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

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 is largely determined by a linearly reductive subgroup scheme of SL(2), with the McKay quiver of relative to its standard module being the Gabriel quiver of the principal block . The graphs underlying these quivers are extended Dynkin diagrams of type or , 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.