Kleinian singularities: some geometry, combinatorics and representation theory

TL;DR

This paper explores the rich interplay between Kleinian singularities, algebraic geometry, combinatorics, and Lie theory, emphasizing the McKay correspondence and root-of-unity calculus, with focus on types A, D, and E.

Contribution

It reviews and extends the understanding of Kleinian singularities, connecting geometric, combinatorial, and algebraic structures, including new insights into type D and open questions in type E.

Findings

Enumerated partition-like objects related to Kleinian singularities

Developed root-of-unity substitution calculus for combinatorial enumeration

Extended aspects of the theory to type D and posed questions for type E

Abstract

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, we also develop aspects of the theory in type D, and end with some questions about the exceptional type E cases.