We give an action of the symmetric group on non-commuting indeterminates in terms of series in the corresponding Mal'cev-Newmann division ring. The action is constructed from the non-Abelian Hirota-Miwa (discrete KP) system. The corresponding companion map, which gives generators of the action, is discussed in the generic case and the corresponding explicit formulas have been found in the periodic reduction. We discuss also briefly connection of the companion to the KP map with context-free languages.

中文翻译：

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非通勤符号的 Coxeter 关系和 KP 映射

我们根据相应的 Mal'cev-Newmann 划分环中的级数给出对称群对非交换不定数的作用。该动作是从非阿贝尔 Hirota-Miwa（离散 KP）系统构建的。在一般情况下讨论了相应的伴随映射，它给出了动作的生成器，并且在周期性归约中找到了相应的显式公式。我们还简要讨论了使用上下文无关语言将伴侣连接到 KP 地图。