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arXiv · 2410.17860

Kleinian singularities: some geometry, combinatorics and representation theory

Lukas Bertsch·Ádám Gyenge·Balázs Szendrői
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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.

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Lukas Bertsch, Ádám Gyenge, Balázs Szendrői. 2024-10-23. Kleinian singularities: some geometry, combinatorics and representation theory. https://arxiv.org/abs/2410.17860

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