17 July 2020 to 14 August 2020
Kavli IPMU, Kashiwa, Japan
Asia/Tokyo timezone

8/14 [Louis-Philippe Thibault ] Tilting objects in singularity categories and levelled mutations

Not scheduled
Online with registration by Aug. 14.

Online with registration by Aug. 14.



In 1989, Reiten and Van den Bergh showed that for every finite subgroup G of SL(2,k), the skew-group algebra k[x,y]#G is Morita equivalent to the preprojective algebra over the extended Coxeter-Dynkin quiver associated to G via the McKay correspondence, thus providing another bridge between Kleinian singularities and representation theory. In the context of Iyama’s higher Auslander-Reiten theory, it is natural to ask whether the same holds true for finite subgroups of SL(n,k) and higher preprojective algebras. In the first part of this talk, I will give a class of subgroups for which the skew-group algebra is not Morita equivalent to a higher preprojective algebra.

We will then move on to study the graded singularity category over the invariant ring k[x_1,…,x_n]^G. When the skew-group algebra is endowed with a grading giving it the structure of a preprojective algebra, Amiot, Iyama and Reiten showed that this category admits a tilting object. In the second part of this talk, we will be motivated by the case where the skew-group algebra does not admit such grading structure. We will explain that, in certain situations, one can use levelled mutations to obtain tilting objects in the graded singularity category.

Presentation Materials