仿射建筑的对径理想腔三元组的几何学
The geometry of triples of antipodal ideal chambers of affine buildings
AI总结：
本文研究仿射建筑中对径理想腔三元组的两种一般性概念，证明仿射一般性对应的重心映射局部常值，给出保证所有对径理想腔三元组满足一般性的仿射外尔群条件，并确定仅三类不可约有限外尔群使所有对径理想腔三元组自动满足理想一般性。
AI中文摘要：
本文研究局部有限仿射建筑$X$中对径理想腔三元组的两种一般性概念：一种在理想边界$X^{\infty}$上定义，称为理想一般性；另一种在仿射建筑$X$内部定义，称为仿射一般性。理想一般性蕴含仿射一般性，但后者更适合构造与仿射一般性对径理想腔三元组相关的重心映射，该视角还可证明此重心映射是局部常值的，因此是连续的。最后，本文给出与$X$相关的仿射外尔群的充分几何条件，以保证$X^{\infty}$的所有对径理想腔三元组均满足理想一般性和仿射一般性，这些条件为构造$X \cup X^{\infty}$中对应$X^{\infty}$非一般性对径理想腔三元组的几何构型（存在时）提供了算法方法。此外，本文对秩至少为2的不可约有限外尔群的计算表明，仅在类型$B_2 = C_2$、$G_2$和$B_3$中，所有对径理想腔三元组自动满足理想一般性。
英文摘要：
In this article, we investigate two notions of genericity for triples of antipodal ideal chambers in a locally finite affine building $X$: one defined at the ideal boundary $X^{\infty}$, which we call ideal-genericity, and the other defined from within the affine building \(X\), which we call affine-genericity. While ideal-genericity implies affine-genericity, the latter is the more suitable notion for constructing a barycenter map associated with affine-generic triples of antipodal ideal chambers. This perspective also allows us to establish that this barycenter map is locally constant, and hence continuous. Finally, we provide sufficient geometric conditions on the affine Weyl group associated with $X$ that guarantee both ideal- and affine-genericity for all triples of antipodal ideal chambers of $X^\infty$. These conditions yield an algorithmic method for constructing geometric configurations in $X \cup X^{\infty}$ (when they exist) that correspond to non-generic triples of antipodal ideal chambers of $X^{\infty}$. Furthermore, our computations for the irreducible finite Weyl groups of rank at least two show that automatic ideal-genericity holds for all triples of antipodal ideal chambers only in types $B_2 = C_2$, $G_2$ and $B_3$.
