论文标题
布鲁哈特的间隔结构和弱点
Interval structures in the Bruhat and weak orders
论文作者
论文摘要
我们研究了Coxeter组的Bruhat和弱顺序，研究了著名的间隔结构的外观 - 晶格，模块化晶格，分布晶格和布尔晶格。我们收集并扩大了主要秩序理想的已知结果，包括对称组的模式特征和枚举。这自然会在任意间隔内进行类似的分析，尽管在此通用性下，结果的表征较少。但是，在对立点中，我们获得了对称组的等级第一的间隔的完整表征，对于这两个posets中的每个结构类型中的每一个，每个结构类型中的每一个。可以列举每个类别，并具有与斐波那契和加泰罗尼亚人数的有趣联系。我们最终提出了有关进一步方向和问题的建议，包括对置换和每个发电机之间形成的间隔的有趣分析。
We study the appearance of notable interval structures -- lattices, modular lattices, distributive lattices, and boolean lattices -- in both the Bruhat and weak orders of Coxeter groups. We collect and expand upon known results for principal order ideals, including pattern characterizations and enumerations for the symmetric group. This segues naturally into a similar analysis for arbitrary intervals, although the results are less characterizing for the Bruhat order at this generality. In counterpoint, however, we obtain a full characterization for intervals starting at rank one in the symmetric group, for each of the four structure types, in each of the two posets. Each category can be enumerated, with intriguing connections to Fibonacci and Catalan numbers. We conclude with suggestions for further directions and questions, including an interesting analysis of the intervals formed between a permutation and each generator in its support.
