Abstract

Felix Klein classified the finite subgroups of SL(2,CC) around 1860; there are two infinite families corresponding to regular polygons in the plane, together with three exceptional groups of order 24, 48 and 120 that are “spinor” double covers of the symmetry groups of the regular polyhedra (the tetrahedron, octahedron and icosahedron). The finite subgroups of SL(3,CC) are also classified (and also SL(n,CC) for higher n), although the problem gets harder and it is not clear how to view the assortment of solutions with any pretence to elegance. The quotient spaces X = CC^2/G by Klein’s finite subgroups G in SL(2,CC) form a very remarkable family of isolated surface singularities, that were studied by Du Val during the 1930s (aided by Coxeter). Du Val’s work was central to the study of algebraic surfaces during the 1970s and 1980s, and played a foundational role in the study of algebraic 3-folds from the 1980s onwards. In the 1980s McKay observed that the representation theory of the group G is reflected in the geometry of the resolution of singularities of X. This correspondence has been generalised to 3-dimensions, with the same proviso concerning the nature of the problem and its solutions.