Towards a McKay Correspondence for Lie Superalgebra gl(m|n)

Abstract

This is from joint work in progress with Mina Aganagic, Jinghang Miao, Spencer Tamagni and Peng Zhou. The classical McKay correspondence started from the miracle that there is a bijection between type ADE irreducible root systems and finite subgroups of SU(2) up to conjugation. Under the correspondence, the symmetry groups of the platonic solids correspond to E_6, E_7 and E_8 root systems. If we let the finite subgroup act on the two-dimensional complex vector space via their embedding in SU(2) and take the quotient of this action, we get isolated surface singularities called ADE singularities. We will first review how the ADE root system can be reconstructed from minimal resolutions and semi-universal deformations of these singularities. Then we'll describe a possible way to extend this to the case of Lie superalgebra gl(m|n). It turns out that the geometry here is the 3-fold singularity xy = z^mw^n. As a 3-fold singularity, it has more than one minimal resolution. This corresponds to the fact that gl(m|n) has many inequivalent root systems.