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Quantum affine Cartan matrices, Poincaré series of binary polyhedral groups, and reflection representations

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Abstract

We first review some invariant theoretic results about the finite subgroups of SU(2) in a quick algebraic way by using the McKay correspondence and quantum affine Cartan matrices. By the way it turns out that some parameters (a, b, h; p, q, r) that one usually associates with such a group and hence with a simply-laced Coxeter–Dynkin diagram have a meaningful definition for the non-simply-laced diagrams, too, and as a byproduct we extend Saito’s formula for the determinant of the Cartan matrix to all cases. Returning to invariant theory we show that for each irreducible representation i of a binary tetrahedral, octahedral, or icosahedral group one can find a homomorphism into a finite complex reflection group whose defining reflection representation restricts to i.

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  1. Department of Mathematics, ETH Zurich, 8092, Zurich, Switzerland

    Ruedi Suter

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  1. Ruedi Suter
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Suter, R. Quantum affine Cartan matrices, Poincaré series of binary polyhedral groups, and reflection representations. manuscripta math. 122, 1–21 (2007). https://doi.org/10.1007/s00229-006-0055-1

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  • DOI: https://doi.org/10.1007/s00229-006-0055-1

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