论文标题
关于莫里尔·根诺德（Morier-Genoud）和ovsienko的排名障碍猜想
On a rank-unimodality conjecture of Morier-Genoud and Ovsienko
论文作者
论文摘要
令alpha =（a，b，...）为组成。考虑关联的poset f（alpha），称为围栏，其覆盖关系为x_1 <x_2 <... <x_ {a+1}> x__ {a+2}> ...> x_ {a+b+1} <x__ {a+b+b+2} <...。我们研究了由F（alpha）的所有低阶理想组成的相关分布晶格L（alpha）。这些晶格在群集代数理论中很重要，它们的等级生成函数可用于定义有理数的Q-Analogues。特别是，我们在最近猜想的是莫里尔·根诺德（Morier-Genoud）和ovsienko上取得了进展，即L（alpha）是单峰的排名。我们表明，如果alpha的一个部分大于其他总和，那么猜想是正确的。我们猜想L（alpha）具有具有嵌套链分解并具有顶部或底部交织的等级序列的更强特性，后者是最近定义的序列属性。我们验证了这些属性是否具有最多三个部分的组成以及所谓的d-Diviend posets的作品，从而概括了克劳森的工作，并简化了甘斯纳的构造。
Let alpha = (a,b,...) be a composition. Consider the associated poset F(alpha), called a fence, whose covering relations are x_1 < x_2 < ... < x_{a+1} > x_{a+2} > ... > x_{a+b+1} < x_{a+b+2} < ... . We study the associated distributive lattice L(alpha) consisting of all lower order ideals of F(alpha). These lattices are important in the theory of cluster algebras and their rank generating functions can be used to define q-analogues of rational numbers. In particular, we make progress on a recent conjecture of Morier-Genoud and Ovsienko that L(alpha) is rank unimodal. We show that if one of the parts of alpha is greater than the sum of the others, then the conjecture is true. We conjecture that L(alpha) enjoys the stronger properties of having a nested chain decomposition and having a rank sequence which is either top or bottom interlacing, the latter being a recently defined property of sequences. We verify that these properties hold for compositions with at most three parts and for what we call d-divided posets, generalizing work of Claussen and simplifying a construction of Gansner.
