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。最近，Szendroi和合作者对Kleinian奇点的Hilbert格式的欧拉特征进行了研究，对a和D类奇点的欧拉特征的生成序列进行了漂亮的描述，作为仿射李代数表示特征的专门化。他们推测，同样的公式也适用于其余的E型病例。此后，Nakajima证明了这一身份，使用了由Szendroi和他的合作者建立的希尔伯特方案作为某些颤振变体的识别。他证明的关键是，量子仿射代数的某些“标准表示”具有“量子维数”，当量子参数q被取为单位的一个特定根时，该“量子维数”特化为1。这个引人注目的事实是由中岛通过一些明确的个案计算得出的。然而，这些专门化是通过对Kirillov和Reshetikhin首先研究的一类特定表征的量子维度专门化的猜想来预测的。这些猜想的动机是来自数学物理的考虑，以前没有与颤振品种有关。本研究计划通过在颤振变化的背景下理解这种专门化来研究标准模块的量子维度专门化。正如Hernandez-Leclerc的工作所提出的那样，这也可能与簇代数理论有关。该项目的最初步骤将侧重于对相关文献进行审查，以确保对这一领域的最新进展有适当的了解。这个提议是由最近发现的量子维度的专门化和颤振品种的欧拉特性之间的联系而提出的。因此，任何关于这些联系的新见解都可能有助于对这两个对象的理解。该项目属于EPSRC“代数”和“几何与拓扑”研究领域的交叉点。我将由Kevin McGerty教授指导，他在牛津大学（数学研究所）工作，资金来自EPSRC优秀奖。此外，该项目有可能与研究所其他学者的工作进行互动，例如Balázs Szendroi教授。

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.