Connections between Singularities and Representation Theory
Lead Research Organisation:
University of Leeds
Department Name: Pure Mathematics
Abstract
The aim of this thesis project is to find new links between singularity theory and representation theory. The main objects of study will be singularities of algebraic varieties coming from group actions (e.g., quotient singularities and discriminants of reflection groups) and categories associated to them, in particular, the category of maximal Cohen-Macaulay modules and cluster categories.
These categories will be investigated with homological methods as well as by combinatorial means, e.g., by studying and classifying McKay graphs associated to the singularities. Ultimately, these categorical invariants should yield new insights into the geometry of the singularities. In particular, they should help to understand the McKay correspondence for reflection groups.
These categories will be investigated with homological methods as well as by combinatorial means, e.g., by studying and classifying McKay graphs associated to the singularities. Ultimately, these categorical invariants should yield new insights into the geometry of the singularities. In particular, they should help to understand the McKay correspondence for reflection groups.
Organisations
| Description | Outcomes have been purely academic. Focus has been on the specific complex reflection groups G(m,p,2) in the wildly known classification of complex reflection groups (by Chevalley-Shepard-Todd) . The main object of study is the discriminant of these groups and the hyperplane arrangement. The discriminant is a singularity which we want to study by using the representations of the group. The main outcome is that we have shown that, for these specific groups, viewing the hyperplane arrangement as a Cohen-Macaulay module over the discriminant gives rise to a non-commutative resolution (NCR) for the discriminant. This is an extension of the result by Buchweitz-Faber-Ingalls in which they show that this is true when considering the true reflection group (that is when the complex reflection group is generated by reflections of order 2). This was done by giving a complete decomposition of the hyperplane arrangement as a Cohen-Macaulay module, via matrix factorizations, using the data of irreducible representations of the group. |
| Exploitation Route | The results so far could be expanded into a larger set of groups, namely all the expectional groups in the classification. Expanding this could lead to greater insight into how to generalise the McKay correspondence to all pseudo reflection groups, in particular the information included in the outcomes suggest that certain representations of the groups hold information about the discriminant. Others might be able to use the outcomes to help study the hyperplane arrangement which is studied in different fields, for example mirror symmetry and algebraic geometry. |
| Sectors | Other |