论文标题
通过牛顿多面体在群集代数中的复发公式，阳性和多层基础
Recurrence formula, positivity and polytope basis in cluster algebras via Newton polytopes
论文作者
论文摘要
在本文中，我们研究了tsss群集代数$ \ Mathcal a $ $ f $ - 多元素的牛顿多面体，并将其推广到由polytopes $ n_ {h} $组成的较大集合中功能$ρ_{h} $对应于$ n_ {h} $。 主要贡献包括 （i）从其$ g $ -vector中群集变量的laurent表达式的{\ em复发构建}获得{\ em recurrence construction}； （ii）证明$ \ wideHat的子集$ \ Mathcal {p} $ {\ Mathcal {p}} $由$ \ wideHat中的laurent多项式组成的{\ nathcal {p}} $是$ \ z trop（y）$ - $ \ nmater的强烈$ \ mathercant $当$ \ a $是带有主系数的集群代数时，不可塑性的劳伦元素多项式。对于群集代数$ \ mathcal a $，超过任意的semifield $ \ mathbb p $一般而言，$ \ nathcal {p} $是一个非常积极的$ \z¶$ - $ basis，用于中间群集群subalgebra $ \ nathcal $ \ nathcal {i_p（a）$ \ $ \ nathcal ca）我们称$ \ mathcal p $为{\ em polytope basis}; （iii）在相应的$ f $ -polynomials，$ g $ - 向量，$ d $ - 向量和集群变量之间构建一些明确的地图以表征其关系。 此外，我们分别提供（i），（ii）和（iii）的三个应用。
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.
