论文标题
手镯断裂和Kostant的分区功能
Broken Bracelets and Kostant's Partition Function
论文作者
论文摘要
受到Amdeberhan，Can和Moll折磨项链的工作的启发，我们将破损的手镯定义为一个线性的构成和未标记的顶点的线性布置，并引入了称为$ n $ stars的概括，该概括是$ n $损坏的手镯的集合，其最终（未标记）的顶点已确定。通过这些组合对象，我们为研究Kostant的分区函数提供了一个新的框架，该框架计算出表达向量的方法的数量，将其视为Lie代数积极根的非负整数线性组合。我们的主要结果表明，（进行反思）具有固定数量的未标记的顶点的破碎手镯数量，具有非连续标记的顶点的上限为Kostant分区函数的值提供了上限的上限，用于$ a $ a $的lie代数的最高元素的倍数。正如Benedetti，Hanusa，Harris，Morales和Simpson所研究的那样，我们将这项工作与多重杂耍序列联系起来，通过提供与$ n $标准的等价关系的对应关系。
Inspired by the work of Amdeberhan, Can, and Moll on broken necklaces, we define a broken bracelet as a linear arrangement of marked and unmarked vertices and introduce a generalization called $n$-stars, which is a collection of $n$ broken bracelets whose final (unmarked) vertices are identified. Through these combinatorial objects, we provide a new framework for the study of Kostant's partition function, which counts the number of ways to express a vector as a nonnegative integer linear combination of the positive roots of a Lie algebra. Our main result establishes that (up to reflection) the number of broken bracelets with a fixed number of unmarked vertices with nonconsecutive marked vertices gives an upper bound for the value of Kostant's partition function for multiples of the highest root of a Lie algebra of type $A$. We connect this work to multiplex juggling sequences, as studied by Benedetti, Hanusa, Harris, Morales, and Simpson, by providing a correspondence to an equivalence relation on $n$-stars.
