A \(\mathbb Z\)-basis for the cluster algebra of type \(\tilde D_4\) (Q2859245)

Cluster algebras weres invented by \textit{S. Fomin} and \textit{A. Zelevinsky} [J. Am. Math. Soc. 15, No. 2, 497--529 (2002; Zbl 1021.16017)] in order to study total positivity in algebraic groups and canonical bases in quantum groups. Cluster categories [\textit{A. B. Buan} et al., Adv. Math. 204, No. 2, 572--618 (2006; Zbl 1127.16011)] are certain categories of representations of finite dimensional algebras which were introduced to ``categorify'' cluster algebras. The Caldero--Chapoton map was introduced in [\textit{P. Caldero} and \textit{F. Chapoton}, Comment. Math. Helv. 81, No. 3, 595--616 (2006; Zbl 1119.16013)] to formalize the connection between the cluster algebras and the cluster categories. Indeed, using the Caldero--Chapoton map, \textit{P. Caldero} and \textit{B. Keller} [Ann. Sci. Éc. Norm. Supér. (4) 39, No. 6, 983--1009 (2006; Zbl 1115.18301)] established a bijection between the indecomposable rigid objects of a cluster category and the cluster variables of the corresponding cluster algebra.NEWLINENEWLINELet \(Q\) be an acyclic quiver with vertex set \(Q_0=\{1,2,\dots, n\}\). Let \(\mathbb{C}Q\) be the path algebra of \(Q\) and denote by \(P_i\) the indecomposable projective \(\mathbb{C} Q\)-module with the simple top \(S_i\) corresponding to \(i\in Q_0\) and \(I_i\) the indecomposable injective \(\mathbb{C} Q\)-module with the simple socle \(S_i\). The cluster category associated to \(Q\) is the orbit category \(\mathcal{C}(Q):=\mathcal{D}^{b}(Q)/[1]\circ\tau^{-1}\) where \(\mathcal{D}^b(Q)\) is the bounded derived category of \(\mathrm{mod} \mathbb{C} Q\) with the shift functor \([1]\) and the AR-translation \(\tau\). Let \(\mathbb{Q}(x_1,\dots, x_n)\) be a transcendental extension of the rational number field \(\mathbb{Q}\).NEWLINENEWLINEThe Caldero-Chapton map of an acyclic quiver \(Q\) is \(X_?^Q: \text{obj} (\mathcal{C}(Q))\rightarrow \mathbb{Q}(x_1, \dots, x_n)\) defined by the following rules [Zbl 1119.16013]:NEWLINENEWLINE (1) If \(M\) is an indecomposable \(\mathbb{C}Q\)-module, then NEWLINE\[NEWLINEX_M^Q=\sum_{\underline{e}}\chi(\mathrm{Gr}_{\underline{e}}(M)) \prod_{i_0 \in Q_0}x_i^{-\langle \underline{e}, s_i \rangle -\langle s_i, \underline{\dim}M-\underline{e} \rangle}.NEWLINE\]NEWLINENEWLINENEWLINE(2) If \(M=TP_i\) is the shift of the projective module associated to \(i\in Q_0\), then \(X_M^Q=x_i.\)NEWLINENEWLINE(3) For any two objects \(M, N\) of \(\mathcal{C}_Q\), we have \(X_{M\oplus N}^Q=X_M^QX_N^Q\).NEWLINENEWLINEHere, we denote by \(\langle -,-\rangle\) the Euler form on \(\mathbb{C}Q\)-module and \(\mathrm{Gr}_{\underline{e}}(M)\) is the \(\underline{e}\)-Grassmannian of \(M\). Note that the indecomposable \(\mathbb{C}Q\)-modules and \(TP_i\) for \(i\in Q_0\) exhaust the indecomposable objects of the cluster category \(\mathcal{C}(Q)\). For any object \(M \in \mathcal{C}(Q),\) \(X_M^Q\) will be called the generalized cluster variable for \(M\).NEWLINENEWLINELet \(R=(r_{ij})\) be a matrix with \(r_{ij}=\dim_{\mathbb{C}}\mathrm{Ext}^1(S_i,S_j)\) for any \(i, j\in Q_0\). For \(v=(v_1, \dots, v_n),\) we set \(x_v=x_1^{v_1} \dots x_n^{v_n}\). The Caldero-Chapton map can be reformulated by the following rules \textit{F. Xu} [Trans. Am. Math. Soc. 362, No. 2, 753--776 (2010; Zbl 1200.16021)]:NEWLINENEWLINE(1) \( X_{\tau P}=X_{P[1]}=x^{\underline{\mathrm{dim}}{P/\mathrm{rad} P}}, X_{\tau^{-1}I}=X_{I[-1]}=x^{\underline{\mathrm{dim}} \mathrm{soc}I}\) for any projective \(\mathbb{C} Q\)-module \(P\) and any injective \(\mathbb{C} Q\)-module \(I\);NEWLINENEWLINE(2) NEWLINE\[NEWLINE X_{M}=\sum_{\underline{e}}\chi(\mathrm{Gr}_{\underline{e}}(M))x^{\underline{e} R+(\underline{\mathrm{dim}}M-\underline{e})R^{\mathrm{tr}}- \underline{\mathrm{dim}}M }. NEWLINE\]NEWLINENEWLINENEWLINELet \(\mathcal{AH}(Q)\) be the subalgebra of \(\mathbb{Q}(x_1, \dots, x_n)\) generated by all \(X_M, X_{\tau P}\), where \(M, P \in \text{mod}-\mathbb{C}{Q}\) and \(P\) is projective. Let \(\mathcal{EH} (Q)\) be the subalgebra of \(\mathcal{AH}(Q)\) generated by all \(X_M\), where \(M\in \text{ind}-\mathbb{C}(Q)\) and \(\text{Ext}_{\mathcal{C}}^1(M,M)=0.\)NEWLINENEWLINEIn the paper under review, the authors obtain a \(\mathbb{Z}\)-basis of \(\mathcal{AH}(Q)\) and proved that \(\mathcal{EH}(Q)\) coincides with \(\mathcal{AH}(Q)\) for the quiver \(Q=\widetilde{D}_4.\) Moreover, the authors prove that coefficients of Laurent expansions of the basis elements are non-negative integers.