This document (together with the ancillary file e6_proof.pdf) is an appendix to the paper [12]. The ancillary file contains the computer generated part of the proof of the main result in [12], giving a complete and general list of tree representations corresponding to exceptional modules over the path algebra of the canonically oriented Euclidean quiver $\widetilde{\mathbb{E}}_6$. The proof (involving induction and symbolic computation with block matrices) was partially generated by a purposefully developed computer software, outputting in a detailed step-by-step fashion as if written "by hand". Tree representations are exhibited using matrices involving only the elements 0 and 1, and all representations enlisted in the ancillary document remain valid over any base field.

中文翻译：

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箭袋 $\widetilde{\mathbb{E}}_6$ 的异常表示的树模块属性的证明

该文档（连同辅助文件 e6_proof.pdf）是论文 [12] 的附录。辅助文件包含 [12] 中主要结果证明的计算机生成部分，给出了一个完整和通用的树表示列表，对应于规范定向欧几里得箭袋的路径代数上的异常模块 $\widetilde{\mathbb{ E}}_6$。证明（涉及块矩阵的归纳和符号计算）部分由专门开发的计算机软件生成，以详细的逐步方式输出，就像“手写”一样。树表示使用仅涉及元素 0 和 1 的矩阵来展示，并且附属文档中登记的所有表示在任何基本字段上都保持有效。