McKay Matrices and Quivers

Professor Georgia Benkart, University of Wisconsin-Madison. Public lecture in frame of the workshop WINART2.

The McKay correspondence is a bijection between the finite subgroups G of SU2 and the simply-laced affine Dynkin diagrams. McKay’s insight was that the quivers determined by tensoring the simple modules of G with the G-module V = C2 exactly correspond to the affine Dynkin diagrams of type A, D, E.

This talk will focus on the McKay correspondence from the point of view of Schur-Weyl duality for any finite group G (or finite-dimensional Hopf algebra) and any finite-dimensional G-module V. This provides results on the tensor product module V⊗k and its G-invariants, and on the centralizer algebra EndG (V⊗k), which often has a nice diagrammatic realization.

It leads (Leeds) to the investigations that our WINART2 group has begun. There are applications to sub-factors of operator algebras, knot and link invariants, and the Potts model in statistical mechanics, and more recently to chip-firing dynamics and Markov chains.