\) associated to a finite-dimensional simple Lie algebra \(\mathfrak{g}\) can be viewed as a deformation of the universal enveloping algebra of \(\mathfrak{g}\). In the early 90’s, Lusztig [3] constructed a remarkable basis of the quantum coordinate ring \(\mathcal{A}_q(\mathfrak{n})\) (dual of the negative half \(U_q(\mathfrak{n})\) arising from a triangular decomposition of \(U_q(\mathfrak{g})\)) called the dual canonical basis. Kashiwara [4] proposed an alternative construction called the upper global basis which was shown to coincide with Lusztig’s dual canonical basis by Grojnowski–Lusztig [5] and Kashiwara–Saito [6]. Evaluating at \(q=1\), the dual canonical basis descends to a basis of the ring of regular functions on the Lie group \({\boldsymbol{N}}\) corresponding to \(\mathfrak{n}\). Other remarkable bases of \(\mathbb{C}[\mathbf{N}]\) were discovered since then, for instance the dual semicanonical basis constructed by Lusztig [7], or the Mirković–Vilonen basis arising from the geometric Satake correspondence [8]. The study of the combinatorial properties of the elements of these bases was one of the main motivations at the origin of the foundation of cluster theory by Fomin and Zelevinsky [9].

The quantum affine algebra associated to \(\mathfrak{g}\), denoted \(U_q(\widehat{\mathfrak{g}})\) can be viewed as a affinization of \(U_q(\mathfrak{g})\), or as quantization of the universal enveloping algebra of the Kac-Moody algebra \(\widehat{\mathfrak{g}}\) associated to \(\mathfrak{g}\), see [10]. In [11], Hernandez and Leclerc discovered an intimate connection between the representation theory of quantum affine algebras and cluster theory. They introduced the notion of monoidal categorification of cluster algebras, which opened the way for the study of various cluster structures using representation-theoretic tools. At the heart of this framework lies the idea that certain distinguished short exact sequences in certain well-chosen monoidal categories yield relations in their Grothendieck rings that can be identified with exchange relations associated to mutations for a suitable cluster structure. Kang–Kashiwara–Kim–Oh [12] obtained important monoidal categorification results with categories of representations of quiver Hecke algebras by constructing certain non trivial braiding operators (\(R\)-matrices). These operators give rise to short exact sequences corresponding to the exchange relations for the cluster structure of \(\mathbb{C}[\mathbf{N}]\). They also provide useful (necessary) criteria for the irreducibility of products of simple modules in these categories. Other examples of monoidal categorifications have also been exhibited by Cautis–Williams [13] via the coherent Satake category. Hernandez–Leclerc’s categories are also crucially involved in the major recent works by Kang–Kashiwara–Kim–Oh-Park [12], [14], [15]. In the original work of Hernandez–Leclerc [11], these monoidal subcategories are constructed for a source–sink orientation. This construction was later generalized to arbitrary height functions in [16] and [15], [17].

Beyond the study of cluster monomials, the categories involved in monoidal categorifications of \(\mathbb{C}[\mathbf{N}]\) also provide powerful tools to study the dual canonical bases of \(\mathbb{C}[\mathbf{N}]\) [18] mentioned earlier, as well as dual canonical basis of Grassmannian cluster algebras [19]. For instance, when \(\mathfrak{g}\) is of simply-laced type, Hernandez–Leclerc [16] introduced a family \(\{ \mathcal{C}_Q\}\) of (monoidal) categories of finite-dimensional representations of \(U_q(\widehat{\mathfrak{g}})\) indexed by the set of orientations of the Dynkin graph of \(\mathfrak{g}\) and showed that for each \(Q\), the complexified Grothendieck ring of \(\mathcal{C}_Q\) is isomorphic to \(\mathbb{C}[\mathbf{N}]\) in such a way that the natural basis of classes of irreducible representations of \(\mathcal{C}_Q\) gets identified with the dual canonical basis of \(\mathbb{C}[\mathbf{N}]\). These categories \(\mathcal{C}_Q\) are all monoidal subcategories of a larger monoidal category \(\mathcal{C}_{\mathbb{Z}}\) that can be viewed as discrete version of the whole category of finite-dimensional representations of \(U_q(\widehat{\mathfrak{g}})\). Frenkel–Reshetikhin’s \(q\)-character morphism [20] yields an injective ring homomorphism from \(K_0(\mathcal{C}_{\mathbb{Z}})\) to the ring of Laurent polynomials in variables \(Y_{i,p}\) where \(i\) runs over the set of vertices of \(Q\) and \(p \in i + 2 \mathbb{Z}\). For reasons related to cluster theory, Hernandez–Leclerc consider a slightly modified version \({\widetilde{\chi}}_q\) of \(\chi_q\) called truncated \(q\)-character, which depends on \(Q\) (more precisely on a choice of a height function adapted to \(Q\)).

Lusztig [21] discovered a deep connection (called the geometric Satake correspondence) between the representation theory of the Lie group \(\mathbf{G}\) of \(\mathfrak{g}\) and the geometry of the affine Grassmannian \(Gr_{\mathbf{G}^{\vee}}\) of its Langlands dual \(\mathbf{G}^{\vee}\). Using this result together with ideas from [22], Mirković and Vilonen [8], [23] constructed a remarkable basis \(\{ b_z \}\) of \(\mathbb{C}[\mathbf{N}]\), called the Mirković–Vilonen basis (or simply MV basis), whose elements are indexed by a family of closed irreducible subvarieties of \(Gr_{\mathbf{G}^{\vee}}\) called (stable) Mirković–Vilonen cycles. Note that the MV basis has been proved to be biperfect in the sense of [24] (cf. [24]). These varieties are endowed with a natural action of a maximal torus \(\mathbf{T}^{\vee}\) in \(\mathbf{G}^{\vee}\) and one can therefore consider the corresponding equivariant homology theory. In their recent work [24], Baumann–Kamnitzer–Knutson introduced a remarkable algebraic homomorphism \(\overline{D}: \mathbb{C}[\mathbf{N}]\longrightarrow \mathbb{C}(\mathfrak{t}^{*})\) where \(\mathfrak{t}\) denotes the Lie algebra of \(\mathbf{T}^{\vee}\) and proved that the evaluation of \(\overline{D}\) on an element \(b_Z\) of the MV basis coincides with the equivariant multiplicity at a well-chosen fixed point of the corresponding MV cycle \(Z\). The morphism \(\overline{D}\) turned out to provide a powerful tool to compare the several distinguished bases of \(\mathbb{C}[\mathbf{N}]\) mentioned above.

The motivation for the present paper stems from our desire to provide an interpretation of \(\overline{D}\) from the perspective of Hernandez–Leclerc’s monoidal categorification and investigate further the connections between the representation theory of quantum affine algebras and the geometric Satake correspondence. In our previous work [25], we constructed for each simply-laced type Lie algebra \(\mathfrak{g}\) and for each height function \(\xi\) on the Dynkin diagram of \(\mathfrak{g}\) a morphism \(\widetilde{D}_{\xi}: \mathcal{Y}_{\mathbb{Z}}^{\leq \xi}\longrightarrow \mathbb{C}(\mathfrak{t}^{*})\) where \(\mathcal{Y}_{\mathbb{Z}}^{\leq \xi}\) is the subtorus of \(\mathcal{Y}_{\mathbb{Z}}\) generated by the \(Y_{i,p}^{\pm 1}, i \in I , p \leq \xi(i)\). The definition of \(\widetilde{D}_{\xi}\) relies on Auslander–Reiten combinatorics and involves the coefficients of the inverse quantum Cartan matrix of \(\mathfrak{g}\). The ring \(\mathcal{Y}_{\mathbb{Z}}^{\leq \xi}\) contains the truncated \(q\)-characters of all the representations in Hernandez–Leclerc’s category \(\mathcal{C}_Q\) where \(Q\) is the Dynkin quiver given by \(\xi\). Our main result from [25] consists in proving that the composition of \(\widetilde{D}_{\xi}\) with the restriction of the truncated \(q\)-character morphism to \(K_0(\mathcal{C}_Q)\) coincides with Baumann–Kamnitzer–Knutson’s morphism \(\overline{D}\) (more precisely its composition with Hernandez–Leclerc’s isomorphism \(\mathbb{C}[\mathbf{N}]\simeq K_0(\mathcal{C}_Q)\)). In other words, \(\overline{D}\) can be viewed as the restriction to \(\mathbb{C}[\mathbf{N}]\) of \(\widetilde{D}_{\xi}\circ {\widetilde{\chi}}_q\). As a consequence, we were able to extend the results and prove several observations and conjectures made by the first author in [26]. Moreover, this allows to extend \(\overline{D}\) to the Grothendieck rings of larger subcategories of \(\mathcal{C}_{\mathbb{Z}}\) also carrying non-trivial cluster structures (see [27]).

