The Littelmann path model via the affine Grassmannian

通过仿射格拉斯曼的 Littelmann 路径模型

• 批准号: 372169579
• 负责人: Dr. Jacinta Torres
• 金额: --
• 依托单位: Instytut Matematyczny
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2017
• 资助国家: 德国
• 起止时间: 2016-12-31 至 2020-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/372169579?language=en
• 关键词: Littelmann; path; model; via; affine

One of the most important open problems in Kazhdan-Lusztig theory is to provide a closed formula for Kazhdan-Lusztig polynomials. These polynomials connect various areas of mathematics. So far, closed formulas exist only in some special cases, such as the well-known Kostka-Foulkes polynomials, which are affine Kazhdan-Lusztig polynomials. They play important roles in the theory of symmetric functions and are used in the geometric study of affine Grassmannians. There is a celebrated closed formula for these polynomials in terms of the charge statistic associated to Young tableaux. These tableaux are classical combinatorial objects which play an important role in the representation theory of the general linear group, the most fundamental example of an algebraic group. One of the main targets of this project is to give a geometric interpretation of the charge statistic, thus reuniting the combinatorics and the geometry. There exists a natural generalisation of Kostka-Foulkes polynomials for any complex reductive group. However, a closed formula exists only in the case of the special linear group. We believe that a geometric interpretation of charge will provide such a formula. The Littelmann path model is a generalisation of Young tableaux for all complex reductive groups. To say that Littelmann paths have attracted a lot of attention in the last twenty years would be a vast understatement. We aim to assign a charge statistic to any Littelmann path. Recently, the Littelmann path model has also been interpreted in terms of the geometry of the affine Grassmannian, using objects called buildings. This interpretation is, however, not complete. The aim is to complete this interpretation, which would deepen the existing connection between Littelmann paths, the geometry of the affine Grassmannian, and the theory of buildings. We believe that this will lead to a geometric definition of charge.Related to this is the second aim of this project. In recent work with Schumann we have proven a conjecture ofNaito-Sagaki giving a branching rule for the decomposition of the restriction of an irreduciblerepresentation of the special linear Lie algebra to the symplectic Lie algebra. This conjecture had been open for over ten years,and the new rule provides a new approach to branching rules for non-Levi subalgebras in termsof Littelmann paths. The subalgebras that we consider are those obtained as fixed-point sets of an automorphism of finite order. If the automorphism is semisimple and of infinite order, the set of fixed points is a Levi subalgebra.Our aim in this project is to describe the building theoretical geometry of "Levi branching", and provide an analogue in the more enigmatic and difficult non-Levi case. We believe that such an interpretation would not only provide branching rules, but also restriction functors in the set-up of the geometric Satake equivalence, which only exist in the case of Levi subalgebras.

Kazhdan-Lusztig理论中最重要的公开问题之一是给出Kazhdan-Lusztig多项式的封闭公式。这些多项式连接了数学的各个领域。到目前为止，封闭公式只存在于某些特殊情况下，如著名的Kostka-Foulkes多项式，它们是仿射Kazhdan-Lusztig多项式。它们在对称函数理论中起着重要的作用，并被用于仿射Grassmannians的几何研究。根据与Young画面相关的电荷统计，这些多项式有一个著名的封闭公式。这些表是经典的组合对象，在一般线性群的表示理论中起着重要的作用，一般线性群是代数群的最基本的例子。这个项目的主要目标之一是对电荷统计进行几何解释，从而将组合学和几何学重新结合起来。对于任意复约群，都存在Kostka-Foulkes多项式的自然推广。然而，封闭公式只存在于特殊的线性群的情况下。我们相信，电荷的几何解释将提供这样一个公式。Littelmann路模型是所有复约化群的Young图景的推广。要说利特曼路径在过去20年里吸引了很多关注，那就太轻描淡写了。我们的目标是给任何Littelmann路径分配一个电荷统计。最近，Littelmann路径模型也被用仿射格拉斯曼几何来解释，使用了被称为建筑物的物体。然而，这种解释并不完整。其目的是完成这一解释，这将加深利特曼路径、仿射格拉斯曼几何和建筑理论之间的现有联系。我们相信，这将导致电荷的几何定义。与此相关的是该项目的第二个目标。在最近与Schumann的工作中，我们证明了Naito-Sagaki的一个猜想，该猜想给出了将特殊的线性李代数的不可约表示分解为辛李代数的限制的分支规则。这一猜想已经提出了十多年，新的规则为非Levi子代数在Littelmann路方面的分支规则提供了一种新的途径。我们所考虑的子代数是作为有限阶自同构的不动点集得到的子代数。如果自同构是半单的无穷阶的，则不动点集是Levi子代数.我们在这个项目中的目的是描述“Levi分支”的建筑理论几何，并在更神秘和更困难的非Levi情形下提供一个类比.我们认为，这样的解释不仅提供了分支规则，而且还提供了只存在于Levi子代数情况下的几何Satake等价的建立中的限制函子。