A Birkhoff-Bruhat Atlas for partial flag varieties
Huanchen Bao, Xuhua He

TL;DR
This paper introduces a Birkhoff-Bruhat atlas for partial flag varieties of Kac-Moody groups, generalizing previous Bruhat atlas constructions and providing new combinatorial and geometric insights.
Contribution
It constructs a Birkhoff-Bruhat atlas for any partial flag variety of a Kac-Moody group, extending the existing Bruhat atlas framework to broader cases.
Findings
Constructed a Birkhoff-Bruhat atlas for all partial flag varieties of Kac-Moody groups.
Developed a combinatorial atlas for the index set of projected Richardson varieties.
Showed that the index set has favorable combinatorial properties.
Abstract
A partial flag variety of a Kac-Moody group has a natural stratification into projected Richardson varieties. When is a connected reductive group, a Bruhat atlas for was constructed by He, Knutson and Lu: 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 -Schubert varieties for a Birkhoff-Bruhat atlas. The notion of the…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Advanced Algebra and Geometry · Algebraic structures and combinatorial models