This paper is devoted to the study of the composition of \(\widetilde{D}_{\xi}\) with Frenkel–Reshetikhin’s original (i.e. non-truncated) \(q\)-character morphism. Firstly, we extend the map \(\widetilde{D}_{\xi}\) to the whole ring \(\mathcal{Y}_{\mathbb{Z}}\) thanks to a periodicity argument. Thus we can study the restriction of \(\widetilde{D}_{\xi}\) to the image of \(K_0(\mathcal{C}_{\mathbb{Z}})\) under \(\chi_q\). A standard module is a tensor product of fundamental modules in a well-chosen order. Our main result can be stated as follows:

Consequently, the image under \(\widetilde{D}_{\xi}\) of the \(q\)-character of an arbitrary module in \(\mathcal{C}_{\mathbb{Z}}\) is always constant. Note that this constant is zero in the case of non-trivial simple modules in the subcategory \(\mathcal{C}_Q\). The crucial ingredient for the proof of this result consists in establishing a remarkable duality between certain distinguished simple modules in \(\mathcal{C}_{\mathbb{Z}}\) called Kirillov–Reshetikhin modules, when \(\mathfrak{g}\) is of classical type, which we believe is of independent interest. The Kirillov–Reshetikhin modules in \(\mathcal{C}_{\mathbb{Z}}\) are denoted \(X_{i,p}^{(k)}\) with \(i \in I\), \(p \in {\mathbb{Z}}\) and \(k \in {\mathbb{Z}}_{\ge 1}\) ( see Section 3.2). We define a dual orientation \(Q^{*}\) given by the height function \(\xi^{*}(i) := - \xi(i^{*})\) where \(i \mapsto i^{*}\) is the involution of the Dynkin diagram such that \(w_0(\alpha_i) = - \alpha_{i^{*}}\). Denoting \({\widetilde{\chi}}_q^{*}\) the corresponding truncation à la Hernandez–Leclerc, we assume \(Q\) is the monotonic orientation as chosen in [25] and show (cf. Theorem 7) that if \(i \in I, 1 \leq k < h\) and \(\xi(i)-2h+2 \leq p \leq \xi(i)\), then the following holds \[\label{eq32:32duality32intro} \widetilde{D}_{\xi}\left( {\widetilde{\chi}}_q \left( X_{i,p}^{(k)} \right) \right) = \pm \left( \psi \circ \widetilde{D}_{\xi^{*}} \right) \left( {\widetilde{\chi}}_q^{*} \left( X_{i^{*}, -p-2h+4}^{(h-k)} \right) \right)\tag{1}\] where \(h\) is the Coxeter number of the Weyl group of \(\mathfrak{g}\), the sign on the right hand side can be made explicit in terms of Auslander–Reiten theoretic quantities [28], and \(\psi\) denote the automorphism of \(\mathbb{C}(\mathfrak{t}^{*})\) given by \(\psi(\alpha_i) = \alpha_{i^{*}}\). We also use several classical facts from Auslander–Reiten combinatorics as key ingredients in the proof of Theorem 1. In particular, our proof may be viewed as a manifestation of some interesting compatibility between mesh relations and T-systems.

In the case where \(\mathfrak{g}\) is of type \(A_n, n \geq 1\), we are able to provide an alternative proof of Theorem 1 that does not rely on 1 , but rather on Mukhin–Young’s path model formulas [29].

As explained above, the map \(\widetilde{D}_{\xi}\) is intimately related to the morphism \(\overline{D}\) introduced by Baumann–Kamnitzer–Knutson [24] in their study of the MV basis and of the equivariant homology of the affine Grassmannian. Given this tight connection with the geometric Satake correspondence, it is natural to believe that the statement of Theorem 1 may admit some interesting geometric interpretation.

The paper is organized as follows. Section 2 is devoted to some recollection on Auslander–Reiten theory. In particular we investigate the combinatorics of the dual quiver \(Q^{*}\) mentioned above. In Section 3, we recall the main features of Hernandez–Leclerc’s categorifications and we provide detailed reminders on the map \(\widetilde{D}_{\xi}\) defined in our previous work [25]. Section 4 is devoted to the proof of the duality 1 between Kirillov–Reshetikhin modules when \(\mathfrak{g}\) is of classical type. Finally, Section 5 contains the final arguments for completing the proof of our main result (Theorem 1).

Acknowledgments↩︎

The research of E.C. was supported by the RTG NSF grant DMS-1645877. JR Li is supported by the Austrian Science Fund (FWF): P-34602, Grant DOI: 10.55776/P34602, and PAT 9039323, Grant-DOI 10.55776/PAT9039323, and the National Natural Science Foundation of China (No. 12471023).

2 Auslander–Reiten combinatorics↩︎

2.1 Brief recollection on Auslander–Reiten theory↩︎

Auslander–Reiten theory was introduced by Auslander and Reiten in [28]. Let \(Q\) be a Dynkin quiver, i.e. an orientation of the Dynkin diagram of a Lie algebra \(\mathfrak{g}\) of type \(A_n (n \geq 1) , D_n (n \geq 4)\) or \(E_n , n=6,7,8\). We use the notation \(i \sim j\) to denote that \(i\) and \(j\) are connected by an edge in the underlying graph of \(Q\). Let \(I\) denote the set of vertices of \(Q\) and \(n:= \sharp I\). We will denote by \(\mathop{\mathrm{Rep}}Q\) the category of finite-dimensional representations of \(Q\) over \(\mathbb{C}\), which we will identify with the category of finite-dimensional right modules over the path algebra \(\mathbb{C}Q\) of \(Q\). Furthermore we will also denote by \(\Delta\) (resp. \(\Delta_+\)) the set of roots (resp. positive roots) associated to \(\mathfrak{g}\), and by \(W\) the corresponding Weyl group.

By Gabriel’s theorem [30], there is a bijection between the set \(\Delta_+\) of positive roots of the root system associated with \(Q\) and the set of isomorphism classes of indecomposable objects in \(\mathop{\mathrm{Rep}}Q\). Moreover, the dimension vectors of objects in \(\mathop{\mathrm{Rep}}Q\) can be identified with elements of the positive root cone \(\bigoplus_{i \in I} \mathbb{Z}_{\geq 0} \alpha_i\). For each \(\beta \in \Delta_+\), we denote by \(M_\beta\) the corresponding indecomposable representation. In particular, its dimension vector satisfies \(\dim M_\beta = \beta\).

For each \(j \in I\), we will respectively denote by \(I_j\) (resp. \(P_j\)) the indecomposable injective (resp. projective) representation of \(Q\), whose dimension vectors are given by \[\mathop{\mathrm{\mathbf{dim}}}I_j = \sum_{k \rightsquigarrow j} \alpha_k, \qquad \mathop{\mathrm{\mathbf{dim}}}P_j = \sum_{l \rightsquigarrow k} \alpha_k,\] where the notation \(k \rightsquigarrow j\) means that there exists an oriented path in \(Q\) starting at \(k\) and ending at \(j\) (\(k \rightsquigarrow j\) includes the stationary path at vertex \(j\)).

A reduced expression \(\mathbf{i}=(i_1,\dots,i_N)\) of \(w_0\) is said to be adapted to \(Q\) if for each \(k=1,\dots,N\), the vertex \(i_k\) is a sink in the quiver \(Q_{k-1}\) defined inductively by \(Q_0=Q\) and \(Q_k\) obtained from \(Q_{k-1}\) by reversing all arrows incident to \(i_k\). The Coxeter transform \(c_Q \in W\) associated to \(Q\) is then given by \(c_Q := s_{i_1}\cdots s_{i_n}\). We also denote by \(h\) the corresponding Coxeter number, i.e. the order of \(c_Q\) in \(W\) (recall that this number is the same for all Coxeter elements in \(W\)). A height function adapted to \(Q\) is a function \(\xi : I \longrightarrow \mathbb{Z}\) such that \(\xi(j) = \xi(i)-1\) whenever there is an arrow from \(i\) to \(j\) in \(Q\).

