Branching law for finite subgroups of SL(3,C) and McKay correspondence

Frédéric Butin (1) , Gadi S. Perets (1)

1 ICJ - Institut Camille Jordan

Given a finite subgroup $\Gamma$ 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 asoociate a generalized McKay matrix $C_\Gamma$. Then, generalizing a method used by B. Kostant for $\mathbf{SL}_2\mathbb{C}$, we decompose $C_\Gamma$ as a sum of products of reflections associated to mutually orthogonal roots: this is a sort of algebraic McKay correspondence in dimension $3$.

Mots clés: branching law; McKay correspondence; finite groups; generalized Cartan matrix; Coxeter element

Domaines: Théorie des représentations [math.RT]

Frédéric Butin, Gadi S. Perets. Branching law for finite subgroups of SL(3,C) and McKay correspondence. Journal of Group Theory, 2013, 17 (2), pp.191-251. ⟨10.1515/jgt-2013-0033⟩. ⟨hal-00412643⟩