Celebratio Mathematica — Benkart

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J. M. Barnes, G. Benkart, and T. Halverson: “McKay centralizer algebras,” Proc. Lond. Math. Soc. (3) 112 : 2 (2016), pp. 375–414. MR 3471253 Zbl 1332.05147 ArXiv 1312.5254 article

For a finite subgroup \( \mathbf{G} \) of the special unitary group \( \mathrm{SU}_2 \), we study the centralizer algebra \[ Z_k(\mathbf{G}) = \operatorname{End}_{\mathbf{G}}(V^{\otimes k}) \] of \( \mathbf{G} \) acting on the \( k \)-fold tensor product of its defining representation \( \mathbf{V} = \mathbb{C}^2 \). These subgroups are in bijection with the simply laced affine Dynkin diagrams. The McKay correspondence relates the representation theory of these groups to the associated Dynkin diagram, and we use this connection to show that the structure and representation theory of \( \mathbf{Z}_k(\mathbb{G}) \) as a semisimple algebra is controlled by the combinatorics of the corresponding Dynkin diagram.

Jeffrey M. Barnes	Related
Thomas Michael Halverson	Related