Quiver varieties and quantum dimensions

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.