Let \(\mathcal{D}_Q:= \mathcal{D}^b(\mathop{\mathrm{Rep}}Q)\) denote the bounded derived category of \(\mathop{\mathrm{Rep}}Q\) and \(\mathop{\mathrm{Ind}}\mathcal{D}_Q\) denote the set of indecomposable objects of \(\mathcal{D}_Q\). The elements of \(\mathop{\mathrm{Ind}}\mathcal{D}_Q\) are of the form \(M[m]\) where \(M\) is an indecomposable object in \(\mathop{\mathrm{Rep}}Q\) and \(m \in \mathbb{Z}\), so that \(\mathop{\mathrm{Ind}}\mathcal{D}_Q\) is in bijection with \(\Delta_+ \times \mathbb{Z}\). We will then set \(\mathop{\mathrm{\mathbf{dim}}}M[m] := (-1)^m \mathop{\mathrm{\mathbf{dim}}}M\) for any indecomposable object \(M\) in \(\mathop{\mathrm{Rep}}Q\) and any \(m \in \mathbb{Z}\), so that \(\mathop{\mathrm{\mathbf{dim}}}N \in \Delta\) for any object \(N \in \mathop{\mathrm{Ind}}\mathcal{D}_Q\). We denote by \(\langle \cdot , \cdot \rangle_Q\) the Euler-Ringel form of \(Q\), which is the bilinear form given by \[\forall M,N \in \mathop{\mathrm{Rep}}Q, \enspace \langle \mathop{\mathrm{\mathbf{dim}}}M , \mathop{\mathrm{\mathbf{dim}}}N \rangle_Q := \dim \mathop{\mathrm{Hom}}_{\mathop{\mathrm{Rep}}Q}(M,N) - \dim \mathrm{Ext}^1_{\mathop{\mathrm{Rep}}Q}(M,N) .\] Using this, it is easy to compute the dimensions of morphism spaces in \(\mathcal{D}_Q\): namely, if \(M\) and \(M'\) are indecomposable objects in \(\mathop{\mathrm{Rep}}Q\) and \(m,m' \in \mathbb{Z}\), then \[\label{eq32:32dim32of32Homs32in32DbRepQ} \dim \mathrm{Hom}_{\mathcal{D}_Q}(M[m],M'[m']) = \begin{cases} (\langle \boldsymbol{\dim}M, \boldsymbol{\dim}M' \rangle )_+& \text{if m=m',} \\ (\langle \boldsymbol{\dim}M, \boldsymbol{\dim}M' \rangle)_- & \text{if m' = m+1,} \\ 0 & \text{otherwise,} \end{cases}\tag{2}\] where \(a_+ := \max(a,0)\) and \(a_- := - \min(a,0)\) for any integer \(a\). We denote \(i+2{\mathbb{Z}}= \{i+2r: r \in {\mathbb{Z}}\}\) for \(i \in I = [n]\), where \(n\) is the rank of \(\mathfrak{g}\). Let \(\widehat{I}_{\mathbb{Z}}\) denote the set of pairs \[\widehat{I}_{\mathbb{Z}}:= \{ (i,p) , i \in I , p \in i + 2 \mathbb{Z} \}\] and let us choose a height function \(\xi\) adapted to \(Q\). Such a choice yields a natural bijection \[\label{eq32:32bijection} (i,p) \in \widehat{I}_{\mathbb{Z}}\longmapsto M_{i,p} \in \mathop{\mathrm{Ind}}\mathcal{D}_Q\tag{3}\] given by \[M_{j, \xi(j)} := I_j, \enspace j \in I, \qquad M_{i,p \pm 2} := \tau_Q^{ \mp 1} M_{i,p} , \enspace (i,p) \in \widehat{I}_{\mathbb{Z}}\] where \(\tau_Q\) stands for the Auslander-Reiten translate in \(\mathcal{D}_Q\). Note that one has \[\label{eq32:32dim32Hom3261320} p>s \enspace \Rightarrow \enspace \dim \mathrm{Hom}_{\mathcal{D}_Q}(M_{i,p},M_{j,s}) = 0\tag{4}\] for any \((i,p), (j,s) \in \widehat{I}_{\mathbb{Z}}\). We then set \[\beta_{i,p} := \mathop{\mathrm{\mathbf{dim}}}M_{i,p} \in \Delta\] for every \((i,p) \in \widehat{I}_{\mathbb{Z}}\). Using standard notations, we will denote by \(i \mapsto i^*\) the involution of \(I\) given by \[\label{eq32:32def32of32istar} \alpha_{i^{*}} := - w_0 (\alpha_i).\tag{5}\]

For each \(i \in I\) we denote by \(d_i\) the following positive number \[\label{eq32:32def32of32di} d_i := \sum_{(j,s) \in \widehat{I}_{\mathbb{Z}}} \dim \mathop{\mathrm{Hom}}_{\mathcal{D}_Q} (M_{i, \xi(i)} , M_{j,s}) .\tag{6}\] Note that this is well-defined as the right hand side contains only finitely many non zero terms in view of 2 . For later purposes, it will be convenient to extend this to all non-negative integers by setting \(d_i := 1\) if \(i=0\) or \(i>n\). Note that for each \(i \in I\) we have that \[\forall p \in i + 2 \mathbb{Z}, \enspace \sum_{(j,s) \in \widehat{I}_{\mathbb{Z}}} \dim \mathop{\mathrm{Hom}}_{\mathcal{D}_Q} (M_{i,p} , M_{j,s}) = d_i\] because \(\dim \mathop{\mathrm{Hom}}_{\mathcal{D}_Q}(M_{i,p},M_{j,s}) = \dim \mathop{\mathrm{Hom}}_{\mathcal{D}_Q}(M_{i,\xi(i)},M_{j,s-p+\xi(i)})\) for each \((j,s) \in \widehat{I}_{\mathbb{Z}}\).

We then have the following:

2.2 Dual Auslander–Reiten theory↩︎

Fix as above an orientation \(Q\) of the Dynkin diagram of \(\mathfrak{g}\) and choose a height function \(\xi\) adapted to \(Q\). We denote by \(Q^{*}\) the Dynkin quiver adapted to the height function \(\xi^{*}\) given by \[\label{eq32:32def32of32xistar} \forall i \in I , \qquad \xi^{*}(i) := - \xi(i^{*})\tag{7}\] where \(i^{*}\) is defined by 5 . For every \(i \in I\), we will denote by \(I_i^{*}\) (resp. \(P_i^{*}\)) the indecomposable injective (resp. projective) representation of \(Q^{*}\) associated to \(i\). We will also denote by \(\beta_{i,p}^{*}\) the corresponding root associated to the vertex \((i,p)\) for every \((i,p) \in \widehat{I}_{\mathbb{Z}}\), namely \[\beta_{i,p}^{*} := \tau_{Q^{*}}^{(\xi^{*}(i)-p)/2} \left( \mathop{\mathrm{\mathbf{dim}}}I_i^{*} \right).\]

2.3 Remark on dual orientations↩︎

We end this section by proving Lemma 6 below. Although we will never use it in the rest of the paper, it sheds light on the motivation for introducing the orientation \(Q^{*}\) and therefore we provide a detailed proof for the reader’s convenience. First recall the following classical fact.

We can now state and prove the desired statement.

3 Reminders on HL categories and the map \({\widetilde{D}}\)↩︎

3.1 Reminders on quantum affine algebras and HL categories↩︎

Let \(\widehat{I}_{\mathbb{Z}}\) denote the set of pairs \[\widehat{I}_{\mathbb{Z}}:= \{ (i,p) , i \in I , p \in i + 2 \mathbb{Z} \}\] and let \(\mathcal{Y}_{\mathbb{Z}}\) denote the torus \[\mathcal{Y}_{\mathbb{Z}}:= \mathbb{Z}[Y_{i,p}^{\pm 1} , (i,p) \in \widehat{I}_{\mathbb{Z}}].\]

Following [16], we denote by \(\mathcal{C}_{\mathbb{Z}}\) the full subcategory of \(\mathcal{C}\) generated by the fundamental representations \(L(Y_{i,p})\) for \(p \in i + 2 \mathbb{Z}\). The simple objects of \(\mathcal{C}_{\mathbb{Z}}\) are the modules \(L(\mathfrak{m})\) where \(\mathfrak{m}\) is a monomial in the \(Y_{i,p}\). We note that the original construction of such monoidal subcategories for a source–sink orientation was developed in [11] and later generalized to arbitrary height functions in [16] and [15], [17].

