Recurrence formula, positivity and polytope basis in cluster algebras via Newton polytopes

Abstract

In this paper, we study the Newton polytopes of $F$-polynomials in a TSSS cluster algebra $\mathcal A$ and generalize them to a larger set consisting of polytopes $N_{h}$ associated to vectors $h\in\Z^{n}$ as well as $\widehat{\mathcal{P}}$ consisting of polytope functions $ρ_{h}$ corresponding to $N_{h}$. The main contribution contains that (i) obtaining a {\em recurrence construction} of the Laurent expression of a cluster variable in a cluster from its $g$-vector; (ii) proving the subset $\mathcal{P}$ of $\widehat{\mathcal{P}}$ consisting of Laurent polynomials in $\widehat{\mathcal{P}}$ is a strongly positive $\Z Trop(Y)$-basis for $\mathcal{U}(\A)$ consisting of certain universally indecomposable Laurent polynomials when $\A$ is a cluster algebra with principal coefficients. For a cluster algebra $\mathcal A$ over arbitrary semifield $\mathbb P$ in general, $\mathcal{P}$ is a strongly positive $\Z¶$-basis for the intermediate cluster subalgebra $\mathcal{I_P(A)}$ of $\mathcal{U(A)}$. We call $\mathcal P$ the {\em polytope basis}; (iii) constructing some explicit maps among corresponding $F$-polynomials, $g$-vectors, $d$-vectors and cluster variables to characterize their relationship. Moreover, we give three applications of (i), (ii) and (iii) respectively.