Quiver varieties and quantum dimensions

量子仿射代数的表示具有丰富的理论，它与数学的许多部分相互作用，包括代数几何和数学物理。Szendroi和他的合作者最近在Kleian奇点上的Hilbert点格式的Euler特征方面的工作导致了对于A和D型奇点的这些Euler特征的生成序列作为仿射李代数的表示特征的特化的美丽的描述。他们猜测，同样的公式也适用于剩余的E型情况。Nakajima后来证明了这一恒等式，使用Szendroi和他的合作者建立的将Hilbert方案标识为某些箭形变种的方法。他证明的关键是声称量子仿射代数的某些‘标准表示’有一个‘量子维度’，当量子参数q被视为单位根时，这个‘量子维度’专门为1。这一显著的事实是由Nakajima通过一些明确的个案计算来证实的。然而，这些专门化是由Kirillov和Reshetikhin最先研究的关于一类特定表示的量子维度专门化的猜想预测的。这些猜想是出于数学物理的考虑，以前与箭图变体无关。这项研究计划通过理解箭图变体背景下的量子维度专门化来调查标准模块的量子维度的专门化。这也可能与Hernandez-Leclerc的工作所提出的簇代数理论有关。该项目的初步步骤将集中于对相关文献的回顾，以确保对这一领域的最新技术水平有一个适当的了解。最近发现了量子维度的专门化与箭牌品种的欧拉特征之间的联系，这一提议得到了推动。因此，对这些联系的任何新的见解都可能有助于对这两个对象的理解。这个项目属于EPSRC“代数”和“几何和拓扑”研究领域的交叉。我将由在牛津大学(数学研究所)工作的凯文·麦克格蒂教授指导，资金来自EPSRC卓越奖。此外，该项目有可能与该研究所的其他学者，如Balázs Szendroi教授，进行互动。

Representations of quantum affine algebras have a rich theory which interacts with many parts of mathematics, including algebraic geometry and mathematical physics. Recent work by Szendroi and collaborators on the Euler characteristics of Hilbert schemes of points on Kleinian singularities led to a beautiful description of the generating series of these Euler characteristics for singularities of types A and D as the specialization of the character of a representation of an affine Lie algebra. They conjectured that the same formula also holds in the remaining type E case.Nakajima has since proved this identity, using the identification of the Hilbert schemes as certain quiver varieties, as established by Szendroi and his collaborators. The key to his proof is the claim that certain 'standard representations' of quantum affine algebras have a 'quantum dimension' which specializes to 1 when the quantum parameter q is taken to be a particular root of unity.This remarkable fact is established by Nakajima using some explicit case-by-case calculations. These specializations, however, are predicted by conjectures on specializations of the quantum dimension of a particular class of representations first studied by Kirillov and Reshetikhin. These conjectures are motivated by considerations coming from mathematical physics and had not previously been related to quiver varieties.This research proposal plans to investigate specializations of quantum dimensions of standard modules by understanding such specializations in the context of quiver varieties. This is also likely to connect with the theory of cluster algebras, as suggested by the work of Hernandez-Leclerc. Initial steps in this project will focus on carrying out a review of relevant literature to ensure a proper understanding of the state of the art in this area.The proposal is prompted by the very recent discovery of connections between specializations of quantum dimensions and Euler characteristics of quiver varieties. Thus, any new insights on these connections are likely to contribute to the understanding of both objects.This project falls within the intersection of the EPSRC 'Algebra' and 'Geometry and Topology' research areas. I will be supervised by Professor Kevin McGerty, working at the University of Oxford (Mathematical Institute) with funding coming from an EPSRC Excellence Award. Furthermore, the project has the potential to interact with the work other academics at the institute, such as Prof. Balázs Szendroi.