论文标题
Birkhoff-Bruhat Atlas，用于部分国旗品种
A Birkhoff-Bruhat Atlas for partial flag varieties
论文作者
论文摘要
一个部分标志品种$ {\ Mathcal {p}} _ k $的kac-moody group $ g $具有自然分层为预测的Richardson品种。当$ g $是一个连接的还原组时，$ {\ Mathcal {p}} _ k $的Bruhat Atlas是由He，Knutson和Lu：$ {\ Mathcal {p}} _ k $构建的，是本地用Schubert varieties建模的一些KAC-Moody Flag spaces。 Bruaht图谱的存在意味着部分标志品种上的一些不错的组合和几何特性，以及将其分解为预测的Richardson品种。 由于组合和几何原因，由于任意Kac-Moody群体的部分标志品种，不存在Bruhat地图集。为了克服障碍，我们介绍了伯克霍夫 - 布鲁哈特地图集的概念。我们将$ J $ -Schubert品种用于Birkhoff-Bruhat Atlas，而不是用于Bruhat Atlas中的Schubert品种。 $ j $ -schubert品种的概念介绍了Birkhoff的分解和Bruhat分解（一个较大的Kac-Moody集团）。本文的主要结果是为任何部分标志品种$ {\ Mathcal {p}} _ k $构建Birkhoff-bruhat地图集。我们还为索引的组合构建一个组合图集，以$ {\ Mathcal {p}} _ k $的预测richardson品种的$ q_k $。结果，我们表明$ q_k $具有一些不错的组合属性。这给出了新的证明，并在$ g $是连接的还原组的情况下概括了威廉姆斯的工作。
A partial flag variety ${\mathcal {P}}_K$ of a Kac-Moody group $G$ has a natural stratification into projected Richardson varieties. When $G$ is a connected reductive group, a Bruhat atlas for ${\mathcal {P}}_K$ was constructed by He, Knutson and Lu: ${\mathcal {P}}_K$ is locally modeled with Schubert varieties in some Kac-Moody flag variety as stratified spaces. The existence of Bruaht atlases implies some nice combinatorial and geometric properties on the partial flag varieties and the decomposition into projected Richardson varieties. A Bruhat atlas does not exist for partial flag varieties of an arbitrary Kac-Moody group due to combinatorial and geometric reasons. To overcome obstructions, we introduce the notion of Birkhoff-Bruhat atlas. Instead of the Schubert varieties used in a Bruhat atlas, we use the $J$-Schubert varieties for a Birkhoff-Bruhat atlas. The notion of the $J$-Schubert varieties interpolates Birkhoff decomposition and Bruhat decomposition of the full flag variety (of a larger Kac-Moody group). The main result of this paper is the construction of a Birkhoff-Bruhat atlas for any partial flag variety ${\mathcal {P}}_K$ of a Kac-Moody group. We also construct a combinatorial atlas for the index set $Q_K$ of the projected Richardson varieties in ${\mathcal {P}}_K$. As a consequence, we show that $Q_K$ has some nice combinatorial properties. This gives a new proof and generalizes the work of Williams in the case where the group $G$ is a connected reductive group.
