Positivity for regular cluster characters in acyclic cluster algebras (Q2909799)

Let \(Q\) be an acyclic quiver which is representation infinite, and \(X_? : \text{rep}(Q) \to \mathbb{Z}[\mathbf{y}, \mathbf{x}^{\pm 1}]\) be the associated cluster character. The first main result of the paper (Theorem 1.1) is that whenever \(M\) is a regular Schur representation of \(Q\), but not quasi-simple, the Laurent polynomial \(X_M\) has non-negative coefficients with respect to the collections of variables \(\mathbf{y}\) and \(\mathbf{x}^{\pm 1}\). As a corollary, we find that all quiver Grassmannians associated to such representations have non-negative Euler characteristic (Corollary 1.2).NEWLINENEWLINEThe second part of the paper concerns coefficient-free cluster algebras associated to tame quivers. The first result here is that for any representation \(M\), the Laurent polynomial \(X_M\) has non-negative coefficients with respect to the variables \(\mathbf{x}^{\pm 1}\) (Theorem 1.4). This is used to establish a certain positivity property for three particular bases of these algebras which have been studied previously in the literature. Namely, in each case, the bases consist of elements which have non-negative coefficients with respect to the collections of initial variables \(\mathbf{x}^{\pm 1}\) (Theorem 1.5).NEWLINENEWLINEThe first main result is demonstrated using generalized Chebyshev polynomials introduced by the author in previous work, [J. Algebr. Combin. 31, No. 4, 501--532 (2010; Zbl 1231.05290)] and [Algeb. Represent. Theory 15, No. 3, 527--549 (2012; Zbl 1253.16012)]. These allow us to understand cluster characters associated to regular modules in terms of their quasi-composition factors. A key fact in the proof is that for \(M\) as above, the quasi-composition factors are rigid, and thus give rise to quiver Grassmannians with non-negative Euler characteristics by recent results of \textit{H. Nakajima} [Kyoto J. Math. 51, 71--126 (2011; Zbl 1223.13013)] and \textit{F. Qin} [J. Reine Angew. Math. 668, 149--190 (2012; Zbl 1252.13013)].NEWLINENEWLINEThe results for coefficient-free algebras associated to tame quivers utilize \(\Delta\)-polynomials, which play a role similar to Chebyshev polynomials. They allow one to understand cluster characters when moving up a homogeneous tube, where the composition factors are not rigid. This technique is the key to the second set of results.