Combinatorial aspects of representation theory and geometry

表示论和几何的组合方面

• 批准号: RGPIN-2016-04872
• 负责人: Thomas, Arthur
• 金额: $ 2.4万
• 依托单位: Université du Québec à Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=684598
• 关键词: Combinatorial; aspects; representation; theory; geometry

Combinatorics is the study of discrete arrangements, such as the different ways of dividing n balls among k different boxes. Often, it is possible to understand such discrete arrangements much better than we could hope to understand the continuously changing phenomena that we typically encounter in real life. However, as it turns out, discrete structures can have important implications, in real life, and also in other areas of mathematics. My proposal consists of using combinatorial techniques to understand mathematical structures which arise in algebra and physics.***The physics I am interested in is the problem of calculating scattering amplitudes. This is quite a natural and fundamental problem. We throw a collection of particles at each other. They interact somehow, and then zoom off. We want to know the outcome. Because we are in a quantum setting, the outcome is uncertain, but there are different possibilities, to each of which a probability can be assigned. The usual approach to calculating scattering amplitudes is to write down what is typically a very large number of Feynman diagrams which encode the possible interactions among the particles, and then add up a contribution from each. It is a surprising fact that when we do this, the answer is simple and symmetrical in a way that the individual terms being summed are not. This suggests that the Feynman diagram approach might be obscuring the fundamental simplicity of what is happening. A new approach to scattering amplitudes has been developed by Nima Arkani-Hamed and a team of collaborators. In their approach, Feynman diagrams are replaced by something which the physicists call "on-shell diagrams", but which had already been studied by Alex Postnikov and other mathematicians under the name of "plabic graphs". By studying the combinatorics and geometry associated to these plabic graphs, in collaboration with Nima Arkani-Hamed and Jaroslav Trnka, I hope to come to a better understanding of this new approach. ***Combinatorics can also be applied in algebraic settings. One way to do this is to start from a combinatorial system and associate some kind of algebra to it. Our understanding of the combinatorics will likely help us to understand such algebras. It is also possible that powerful algebraic tools may then also improve our understanding of the combinatorics from which we set out. To be a bit more specific, I am interested in links between the combinatorics of reflection groups (a generalization of permutations) and the representation theory of preprojective algebras and their quotients. **

组合学是对离散布置的研究，例如将N球分隔在不同盒子之间的不同方式。 通常，可以比我们希望理解我们在现实生活中通常遇到的不断变化的现象要好得多。 但是，事实证明，离散的结构在现实生活以及其他数学领域都可能具有重要的含义。 我的建议包括使用组合技术来了解代数和物理学中出现的数学结构。***我感兴趣的物理学是计算散射幅度的问题。 这是一个自然而根本的问题。 我们互相投入一系列粒子。 他们以某种方式互动，然后缩小。 我们想知道结果。 因为我们处于量子设置，所以结果不确定，但是有不同的可能性，可以分配概率。 计算散射幅度的通常方法是写下通常是大量的Feynman图，这些图表编码粒子之间可能的相互作用，然后从每个粒子中添加贡献。 令人惊讶的事实是，当我们这样做时，答案是简单而对称的，而单个术语则不是。 这表明Feynman图的方法可能掩盖了正在发生的事情的基本简单性。 Nima Arkani-Hamed和一组合作者已经开发了一种新的散射幅度方法。 在他们的方法中，Feynman图被物理学家称为“壳图”的东西取代，但是Alex Postnikov和其他数学家已经以“ Plabic图”的名义研究了这些图。 通过与Nima Arkani-Hamed和Jaroslav Trnka合作，研究与这些具有这些知识图相关的组合和几何形状，我希望对这种新方法有更好的了解。 ***组合学也可以在代数设置中应用。 一种方法是从组合系统开始，并将某种代数与之相关联。 我们对组合学的理解可能会帮助我们理解此类代数。 强大的代数工具也有可能还可以提高我们对我们列出的组合学的理解。 更具体地说，我对反射组组合（置换术的概括）与前反射式代数及其商的表示理论之间的联系感兴趣。 **