Kostant’s partition function is a vector partition function that counts the number of ways one can express a weight of a Lie algebra \(\mathfrak {g}\) as a nonnegative integral linear combination of the positive roots of \(\mathfrak {g}\). Multiplex juggling sequences are generalizations of juggling sequences that specify an initial and terminal configuration of balls and allow for multiple balls at any particular discrete height. Magic multiplex juggling sequences generalize further to include magic balls, which cancel with standard balls when they meet at the same height. In this paper, we establish a combinatorial equivalence between positive roots of a Lie algebra and throws during a juggling sequence. This provides a juggling framework to calculate Kostant’s partition functions, and a partition function framework to compute the number of juggling sequences. From this equivalence we provide a broad range of consequences and applications connecting this work to polytopes, posets, positroids, and weight multiplicities.

Kostant的分区函数是矢量分区函数，它计算人们可以表达Lie代数\（\ mathfrak {g} \）作为\（\ mathfrak {g}的正根的非负整数线性组合的方式的数量\）。多重处理序列是处理序列的概括，它指定了球的初始和最终配置，并允许在任何特定的离散高度上放置多个球。魔术多重杂耍序列可以进一步概括为包括魔术球，当它们在相同高度相遇时，这些魔术球会消失。在本文中，我们建立了李代数的正根与杂耍序列之间的投掷之间的组合等价。这提供了一个计算Kostant的分区函数的杂耍框架，以及一个计算杂耍序列数的分区函数框架。通过这种等效，我们提供了广泛的结果和应用，可将这项工作与多面体，波状体，正体和体重多重性联系起来。