Work

Kleinian Singularities: Some Geometry, Combinatorics and Representation Theory

Year: 2024

Type: article

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

Source: Jahresbericht der Deutschen Mathematiker-Vereinigung

DOI: 10.1365/s13291-024-00291-5

Authors: Lukas Bertsch, Ádám Gyenge, Balázs Szendrői

Institutions: University of Vienna, Budapest University of Technology and Economics

Language: English

Topic: Algebraic structures and combinatorial models

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,090

Funders: Universität Wien, Magyar Tudományos Akadémia