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

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

2005-03-24

arxiv.org/abs/math/0503542v1

Manuscripta mathematica 122 (2007), no. 1, 1-21

Mathematics

Representation Theory

19 pages

Scientific paper

10.1007/s00229-006-0055-1

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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