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

组织者

吴杰

演讲者

李方

时间

2022年01月17日 10:30 至 11:30

地点

中会议室一

摘要

In this talk, we study the Newton polytopes of F-polynomials in totally sign-skew-symmetric cluster algebras and generalize them to a larger set consisting of polytopes N(h) associated to vectors h in Z^n as well as S consisting of polytope functions \rho_h corresponding to N(h). The main contribution contains that (i) obtaining a recurrence construction of the Laurent expression of a cluster variable in a cluster from its g-vector; (ii) proving the subset P of S is a strongly positive basis of U(A) consisting of certain indecomposable universally positive elements, which is called as the polytope basis; (iii) constructing some explicit maps among corresponding F-polynomials, g-vectors and d-vectors to characterize their relationship. As an application of (i), we give an affirmation to the positivity conjecture of cluster variables in a totally sign-skew-symmetric cluster algebra, which in particular provides a new method different from that given in \cite{GHKK} to present the positivity of cluster variables in the skew-symmetrizable case. As another application, a conjecture on Newton polytopes posed by Fei is answered affirmatively. For (ii), we know that in rank 2 case, $\mathcal{P}$ coincides with the greedy basis introduced by Lee, Li and Zelevinsky. Hence, we can regard $\mathcal{P}$ as a natural generalization of the greedy basis in general rank. As an application of (iii), the positivity of denominator vectors associated to non-initial cluster variables, which was first come up as a conjecture in \cite{FZ4}, is proved in a totally sign-skew-symmetric cluster algebra. This is a joint work with Jie Pan.