Frenkel–Reshetikhin’s \(q\)-character morphism [20] yields an injective ring homomorphism \[\chi_q : \mathcal{C}_{\mathbb{Z}}\longrightarrow \mathcal{Y}_{\mathbb{Z}}.\] Using standard notation, we set \[A_{i,p+1} := Y_{i,p}Y_{i,p+2} \prod_{j \sim i} Y_{j,p+1}^{-1}\] for each \((i,p) \in \widehat{I}_{\mathbb{Z}}\). It is then a known fact that for a simple object \(M = L(\mathfrak{m})\) in \(\mathcal{C}_{\mathbb{Z}}\), the renormalized \(q\)-character of \(M\) defined as \(\mathfrak{m}^{-1} \chi_q(M)\) is a polynomial in the \(A_{i,p+1}^{-1} , (i,p) \in \widehat{I}_{\mathbb{Z}}\) with constant term \(1\), see [20] and [33].

As in the previous section, we fix an orientation \(Q\) of the Dynkin diagram of \(\mathfrak{g}\) and \(\xi\) a height function adapted to \(Q\). Then we set \[\widehat{I}_{\mathbb{Z}}^{ \leq \xi}:= \{ (i,p) \in \widehat{I}_{\mathbb{Z}}, p \leq \xi(i) \}\] and we denote by \(\mathcal{C}_{\mathbb{Z}}^{\leq \xi}\) the smallest full monoidal subcategory of \(\mathcal{C}_{\mathbb{Z}}\) containing all the fundamental representations \(L(Y_{i,p}) , (i,p) \in \widehat{I}_{\mathbb{Z}}^{ \leq \xi}\). Given a reduced expression \(\mathbf{i}_Q\) of \(w_0\) adapted to \(Q\), one can define a semi-infinite word denoted \(\widehat{\mathbf{i_Q}} := (i_1,i_2, \ldots , )\) by setting \(i_{k+mN} := i_k^{*m}\) i.e. the vertex obtained by applying \(m\) times the involution \(*\) to \(i_k\), for every \(1 \leq m \leq N\) and every \(m \geq 0\). Using standard notations, we set \[k_+ := \min\{l>k \mid i_l = i_k\} \qquad k_- := \max (\{l<k \mid i_l=i_k\} \sqcup \{0\}) .\] We also have a bijection \[\varphi : \widehat{I}_{\mathbb{Z}}^{ \leq \xi}\longrightarrow \mathbb{Z}_{\geq 1}\] such that \(\varphi(i,p-2) = \varphi(i,p)_+\). We also denote by \(\mathcal{C}_Q\) the smallest full monoidal subcategory of \(\mathcal{C}_{\mathbb{Z}}^{\leq \xi}\) containing all the fundamental representations \(L(Y_{i,p}) , (i,p) \in \widehat{I}_Q\) where \[\widehat{I}_Q := \{ (i,p) \in \widehat{I}_{\mathbb{Z}}, \xi(i^{*})-h \leq p \leq \xi(i) \} .\] The assignment \[\label{eq32:32grading} Y_{i,p} \in \mathcal{Y}_{\mathbb{Z}}\longmapsto \beta_{i,p} \in \Delta\tag{8}\] endows \(\mathcal{Y}_{\mathbb{Z}}\) with a grading by the root lattice, such that \(A_{j,s+1}\) has a degree \(0\) for each \((j,s) \in \widehat{I}_{\mathbb{Z}}\) by the mesh relation following from the fact that there is a distinguished triangle \[\tau_Q M_{i,p} \longrightarrow \bigoplus_{j \sim i}M_{j,p+1} \longrightarrow M_{i,p} \longrightarrow \tau_Q M_{i,p}[1]\] in \(\mathcal{D}_Q\). In particular, the \(q\)-character of any simple module in \(\mathcal{C}_{\mathbb{Z}}\) is homogeneous with respect to the grading 8 . Moreover,  8 induces a bijection between \(\{ Y_{i,p} , (i,p) \in \widehat{I}_Q \}\) and \(\Delta_+\). Finally, following [27], we define for each \(M \in \mathcal{C}_{\mathbb{Z}}^{\leq \xi}\) the truncated \(q\)-character \({\widetilde{\chi}}_q(M)\) of \(M\) as the element of \(\mathcal{Y}_{\mathbb{Z}}\) obtained from \(\chi_q(M)\) by dropping all the terms involving variables \(Y_{i,p}\) with \(p> \xi(i)\). This gives another injective ring homomorphism \[{\widetilde{\chi}}_q : \mathcal{C}_{\mathbb{Z}}^{\leq \xi}\longrightarrow \mathcal{Y}_{\mathbb{Z}}^{\leq \xi}:=\mathbb{Z}[Y_{i,p}^{\pm 1} , (i,p) \in \widehat{I}_{\mathbb{Z}}^{ \leq \xi}] .\] Truncated \(q\)-character map is injective because \(q\)-character map is injective and the \(q\)-character of a module is determined by its highest \(l\)-weight and also by its truncated \(q\)-character [20], [33].

3.2 Kirillov–Reshetikhin modules↩︎

A distinguished family of simple objects in \(\mathcal{C}_{\mathbb{Z}}\) will play a central role in this work: the Kirillov–Reshetikhin modules (KR modules). They are the simple modules of the form \[X_{i,p}^{(k)} := L \left( Y_{i,p} \cdots Y_{i,p+2k-2} \right) \quad (i,p) \in \widehat{I}_{\mathbb{Z}}, k \geq 1 .\] The \(q\)-characters of the KR modules satisfy the following remarkable identities called T-systems: \[\chi_q(X_{i,p}^{(k)}) \chi_q(X_{i,p+2}^{(k)}) = \chi_q(X_{i,p}^{(k+1)}) \chi_q(X_{i,p+2}^{(k)}) + \prod_{j \sim i} \chi_q(X_{j,p+1}^{(k)}) .\] Once a height function \(\xi\) has been fixed as above, it was shown by Hernandez and Leclerc [27] that the Grothendieck ring of \(\mathcal{C}_{\mathbb{Z}}^{\leq \xi}\) carries a cluster algebra structure, and that the classes of the KR modules are cluster variables in this cluster algebra. The T-systems are interpreted as exchange relations associated to a well-chosen mutation sequence, starting with an initial cluster given by the KR module of the form \[\label{eq32:32initial32KRs} X_{i,p} := X_{i,p}^{( (\xi(i)-p+2)/2 )}, \quad (i,p) \in \widehat{I}_{\mathbb{Z}}^{ \leq \xi}.\tag{9}\]

3.3 The map \(\widetilde{D}_{\xi}\)↩︎

This subsection is devoted to necessary background involving the constructions and results from our previous work [25]. Firstly, recall the coefficients of the inverse quantum Cartan matrix of \(\mathfrak{g}\) (assumed to be of simply-laced type). The quantum Cartan matrix \(C(z)\) of \(\mathfrak{g}\) is defined as \[C_{i,j}(z) := \begin{cases} z+z^{-1}, & \text{if i=j,} \\ -1, & \text{if i \sim j,} \\ 0, & \text{otherwise.} \end{cases}\] Here \(z\) is an indeterminate and the notation \(i \sim j\) means that there is an edge relating \(i\) and \(j\) in the Dynkin graph of \(\mathfrak{g}\). We denote by \(\widetilde{C}_{i,j}(z)\) the entries of the inverse of \(C(z)\). For example, when \(\mathfrak{g}\) is of type \(D_4\), the series \(\widetilde{C}_{i,j}(z)\) are given by \[\begin{align} \widetilde{C}_{1,1}(z) &= z + z^5 - z^7 - z^{11} + z^{13} + z^{17} - z^{19} + O(z^{21}) \\ \widetilde{C}_{1,2}(z) &= z^2 + z^4 - z^8 - z^{10} + z^{14} + z^{16} - z^{20} + O(z^{22}) \\ \widetilde{C}_{1,3}(z) &= z^3 - z^9 + z^{15} - z^{21} + O(z^{23}) \\ \widetilde{C}_{1,4}(z) &= z^3 - z^9 + z^{15} - z^{21} + O(z^{23}) \\ \widetilde{C}_{2,2}(z) &= z + 2 z^3 + z^5 - z^7 - 2 z^9 - z^{11} + z^{13} + 2 z^{15} + z^{17} - z^{19} + O(z^{21}) \\ \widetilde{C}_{2,3}(z) &= z^2 + z^4 - z^8 - z^{10} + z^{14} + z^{16} - z^{20} + O(z^{22}) \\ \widetilde{C}_{2,4}(z) &= z^2 + z^4 - z^8 - z^{10} + z^{14} + z^{16} - z^{20} + O(z^{22}) \\ \widetilde{C}_{3,3}(z) &= z + z^5 - z^7 - z^{11} + z^{13} + z^{17} - z^{19} + O(z^{21}) \\ \widetilde{C}_{3,4}(z) &= z^3 - z^9 + z^{15} - z^{21} + O(z^{23}) \\ \widetilde{C}_{4,4}(z) &= z + z^5 - z^7 - z^{11} + z^{13} + z^{17} - z^{19} + O(z^{21}). \end{align}\]

We then define the integers \(\widetilde{C}_{i,j}(m)\) for every \(i,j \in I\) and \(m \geq 0\) by \[\widetilde{C}_{i,j}(z) = \sum_{m \geq 1} \widetilde{C}_{i,j}(m)z^m .\] The definition is extended to all \(m \in \mathbb{Z}\) by setting \(\widetilde{C}_{i,j}(m) := 0\) for all \(m \leq 0\). The following result is due to Hernandez–Leclerc [16].

