Abstract
Recently, J. McKay [7] has observed that the irreducible complex representations of the binary polyhedral groups can be arranged in order to form the vertices of a Euclidean diagram in such a way that the tensor product of any irreducible representation M with the standard two-dimensional representation is the direct sum of the irreducible representations which are the neighbors of M in the diagram, and he asked for an explanation. In this note, we will show that any self-dual two-dimensional representation gives rise to a generalized Euclidean diagram, and that this in fact can be used to give a proof of the classification theorem of the binary polyhedral groups which at the same time furnishes a list of the irreducible representations and also gives the minimal splitting field.
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Happel, D., Preiser, U. & Ringel, C.M. Binary polyhedral groups and Euclidean diagrams. Manuscripta Math 31, 317–329 (1980). https://doi.org/10.1007/BF01303280
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DOI: https://doi.org/10.1007/BF01303280