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

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.

32 pages

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