Kleinian Singularities: Some Geometry, Combinatorics and Representation Theory

Bertsch, Lukas and Gyenge, Ádám and Szendrői, Balázs (2024) Kleinian Singularities: Some Geometry, Combinatorics and Representation Theory. JAHRESBERICHT DER DEUTSCHEN MATHEMATIKER-VEREINIGUNG. ISSN 0012-0456

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Abstract

We review the relationship between discrete groups of symmetries of Euclidean threespace, 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.

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