In our previous work [25], we defined a morphism of \(\mathbb{C}\)-algebras \(\widetilde{D}_{\xi}: \mathbb{C} \otimes \mathcal{Y}_{\mathbb{Z}}^{\leq \xi}\longrightarrow \mathbb{C}(\mathfrak{t}^{*})\) by setting \[\forall (i,p) \in \widehat{I}_{\mathbb{Z}}^{ \leq \xi}, \enspace \widetilde{D}_{\xi}(Y_{i,p}) := \prod_{ \substack{j \in I \\ s \in \xi(j)- 2 \mathbb{Z}_{\geq 0}} } \beta_{j,s}^{\tilde{C}_{i,j}(s-p-1) - \tilde{C}_{i,j}(s-p+1)} .\]

The following properties can be stated and proved as in [25] without any modification.

The only results from [25] that get slightly modified when considering the present definition of \(\widetilde{D}_{\xi}\) are Proposition 11.1 and Corollary 11.2. Hence we reproduce detailed proofs below for the reader’s convenience.

From this, we can derive the following consequences:

Corollary 2 has the important consequence that it allows to extend \(\widetilde{D}_{\xi}\) to the whole torus \(\mathcal{Y}_{\mathbb{Z}}\) by setting, for any \((i,p) \in \widehat{I}_{\mathbb{Z}}\): \[Y_{i,p} \longmapsto \widetilde{D}_{\xi}(Y_{i,p-2mh}) \quad \text{where m \geq 0 is such that p-2mh \leq \xi(i) .}\] Note that Lemma 7 can now be extended to the whole torus \(\mathcal{Y}_{\mathbb{Z}}\).

3.4 Composition with truncated \(q\)-characters↩︎

Recall that \({\widetilde{\chi}}_q\) denotes the truncated \(q\)-character morphism associated to the height function \(\xi\) previously fixed. Recall also the category \(\mathcal{C}_Q\) from Section 3.1. Hernandez–Leclerc [16] constructed an isomorphism of algebras \[\iota_Q : \mathbb{C} \otimes K_0(\mathcal{C}_Q) \xrightarrow{\simeq} \mathbb{C}[\mathbf{N}]\] inducing a bijective correspondence between the classes of simple modules in \(\mathcal{C}_Q\) and the elements of the dual canonical basis of \(\mathbb{C}[\mathbf{N}]\). In view of Remark 3, we can recall the main result we obtained in [25] using the same notation \(\widetilde{D}_{\xi}\).

The following lemmas will also be useful later. Recall the notation \(X_{i,p}\) from 9 .

4 Duality between Kirillov–Reshetikhin modules↩︎

This section is devoted to the key result allowing to prove the main result of this paper. It consists in exhibiting a certain duality between the values taken on certain Kirillov–Reshetikhin modules by the composition \(\widetilde{D}_{\xi}\circ {\widetilde{\chi}}_q\) when \(\xi\) (as well as the corresponding truncation \({\widetilde{\chi}}_q\)) is respectively adapted to a Dynkin quiver \(Q\) and its dual \(Q^{*}\).

In what follows, we will sometimes write \(\overline{D}(M)\) instead of \(\overline{D}([M])\).

4.1 Preliminaries and statement of the main result↩︎

Recall that the truncation of the \(q\)-character heavily depends on the choice of height function. As there will be two height functions (\(\xi\) and \(\xi^{*}\)) involved in the following sections, we will use the notation \({\widetilde{\chi}}_q^{*}\) to refer to the truncation of the \(q\)-character morphism given by the height function \(\xi^{*}\), in contrast with \({\widetilde{\chi}}_q\) which is the truncation given by \(\xi\). Assume \(Q = (Q_{0}^{op})^{*}\) where \(Q_0\) is the monotonic orientation defined as in Section 7 in [25] and let us fix a height function \(\xi\) adapted to \(Q\). Let \(\overline{D}, \overline{D}^{*}: K_0(\mathcal{C}_{\mathbb{Z}}) \longrightarrow \mathbb{C}(\mathfrak{t}^{*})\) denote the maps defined as follows: \[\overline{D}(L(Y_{i,p})) := \begin{cases} {\widetilde{D}}_{\xi}( {\widetilde{\chi}}_q(L(Y_{i,p}))), & \text{if p \leq \xi(i),} \\ 0, & \text{otherwise,} \end{cases}\] \[\overline{D}^{*} (L(Y_{j,s})) := \begin{cases} \left( \psi \circ {\widetilde{D}}_{\xi^{*}} \right)( {\widetilde{\chi}}_q^{*}(L(Y_{j,s}))), & \text{if s \leq \xi^{*}(j),} \\ 0, & \text{otherwise.} \end{cases}\] Here \(\psi\) denotes the automorphism of \(\mathbb{C}(\mathfrak{t}^{*})\) given by \[\forall i \in I, \qquad \psi(\alpha_i) := \alpha_{i^{*}} .\] The reason for this choice of notation is given by Theorem 5: we can view \(\widetilde{D}_{\xi}\circ {\widetilde{\chi}}_q\) as an extension to a larger domain of Baumann–Kamnitzer–Knutson’s morphism \(\overline{D}\). Hence by convenience, we still denote by \(\overline{D}\) the map defined above on the Grothendieck ring of \(\mathcal{C}_{\mathbb{Z}}\). Recall the notation \(d_i\) from (6 ). Finally, set \([X_{i,p}^{(0)}] := 1\) for any \((i,p) \in \widehat{I}_{\mathbb{Z}}\). The main result established in this section is the following:

Given a simple Lie algebra \(\mathfrak{g}\) of type \(A_n , n\geq 1\) or \(D_n, n \geq 4\) and denoting by \(I\) the set of vertices of the corresponding Dynkin diagram, we will say that \(i \in I\) is spin if \(\mathfrak{g}\) is of type \(D_n\) and \(i \in \{n-1,n\}\) and that \(i\) is generic (resp. strictly generic) if \(\mathfrak{g}\) is of type \(A_n\) or if \(\mathfrak{g}\) is of type \(D_n\) and \(i \leq n-2\) (resp. \(i<n-2\)). Furthermore, in all what follows we will use the following notations. Let \(\eta : \mathbb{Z} \longrightarrow \mathbb{Z}\) denote the map \[\eta : s \longmapsto -s-2h+4 .\] Note that \(\eta\) induces a bijection \[\eta : \{ s , \xi(i^{*})-h \geq s > \xi(i)-2h+2 \} \longrightarrow \{ s' , \xi^{*}(i^{*}) \geq s' > \xi^{*}(i)-h+2 \} .\] We also define a map \(\rho : \{ 0, \ldots , h\} \longrightarrow \{ 0, \ldots , h \}\) by setting \[\rho : k \longmapsto h-k .\] We denote by \(\overline{\mathcal{C}_Q}\) the smallest full monoidal subcategory of \(\mathcal{C}_{\mathbb{Z}}\) containing the fundamental representations \(L(Y_{i,p}) , i \in I , \xi(i) \geq p \geq \xi(i)-2h+2\). Note that \(\overline{\mathcal{C}_Q}\) contains \(\mathcal{C}_Q\) as a full monoidal subcategory. We moreover introduce an algebra \(\mathcal{A}_{\xi}\) which should be regarded as a dual of \(K_0(\overline{\mathcal{C}_Q})\), in the sense that it is generated by classes of KR modules obtained from those of \(\overline{\mathcal{C}_Q}\) by performing the substitutions \((i,p,k) \mapsto (i^{*}, \eta(p), \rho(k))\). More precisely, we define \(\mathcal{A}_{\xi}\) as the sub-algebra of the fraction field of \(K_0(\mathcal{C}_{{\mathbb{Z}}})\) generated by the classes of \[\Sigma = \Sigma_1 \sqcup \Sigma_2\] where \[\Sigma_1 := \{ [X_{j,s}^{(k)}]^{\pm 1} ,\xi^{*}(j) \geq s > \xi^{*}(j)-2h+2 , k = (\xi^{*}(j)-s+2)/2 \}\] and \[\Sigma_2 := \{ [X_{j,s}^{(h)}]^{\pm 1} ,\xi^{*}(j) \geq s > \xi^{*}(j)-2h+2 \}.\]

