Quantum cluster variables via Serre polynomials. With an appendix On the integral cluster homology by Bernhard Keller. (Q2904014)

For skew-symmetric acyclic quantum cluster algebras, the authors express the quantum \(F\)-polynomials and the quantum cluster monomials in terms of Serre polynomials of quiver Grassmannians of rigid modules. As byproducts, the authorsobtain the existence of counting polynomials for these varieties and the positivity conjecture with respect to acyclic seeds. These results complete previous work by Caldero and Reineke and confirm a recent conjecture by Rupel.NEWLINENEWLINECluster algebras were 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 provide a combinatorial approach to canonical bases and total positivity. They are commutative subalgebras of Laurent polynomial rings and later, their quantum deformations-quantum cluster algebras were introduced in [\textit{A. Berenstein} and \textit{A. Zelevinsky}, Adv. Math. 195, No. 2, 405--455 (2005; Zbl 1124.20028)].NEWLINENEWLINEIn the present paper, the authors restrict the attention to quantum cluster variables, which are certain generators of quantum cluster algebras defined recursively by mutations. Recall that the commutative cluster variables can be expressed in terms of the Euler characteristics of quiver Grassmannians through the Caldero-Chapoton (CC-) formula [\textit{P. Caldero} and \textit{F. Chapoton}, Comment. Math. Helv. 81, No. 3, 595--616 (2006; Zbl 1119.16013)]. Then naturally one may ask for a refined CC-formula to compute quantum cluster variables. Notice that the coefficients of quantum cluster variables are Laurent polynomials in \(q^{\frac{1}{2} }\), such that (1) they specialize to the Euler characteristics of quiver Grassmannians under the quasi-classical limit \(q^{\frac{1}{2} }\rightarrow 1\), and (2) they are invariant under the involution \(q^{\frac{1}{2}} \mapsto q^{-\frac{1}{2}}\).NEWLINENEWLINEThe authors propose a refined CC-formula for quantum cluster algebra via Serre polynomials as below. Let \(\tilde{Q}\) be an ice quiver, i.e. a quiver with frozen vertices. Assume that the matrix of \(\tilde{Q}\) can be completed into a unitally compatible pair, to which we associate a quantum cluster algebra \(\mathcal{A}^q\). Further suppose that the non-frozen part \(Q\) of \(\tilde{Q}\) is acyclic. Endow \(\tilde{Q}\) with a generic potential and consider the associated presentable cluster category \(\mathcal{D}\) and the generalized cluster category \(\mathcal{C}\).NEWLINENEWLINEDefinition 1.2.1 [refined CC-formula] For any coefficient-free rigid object \(M\) of \(\mathcal{D}\), denote by \(m\) the class of \(\text{Ext}^1_{\mathcal{C}}(T,M)\) in \(\text{K}_0(\text{mod} kQ)\), and associate to \(M\) the following element in the quantum torus \(\mathcal{T}\): NEWLINE\[NEWLINEX_M=\sum_e E(\text{Gr}_e (\text{Ext}^1_{\mathcal{C}}(T,M))) q^{-\frac{1}{2} \langle e,m-e \rangle} X^{\text{ind}_T(M)-\phi(e)}.NEWLINE\]NEWLINENEWLINENEWLINEHere the symbol \(E(\text{Gr}_e (\text{Ext}^1_{\mathcal{C}}(T,M)))\) denotes the Serre polynomial of \(\text{Gr}_e (\text{Ext}^1_{\mathcal{C}}(T,M))\), i.e. NEWLINE\[NEWLINEE(\text{Gr}_e (\text{Ext}^1_{\mathcal{C}}(T,M)))=\sum_i (-1)^i \text{dim} H^i(\text{Gr}_e (\text{Ext}^1_{\mathcal{C}}(T,M)))q^{\frac{1}{2}i}.NEWLINE\]NEWLINENEWLINENEWLINEThe main result of the paper claimed that all the quantum cluster variables, and further all the quantum cluster monomials of \(\mathcal{A}^q\), take this form. Let \(\mathbb{T}_n\) be an \(n\)-regular tree, whose vertices label both the (quantum) seeds and the rigid objects which serve as their categorical counterpart. The main theorem is an explicit decategorification formula:NEWLINENEWLINETheorem 1.2.2. [Main theorem] For any vertex \(t\) of \(\mathbb{T}_n\) and any \(1\leq i\leq n\), we have NEWLINE\[NEWLINE X_{T_i(t)}=X_i(t). NEWLINE\]NEWLINE Moreover, the map taking an object \(M\) to \(X_M\) induces a bijection from the set of isomorphism classes of coefficient-free rigid objects of \(\mathcal{D}\) to the set of quantum cluster monomials of \(\mathcal{A}^q\).NEWLINENEWLINEAs one byproduct of the main theorem, the authors deduce the following important result about positivity: The cluster monomials in \(\mathcal{A}^{\mathbb{Z}}\) have non-negative coefficients in their expansions as Laurent polynomials in the initial variables. Recently, \textit{D. Rupel} [Int. Math. Res. Not. 2011, No. 14, 3207--3236 (2011; Zbl 1237.16013)] posed a conjecture that quantum cluster variables could be expressed in terms of counting polynomials of quiver Grassmannians, and proved it for those variables which lie in almost acyclic clusters using combinatorial methods. Clearly, at least in the case of non valued quivers, this conjecture is confirmed by the main theorem of the paper, since the Serre polynomials are also the counting polynomials.