论文标题
上集群代数上的估值配对
The valuation pairing on an upper cluster algebra
论文作者
论文摘要
众所周知，许多（上）集群代数不是独特的分解域。我们在任何给定的种子$ t $上展示了局部分解属性：完整等级上群集代数中的任何非零元素都可以独特地写成$ t $中的群集单元的产物，而另一个元素则不能被$ t $中的群集变量划分。我们的方法基于在上层群集代数上引入估值配对：它计算了任何给定元素的因素化之间的群集变量的最大多重性。 我们应用估值配对来获得许多有关阶乘，$ d $ - 向量，$ f $ - 多元素式和群集泊松变量组合的结果。特别是，我们获得了完整的等级和原始的上层代数是阶乘。使用估值配对的$ D $ - 向量的说明；非初始集群变量中的群集单位群由其$ f $ - 多项式式确定；非初始群集变量的$ f $ - 多项式不可约；群集泊松变量参数化相应上群集代数的交换对。
It is known that many (upper) cluster algebras are not unique factorization domains. We exhibit the local factorization properties with respect to any given seed $t$: any non-zero element in a full rank upper cluster algebra can be uniquely written as the product of a cluster monomial in $t$ and another element not divisible by the cluster variables in $t$. Our approach is based on introducing the valuation pairing on an upper cluster algebra: it counts the maximal multiplicity of a cluster variable among the factorizations of any given element. We apply the valuation pairing to obtain many results concerning factoriality, $d$-vectors, $F$-polynomials and the combinatorics of cluster Poisson variables. In particular, we obtain that full rank and primitive upper cluster algebras are factorial; an explanation of $d$-vectors using valuation pairing; a cluster monomial in non-initial cluster variables is determined by its $F$-polynomial; the $F$-polynomials of non-initial cluster variables are irreducible; and the cluster Poisson variables parametrize the exchange pairs of the corresponding upper cluster algebra.
