Critical Groups of Mckay-cartan Matrices

This thesis investigates the critical groups of McKay-Cartan matrices, a certain type of avalanche-finite matrix associated to a faithful representation γ of a finite group G. It computes the order of the critical group in terms of the character values of γ, and gives some restrictions on its subgroup structure. In addition, the existence of a certain Smith normal form over Z[t] is shown to imply a nice form for the critical group. This is used to compute the critical group for the reflection representation of Sn. In the case where im(γ) ⊂ SL(n,C) it discusses for which pairs (G, γ) the critical group is isomorphic to the abelianization of G, including explicit calculations demonstrating these isomorphisms for the finite subgroups of SU(2) and SO(3,R). In this case it also identifies a subset of the superstable configurations, answering a question posed in [3]. Submitted under the supervision of Professor Victor Reiner to the University Honors Program at the University of Minnesota–Twin Cities in partial fulfilment of the requirements for the degree of Bachelor of Science summa cum laude in Mathematics. Date: June 12, 2016. 1