AI 中文总结
研究通过引入基于仿射考克斯特复形分解的坐标系描述$\tilde{A}_2$阴影，给出将简约字转坐标的算法，利用此框架识别阴影几何对称性，得出阴影基数公式及成员资格标准，有效描述了$\tilde{A}_2$阴影。
AI 中文摘要
我们通过引入基于仿射考克斯特复形分解为隧道和通道的坐标系，对$\tilde{A}_2$型仿射考克斯特复形中的阴影进行了明确的几何和代数描述。还提供了将任意简约字转换为坐标的算法。利用此框架，识别了阴影的几何对称性，表明它们由底层通道结构控制。由此得出了取决于坐标奇偶性的$\tilde{A}_2$中阴影基数的明确分段公式。此外，通过计数函数建立了阴影成员资格的简单标准，该函数可检测通道内的允许位置。这些结果为仿射型$\tilde{A}_2$中的阴影提供了具体且计算有效的描述，弥合了组合定义与几何实现之间的差距。
英文摘要
We develop an explicit geometric and algebraic description of shadows in the affine Coxeter complex of type $\tilde{A}_2$, by introducing a coordinate system based on a decomposition of the complex into tunnels and channels. We also provide an algorithm for converting arbitrary reduced words into coordinates. Using this framework, we identify geometric symmetries of shadows and show that they are governed by the underlying channel structure. This allows us to derive explicit, piecewise formulas for the cardinality of shadows in $\tilde{A}_2$, depending on the parity of the coordinates. Furthermore, we establish a simple criterion for shadow membership via a counting function that detects admissible positions within channels. These results provide a concrete and computationally effective description of shadows in affine type $\tilde{A}_2$, bridging the gap between combinatorial definitions and geometric realisations.
Comments37 pages, comments welcome
