Quantum affine Cartan matrices,Poincaré series of binary polyhedral groups,and reflection representations |
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Authors: | Ruedi Suter |
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Institution: | (1) Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland |
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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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Keywords: | 20C15 20F55 |
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