Kleinian Singularities: Some Geometry, Combinatorics and Representation Theory

We review the relationship between discrete groups of symmetries of Euclidean three-space, constructions in algebraic geometry around Kleinian singularities including versions of Hilbert and Quot schemes, and their relationship to finite-dimensional and affine Lie algebras via the McKay correspondence. We focus on combinatorial aspects, such as the enumeration of certain types of partition-like objects, reviewing in particular a recently developed root-of-unity-substitution calculus. While the most complete results are in type $A$ $A$ , we also develop aspects of the theory in type $D$ $D$ , and end with some questions about the exceptional type $E$ $E$ cases.