The strategy of the proof is the following. First, we show that 12 holds for the KR modules belonging to the initial seed, i.e.in the case \(k \geq 1\) and \(p = \xi(i)-2k+2\). This is what is done in Section 4.2. Then, still considering \(p= \xi(i)-2k+2\), we show that the desired equality holds when \(k=0\). This amounts to prove that \(\overline{D}^{*}\left( X_{i^{*}, \xi^{*}(i^{*}) -2h + 4}^{(h)} \right) = \pm 1\), which will be the purpose of Section 4.3. Finally, we will conclude using the T-systems in \(K_0(\overline{\mathcal{C}_Q})\) (resp. \(\mathcal{A}_{\xi}\)).

4.2 Classes of KR modules in \(\Sigma_1\)↩︎

In this subsection, we establish the following special case of Theorem 7 which deals with classes of the subset \(\Sigma_1\). Note that it is valid for any orientation \(Q\) of the Dynkin diagram of \(\mathfrak{g}\).

4.3 Classes of KR modules in \(\Sigma_2\)↩︎

In this subsection we investigate the values of \(\overline{D}^{*}\) on the classes in the subset \(\Sigma_2\), assuming from now on that \(Q\) is monotonic. We establish the following:

For each \(i \in I\) and each \(j \ge 1\) we denote \[\beta_i^{(j)} := \tau^{j-1}(\gamma_i) \qquad \text{where \gamma_i := \boldsymbol{\dim} I_i.}\] We work in the category \(\mathcal{C}_{\xi^{*}}\) where \(\xi^{*}\) is a height function adapted to \(Q^{*} := Q_0^{op}\). For each \(i \in I\) and \(k \geq 1\) we set \[P_i^{(k)} := \overline{D}^{*} \left( X_{i, \xi^{*}(i)-2k+2}^{(k)} \right) .\] Thus the \(P_i^{(k)}\) are the images under \(\overline{D}^{*}\) of the classes of the KR modules belonging to the initial seed in the category \(\mathcal{C}_{\xi^{*}}\). The strategy now consists in using the T-systems relating classes of KR modules, together with the mesh relations after applying \(\overline{D}^{*}\). In order to do so, we need to introduce the following elements of the root lattice. We set \[\forall i \in I , \forall j \geq 0, \gamma_{i,j} := \begin{cases} \beta_{\sigma^{n-k}(i)}^{(n-1)}, & \text{if j \equiv n-1 \mod h and i is spin,} \\ \sum_{l=1}^j \beta_1^{(l)}, & \text{otherwise.} \end{cases}\] We also set \(\gamma_{i,j} := 0\) if \(j \leq 0\), for every \(i \in I\). Then we set \[\forall i \leq n-1, \forall k \geq 1, \enspace R_i^{(k)} := (-1)^{i} \prod_{j=k-i}^{k-1} \gamma_{i,j} .\] Note that \(R_i^{(k)} = 0\) if \(k \leq i\). We begin by establishing the following:

We begin with the following identity, that follows from the mesh relations.

Now we set \(Q_i^{(k)} := R_i^{(k)} P_i^{(k-1)}\) for every \(i \in I\) and \(k \geq 1\), and we show that the rational functions \(Q_i^{(k)}\) and \(P_i^{(k)}\) are related by identities corresponding to T-systems.

From this, we can now derive the two following important consequences.

4.4 Duality and T-systems↩︎

Recall from Section 4.1 the algebra \(\mathcal{A}_{\xi}\) as well as the maps \(\overline{D}, \overline{D}^{*}\). Let \(x_1, \ldots , x_n\) be formal variables. We define a map \[\mathbf{D} : \mathcal{A}_{\xi} \longrightarrow \mathbb{C}(\alpha_1, \ldots , \alpha_n)(x_1, \ldots , x_n)\] as follows. For the classes in \(\Sigma_1\) we set \[\mathbf{D}(X_{j,s}^{(k)}) := \overline{D}^{*}\left( X_{j , s}^{(k)} \right), \qquad k = (\xi^{*}(j)-s+2)/2\] In order to define the images of \(\mathbf{D}\) on the classes of \(\Sigma_2\), we need the following piece of notation. For any \(j \in I\) and \(m \geq 0\), we set \[z_{j,m} := \begin{cases} \frac{x_m}{x_{j+m}} & \text{if \mathfrak{g} is of type A_n,} \\ \frac{x_{m+j-n+1}x_m}{x_{n-1}x_n} & \text{if \mathfrak{g} is of type D_n and j \leq n-2,} \\ \frac{x_m}{x_{\sigma^m(j)}} & \text{if \mathfrak{g} is of type D_n and j \in \{n-1,n\},} \end{cases}\] where \(x_{n+m}=x_{m-1}\) for \(m \ge 1\) and \(x_0=1\). For \(j \in I\), \(s, l \in {\mathbb{Z}}\), we define \[\begin{align} \label{eq:definition32of32zeta32jsh} \zeta(j,s,l) := \frac{s-\xi^{*}(j)+2l-2}{2}. \end{align}\tag{13}\]

Then we define the images of \(\mathbf{D}\) on the classes of \(\Sigma_2\) as follows: \[\begin{align} \label{eq:definition32of32D32of32Xjsh} \mathbf{D}(X_{j,s}^{(h)}) := z_{j,\zeta(j,s,h)}. \end{align}\tag{14}\] We begin with a preliminary lemma.

We now introduce the following renormalizations: for every \((j,s)\) such that \(\xi^{*}(j) \geq s > \xi^{*}(j)-2h+2\) and \(1 \leq l < h\) we set \[\begin{align} \label{eq:def32of32Dprimejs} {D'}_{j,s}^{(l)} := z_{j, \zeta(j,s,l)}^{-1} \mathbf{D} \left( X_{j,s}^{(l)} \right). \end{align}\tag{16}\] The next Proposition shows that the \({D'}_{j,s}^{(l)}\) satisfy identities dual to the T-systems satisfied by the KR modules \(X_{i,p}^{(k)} , 1 \leq k <h, \xi(i) \geq p > \xi(i)-2h+2\) in \(K_0(\overline{\mathcal{C}_Q})\).

5 Proof of the main result↩︎

In this section, we prove the main result Theorem 1.

