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Polyhedral groups, McKay quivers, and the finite algebraic groups with tame principal blocks

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Abstract

Given an algebraically closed field k of characteristic p≥3, we classify the finite algebraic k-groups whose algebras of measures afford a principal block of tame representation type. The structure of such a group \(\mathcal{G}\) is largely determined by a linearly reductive subgroup scheme \(\hat{\mathcal{G}}\) of SL(2), with the McKay quiver of \(\hat{\mathcal{G}}\) relative to its standard module being the Gabriel quiver of the principal block \(\mathcal{B}_0(\mathcal{G})\). The graphs underlying these quivers are extended Dynkin diagrams of type \(\tilde{A}, \tilde{D}\) or \(\tilde{E}\), and the tame blocks are Morita equivalent to generalizations of the trivial extensions of the radical square zero tame hereditary algebras of the corresponding type.

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  1. Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, 33501, Bielefeld, Germany

    Rolf Farnsteiner

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Farnsteiner, R. Polyhedral groups, McKay quivers, and the finite algebraic groups with tame principal blocks. Invent. math. 166, 27–94 (2006). https://doi.org/10.1007/s00222-006-0506-z

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