Starting from higher dimensional adinkras constructed with nodes referenced by Dynkin Labels, we define “adynkras.” These suggest a computationally direct way to describe the component fields contained within supermultiplets in all superspaces. We explicitly discuss the cases of ten dimensional superspaces. We show this is possible by replacing conventional $\theta$-expansions by expansions over Young Tableaux and component fields by Dynkin Labels. Without the need to introduce $\sigma$-matrices, this permits rapid passages from Adynkras → Young Tableaux → Component Field Index Structures for both bosonic and fermionic fields while increasing computational efficiency compared to the starting point that uses superfields. In order to reach our goal, this work introduces a new graphical method, “tying rules,” that provides an alternative to Littlewood’s 1950 mathematical results which proved branching rules result from using a specific Schur function series. The ultimate point of this line of reasoning is the introduction of mathematical expansions based on Young Tableaux and that are algorithmically superior to superfields. The expansions are given the name of “adynkrafields” as they combine the concepts of adinkras and Dynkin Labels.

中文翻译：

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向 adynkrafields 进阶：10D 分量场的年轻画面，$\mathcal{N}=1$ 标量超场

从由 Dynkin 标签引用的节点构造的更高维 adinkras 开始，我们定义了“adynkras”。这些提出了一种计算直接的方式来描述包含在所有超空间中的超倍数中的分量场。我们明确讨论了十维超空间的情况。我们通过对 Young Tableaux 的扩展和 Dynkin Labels 的组件字段替换传统的 $\theta$ 扩展来证明这是可能的。无需引入 $\sigma$-矩阵，这允许从 Adynkras → Young Tableaux → 用于玻色子和费米子场的分量场索引结构的快速通道，同时与使用超场的起点相比提高计算效率。为了达到我们的目标，这项工作引入了一种新的图形方法，“捆绑规则，”这提供了 Littlewood 1950 年数学结果的替代方案，该结果证明分支规则是使用特定的 Schur 函数系列产生的。这条推理线的最终点是引入了基于 Young Tableaux 的数学扩展，并且在算法上优于超场。这些扩展被命名为“adynkrafields”，因为它们结合了 adinkras 和 Dynkin 标签的概念。