5.1 Examples↩︎

Before presenting the general proof, we provide several examples illustrating that \(\widetilde{D}_{\xi}\left( \chi_q(M) \right) = 0\) is highly non-trivial. In type \(A_3\), take an orientation \(1 \to 2 \to 3\) and a height function \(\xi(1)=3\), \(\xi(2)=2\), \(\xi(3)=1\). Then \(\tau=s_1s_2s_3\) and \[\begin{align} & {\widetilde{D}}_{\xi}(Y_{1,-1}) = \frac{\alpha_2 (\alpha_1 + \alpha_2)}{\alpha_3 (\alpha_2 + \alpha_3) (\alpha_1 + \alpha_2 + \alpha_3)}, \quad {\widetilde{D}}_{\xi}(Y_{1,-3}) = - \alpha_3 (\alpha_2 + \alpha_3) (\alpha_1 + \alpha_2 + \alpha_3), \\ & {\widetilde{D}}_{\xi}(Y_{2,0}) = \frac{\alpha_1}{\alpha_2 (\alpha_2 + \alpha_3) (\alpha_1 + \alpha_2 + \alpha_3)}, \quad {\widetilde{D}}_{\xi}(Y_{2,-2}) = \frac{\alpha_2 (\alpha_1 + \alpha_2) (\alpha_1 + \alpha_2 + \alpha_3)}{\alpha_3}, \\ & {\widetilde{D}}_{\xi}(Y_{2,-4}) = \alpha_3 (\alpha_2 + \alpha_3), \quad {\widetilde{D}}_{\xi}(Y_{3,-1}) = - \frac{\alpha_1 (\alpha_1 + \alpha_2) (\alpha_1 + \alpha_2 + \alpha_3)}{\alpha_2 (\alpha_2 + \alpha_3)}, \\ & {\widetilde{D}}_{\xi}(Y_{3,-3}) = - \frac{\alpha_2 (\alpha_2 + \alpha_3)}{\alpha_3}. \end{align}\] The \(q\)-character of \(L(Y_{2,-4})\) is \[\begin{align} \chi_q(L(Y_{2,-4})) = Y_{2,-4} + \frac{Y_{1,-3} Y_{3,-3}}{Y_{2,-2}} + \frac{Y_{3,-3}}{Y_{1,-1}} + \frac{Y_{1,-3}}{Y_{3,-1}} + \frac{Y_{2,-2}}{Y_{1,-1} Y_{3,-1}} + \frac{1}{Y_{2,0}}. \end{align}\] We check directly that \({\widetilde{D}}_{\xi}(\chi_q(L(Y_{2,-4})))=0\). In type \(D_4\), take an orientation and a height function \(\xi(1)=-1\), \(\xi(2)=-2\), \(\xi(3)=\xi(4)=-3\). Then \(\tau=s_1s_2s_3s_4\) and \[\begin{align} {\widetilde{D}}_{\xi}(Y_{1,-3}) &= \frac{\alpha_1}{\alpha_2 (\alpha_1 + \alpha_2)}, \\ {\widetilde{D}}_{\xi}(Y_{1,-5}) &= \frac{\alpha_2}{\alpha_1 (\alpha_1 + \alpha_2 + \alpha_4) (\alpha_1 + \alpha_2 + \alpha_3) (\alpha_1 + \alpha_2 + \alpha_3 + \alpha_4) (\alpha_1 + 2\alpha_2 + \alpha_3 + \alpha_4)}, \\ {\widetilde{D}}_{\xi}(Y_{1,-7}) &= -\,\frac{ \alpha_1 (\alpha_1 + \alpha_2) (\alpha_1 + \alpha_2 + \alpha_4) (\alpha_1 + \alpha_2 + \alpha_3) (\alpha_1 + \alpha_2 + \alpha_3 + \alpha_4) }{ \alpha_2 (\alpha_2 + \alpha_4) (\alpha_2 + \alpha_3) (\alpha_2 + \alpha_3 + \alpha_4) }, \end{align}\] \[\begin{align} {\widetilde{D}}_{\xi}(Y_{2,-2}) &= \frac{1}{\alpha_1 (\alpha_1 + \alpha_2)}, \\ {\widetilde{D}}_{\xi}(Y_{2,-4}) &= \frac{1}{ \alpha_2 (\alpha_1 + \alpha_2) (\alpha_1 + \alpha_2 + \alpha_4) (\alpha_1 + \alpha_2 + \alpha_3) (\alpha_1 + 2\alpha_2 + \alpha_3 + \alpha_4) }, \\ {\widetilde{D}}_{\xi}(Y_{2,-6}) &= \frac{ \alpha_1 (\alpha_1 + \alpha_2) }{ (\alpha_2 + \alpha_4) (\alpha_2 + \alpha_3) (\alpha_2 + \alpha_3 + \alpha_4) (\alpha_1 + \alpha_2 + \alpha_3 + \alpha_4) (\alpha_1 + 2\alpha_2 + \alpha_3 + \alpha_4) }, \\ {\widetilde{D}}_{\xi}(Y_{2,-8}) &= \frac{ \alpha_2 (\alpha_1 + \alpha_2) (\alpha_1 + \alpha_2 + \alpha_4) (\alpha_1 + \alpha_2 + \alpha_3) (\alpha_1 + 2\alpha_2 + \alpha_3 + \alpha_4) }{ \alpha_4 \alpha_3 (\alpha_2 + \alpha_3 + \alpha_4) }, \end{align}\] \[\begin{align} {\widetilde{D}}_{\xi}(Y_{3,-3}) &= \frac{1}{ \alpha_1 (\alpha_1 + \alpha_2) (\alpha_1 + \alpha_2 + \alpha_3) }, \\ {\widetilde{D}}_{\xi}(Y_{3,-5}) &= \frac{ \alpha_1 (\alpha_1 + \alpha_2 + \alpha_3) }{ \alpha_2 (\alpha_2 + \alpha_4) (\alpha_1 + \alpha_2 + \alpha_4) (\alpha_1 + 2\alpha_2 + \alpha_3 + \alpha_4) }, \\ {\widetilde{D}}_{\xi}(Y_{3,-7}) &= \frac{ \alpha_2 (\alpha_2 + \alpha_4) (\alpha_1 + \alpha_2) (\alpha_1 + \alpha_2 + \alpha_4) }{ \alpha_3 (\alpha_2 + \alpha_3) (\alpha_2 + \alpha_3 + \alpha_4) (\alpha_1 + \alpha_2 + \alpha_3) (\alpha_1 + \alpha_2 + \alpha_3 + \alpha_4) }, \end{align}\] \[\begin{align} {\widetilde{D}}_{\xi}(Y_{4,-3}) &= \frac{1}{ \alpha_1 (\alpha_1 + \alpha_2) (\alpha_1 + \alpha_2 + \alpha_4) }, \\ {\widetilde{D}}_{\xi}(Y_{4,-5}) &= \frac{ \alpha_1 (\alpha_1 + \alpha_2 + \alpha_4) }{ \alpha_2 (\alpha_2 + \alpha_3) (\alpha_1 + \alpha_2 + \alpha_3) (\alpha_1 + 2\alpha_2 + \alpha_3 + \alpha_4) }, \\ {\widetilde{D}}_{\xi}(Y_{4,-7}) &= \frac{ \alpha_2 (\alpha_2 + \alpha_3) (\alpha_1 + \alpha_2) (\alpha_1 + \alpha_2 + \alpha_3) }{ \alpha_4 (\alpha_2 + \alpha_4) (\alpha_2 + \alpha_3 + \alpha_4) (\alpha_1 + \alpha_2 + \alpha_4) (\alpha_1 + \alpha_2 + \alpha_3 + \alpha_4) }. \end{align}\] The \(q\)-character of \(L(Y_{2,-8})\) is \[\begin{align} & \chi_q(L(Y_{2,-8})) = Y_{2,-8} + \frac{Y_{1,-7} Y_{3,-7} Y_{4,-7}}{Y_{2,-6}} + \frac{Y_{3,-7} Y_{4,-7}}{Y_{1,-5}} + \frac{Y_{1,-7} Y_{4,-7}}{Y_{3,-5}} + \frac{Y_{1,-7} Y_{3,-7}}{Y_{4,-5}} \\ & + \frac{Y_{2,-6} Y_{4,-7}}{Y_{1,-5} Y_{3,-5}} + \frac{Y_{2,-6} Y_{3,-7}}{Y_{1,-5} Y_{4,-5}} + \frac{Y_{1,-7} Y_{2,-6}}{Y_{3,-5} Y_{4,-5}} + \frac{Y_{4,-7} Y_{4,-5}}{Y_{2,-4}} + \frac{Y_{2,-6}^2}{Y_{1,-5} Y_{3,-5} Y_{4,-5}} + \frac{Y_{3,-7} Y_{3,-5}}{Y_{2,-4}} \\ & + \frac{Y_{1,-7} Y_{1,-5}}{Y_{2,-4}}+ \frac{Y_{4,-7}}{Y_{4,-3}} + \frac{Y_{2,-6}}{Y_{4,-5} Y_{4,-3}} + \frac{Y_{3,-7}}{Y_{3,-3}} + \frac{Y_{2,-6}}{Y_{3,-5} Y_{3,-3}} + 2 \frac{Y_{2,-6}}{Y_{2,-4}} + \frac{Y_{1,-5} Y_{3,-5} Y_{4,-5}}{Y_{2,-4}^2} \\ & + \frac{Y_{1,-7}}{Y_{1,-3}} + \frac{Y_{2,-6}}{Y_{1,-5} Y_{1,-3}} + \frac{Y_{3,-5} Y_{4,-5}}{Y_{1,-3} Y_{2,-4}} + \frac{Y_{1,-5} Y_{4,-5}}{Y_{2,-4} Y_{3,-3}} + \frac{Y_{1,-5} Y_{3,-5}}{Y_{2,-4} Y_{4,-3}} + \frac{Y_{4,-5}}{Y_{1,-3} Y_{3,-3}} \\ & + \frac{Y_{3,-5}}{Y_{1,-3} Y_{4,-3}} + \frac{Y_{1,-5}}{Y_{3,-3} Y_{4,-3}} + \frac{Y_{2,-4}}{Y_{1,-3} Y_{3,-3} Y_{4,-3}} + \frac{1}{Y_{2,-2}}. \end{align}\] We check directly that \({\widetilde{D}}_{\xi}(\chi_q(L(Y_{2,-8})))=0\).

