标题： 类型为$B$的球面建筑中最大非对立腔的集合 显示英文标题

标题： The largest sets of non-opposite chambers in spherical buildings of type $B$

摘要： 关于有限球形建筑中非对立旗的大族的研究，是极值组合学领域一条研究长线中的最新补充，它扩展了向量空间和极小空间中的经典结果。 这一研究方向属于 Erdős-Ko-Rado (EKR) 问题的范畴，但在代数层面上比上述经典结果更具挑战性。 从建筑理论的角度来看，这可以被视为蒂奇斯关于球形建筑中心猜想的一种变体，其中我们将凸性假设替换为极大性条件。 在之前的工作中，通过对非例外建筑的 Iwahori-Hecke 代数应用特征值和表示论技术，得到了非对立旗族大小的一般上界。 最近，通过 Heering、Lansdown 和 Metsch 的工作，在类型为$A_n$且$n$为奇数的情况下，达到了这个上界的家庭被分类。 对于类型为$B$的建筑，对应的 Iwahori-Hecke 代数更为复杂，并且非平凡地依赖于基础极小空间的类型和秩。 然而，我们能够找到一种基于反设计的统一方法，并在所有情况下（除了类型$^2A_{4n-3}$）获得关于房间（即极大旗）的分类结果。 显示英文摘要

摘要： The investigation into large families of non-opposite flags in finite spherical buildings has been a recent addition to a long line of research in extremal combinatorics, extending classical results in vector and polar spaces. This line of research falls under the umbrella of Erd\H{o}s-Ko-Rado (EKR) problems, but poses some extra difficulty on the algebraic level compared to aforementioned classical results. From the building theory point of view, it can be seen as a variation of the center conjecture for spherical buildings due to Tits, where we replace the convexity assumption by a maximality condition. In previous work, general upper bounds on the size of families of non-opposite flags were obtained by applying eigenvalue and representation-theoretic techniques to the Iwahori-Hecke algebras of non-exceptional buildings. More recently, the classification of families reaching this upper bound in type $A_n$, for $n$ odd, was accomplished by Heering, Lansdown, and Metsch. For buildings of type $B$, the corresponding Iwahori-Hecke algebra is more complicated and depends non-trivially on the type and rank of the underlying polar space. Nevertheless, we are able to find a uniform method based on antidesigns and obtain classification results for chambers (i.e.\ maximal flags) in all cases, except type $^2A_{4n-3}$.