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

讲座时间 Datetime: 2022年9月26日，星期一，10:10-11:00

地点 Venue: 腾讯线上会议：946-628-9201

主持人 Host：黄敏

报告人 Speaker: 李方 

单位 Affiliation: 浙江大学

报告摘要 Abstract:

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 polytope functions  $\rho_h$ corresponding to polytopes $N_h$ associated to vectors $h\in Z^n$. 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) constructing a strongly positive basis of the upper cluster algebra $U(A)$  from the set of polytope functions consisting of certain indecomposable universally positive elements, which is called as the polytope basis; (iii) building the relationship via some explicit maps among  cluster variables, F-polynomials, g-vectors and d-vectors.   As an application of (i), we give an affirmation for the positivity conjecture of Laurent expressions of cluster variables in a totally sign-skew-symmetric cluster algebra, which in particular provides a new method different with that given in [GHKK] to present the positivity of cluster variables in the skew-symmetrizable case. 

As an application of (iii), we give an affirmation for the positivity conjecture of d-vectors of cluster variables in a totally sign-skew -symmetric cluster algebra.   This work is joint with Jie Pan.