5.2 Proof of the main result in the case \(Q = {Q_0^{op}}^{*}\)↩︎

In this subsection, we prove that Theorem 1 holds when \(Q = Q_0\) is the monotonic orientation considered in Section 4. For simplicity, we write \({\widetilde{D}}\) for \({\widetilde{D}}_{\xi}\) where \(\xi\) is a height function adapted to \(Q\). We begin by showing that \[{\widetilde{D}}\left( \chi_q \left( L(Y_{i, \xi(i)-2h+2}) \right) \right) = 0\] for every \(i \in I\). By Theorem 7, we have that \[\label{eq32:32duality32in32the32proof32of32main32result} \overline{D}\left( L(Y_{i, \xi(i)-2h+2}) \right) = \pm \overline{D}^{*}\left( X_{j,\xi^{*}(j)+2}^{(h-1)} \right)\tag{18}\] where \(j := i^{*}\). The class involved on the right hand side can be decomposed as a linear combination of classes of standard modules in \(\mathcal{C}_{\mathbb{Z}}\), all of which are non-trivial (i.e. not isomorphic to the trivial representation). Indeed, recall from Section 3.1 that \(\chi_q \left( X_{j,\xi^{*}(j)+2}^{(h-1)} \right)\) is homogeneous with respect to the grading 8 . Its degree is thus given by the dominant monomial, which here gives \[\beta_{j,\xi^{*}(j)+2} + \cdots + \beta_{j, \xi^{*}(j)+2h-2} = \left( \sum_{k=0}^{h-2} \tau^k \right) (\beta_{j,\xi^{*}(j)+2})\] which is non zero because the order of \(\tau\) is \(h\). Hence all the standard modules involved in the decomposition of \(X_{j,\xi^{*}(j)+2}^{(h-1)}\) have \(q\)-characters of non-zero degree with respect to 8 and hence are not isomorphic to the trivial representation. Furthermore, each of these standard modules is a tensor product of fundamental representations at least one of which is of the form \(L(Y_{k,r})\) with \(r > \xi^{*}(k)\). In view of the definition of \(\overline{D}^{*}\), it follows that the right hand side of 18 is zero.

On the other hand, the truncated \(q\)-character of \(L(Y_{i, \xi(i)-2h+2})\) coincides with its entire \(q\)-character. Indeed, the lowest \(l\)-weight monomial is \(Y_{i^{*},p}\) where \(p = \xi(i)-h+2 < \xi(i)\) and thus all the terms of the \(q\)-character of \(L(Y_{i, \xi(i)-2h+2})\) belong to \(\mathcal{Y}_{\mathbb{Z}}^{\leq \xi}\) (cf. Section 3.1). Hence the left hand side of 18 is actually \({\widetilde{D}}\left( \chi_q \left( L(Y_{i, \xi(i)-2h+2}) \right) \right)\) which proves the desired result in the case of the fundamental representations \(L(Y_{i, \xi(i)-2h+2}) , i \in I\). We claim this implies that the same statement holds for all fundamental representations in \(\mathcal{C}_{\mathbb{Z}}\). Indeed, recall from Section 3.1 that for every \((i,p) \in \widehat{I}_{\mathbb{Z}}\), \(Y_{i,p}^{-1} \chi_q \left( L(Y_{i,p}) \right)\) is a polynomial (with constant term \(1\)) in the \(A_{j,s}^{-1} , (j,s-1) \in \widehat{I}_{\mathbb{Z}}\) such that, given \(p,p' \in i + 2 \mathbb{Z}\), \(Y_{i,p'}^{-1} \chi_q \left( L(Y_{i,p'}) \right)\) is obtained from \(Y_{i,p}^{-1} \chi_q \left( L(Y_{i,p}) \right)\) by performing the substitution \(A_{j,s} := A_{j,s+p'-p}\) for all \((j,s)\). Consequently, Corollary 3 implies that \[{\widetilde{D}}\left(Y_{i,p}^{-1} \chi_q \left( L(Y_{i,p}) \right) \right) ={\widetilde{D}}\left(Y_{i,p'}^{-1} \chi_q \left( L(Y_{i,p'}) \right) \right) |_{\alpha_k := \tau^{ \frac{p'-p}{2}}(\alpha_k)}.\] As \(\tau\) acts as an automorphism of \(\mathfrak{t}^{*}\), this implies that if \({\widetilde{D}}\left( \chi_q \left( L(Y_{i,p}) \right) \right)=0\) for one particular \(p \in i + 2 \mathbb{Z}\), then the same will hold for all \(p' \in i + 2 \mathbb{Z}\). This concludes the proof.

5.3 Arbitrary orientation↩︎

In this section we prove that if the statement \[\forall j,s \quad \widetilde{D}_{\xi}(\chi_q(L(Y_{j,s})))=0\] holds for a height function \(\xi\) adapted to an orientation \(Q\), then it also holds for any height function adapted to any arbitrary orientation. Recall that for a quiver \(Q\) and a vertex \(i\) of \(Q\), we denote by \(\gamma_i^{Q}\) the dimension vector of the indecomposable injective \(\mathbb{C}Q\)-module at \(i\); furthermore, we let \(\mathbf{s}_iQ\) denote the quiver defined by reversing all the arrows adjacent to \(i\) in \(Q\). Given that any Dynkin quiver can be obtained from a given one by performing a certain sequence of transformations \(\mathbf{s}_i\) where \(i\) is a source of the quiver obtained at each step, it is enough to prove that if the statement holds for an orientation \(Q\), then it holds for all orientations \(\mathbf{s}_iQ\) such that \(i\) is a source of \(Q\). This is achieved by establishing the following:

We also get a simple relationship for the variables \(A_{i,p}\), namely \[{\widetilde{D}}_{\mathbf{s}_i Q}(A_{j,s})(\alpha_1, \ldots , \alpha_n) = {\widetilde{D}}_{Q}(A_{j,s})(s_i \alpha_1, \ldots, s_i \alpha_n) .\] In particular, as the \(q\)-character of any fundamental representation can be written as \(Y_{j,s}\) times a polynomial in the variables \(A_{i,p}^{-1}\), we get that \[\forall j,s \quad {\widetilde{D}}_{\mathbf{s}_i Q}(\chi_q(L(Y_{j,s})))(\alpha_1, \ldots , \alpha_n) = (-\alpha_i)^{-(\beta_{j,s},\alpha_i)} {\widetilde{D}}_{Q}(\chi_q(L(Y_{j,s})))(s_i \alpha_1, \ldots, s_i \alpha_n).\] As the linear transformation \(s_i\) is invertible, we conclude that \[{\widetilde{D}}_{\xi}(\chi_q(L(Y_{j,s})))=0 \quad \Rightarrow \quad {\widetilde{D}}_{\mathbf{s}_i \xi}(\chi_q(L(Y_{j,s})))=0\] for all \(j,s\).

5.4 Exceptional types↩︎

In type \(E_6, E_7, E_8\), we start from the initial seed and assign \({\widetilde{D}}\) to the cluster variables and frozen variables in the seed. We apply mutation sequences which produce Kirillov–Reshetikhin modules (in particular, fundamental modules) to the initial seed. We have checked that \({\widetilde{D}}\) of full \(q\)-characters of fundamental modules are all zero. This implies that for any non-trivial standard module \(M\) in \(\mathcal{C}_{\mathbb{Z}}\) of type \(E_6, E_7, E_8\), we have that \(\widetilde{D}_{\xi}\left( \chi_q(M) \right) = 0\). The SageMath codes for these computations could be found in:

https://github.com/lijr07/D-tilde-of-fundamental-modules-in-type-E.

References↩︎