McKay correspondence and the branching law for finite subgroups of $\mathbf{SL}_3\mathbb{C}$

TL;DR

This paper extends the McKay correspondence to finite subgroups of SL(3,C) by analyzing representation decompositions and associating a generalized Cartan matrix, linking algebraic structures to geometric group actions.

Contribution

It introduces a new algebraic framework for the McKay correspondence in three dimensions using Coxeter elements and generalized Cartan matrices.

Findings

Decomposition of SL(3,C) representations under finite subgroup actions

Construction of a generalized Cartan matrix for each subgroup

Expression of Coxeter elements as products of special reflections

Abstract

Given $\Gamma$ a finite subgroup of $\mathbf{SL}_3\mathbb{C}$, we determine how an arbitrary finite dimensional irreducible representation of $\mathbf{SL}_3\mathbb{C}$ decomposes under the action of $\Gamma$. To the subgroup $\Gamma$ we attach a generalized Cartan matrix $C_\Gamma$. Then, inspired by B. Kostant, we decompose the Coxeter element of the Kac-Moody algebra attached to $C_\Gamma$ as a product of reflections of a special form, thereby suggesting an algebraic form for the McKay correspondence in dimension 3.