TY - JOUR
T1 - Kleinian Singularities
T2 - Some Geometry, Combinatorics and Representation Theory
AU - Bertsch, Lukas
AU - Gyenge, Ádám
AU - Szendrői, Balázs
N1 - Publisher Copyright:
© The Author(s) 2024.
PY - 2024/12
Y1 - 2024/12
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85210154502&partnerID=8YFLogxK
U2 - 10.1365/s13291-024-00291-5
DO - 10.1365/s13291-024-00291-5
M3 - Article
AN - SCOPUS:85210154502
SN - 0012-0456
VL - 126
SP - 213
EP - 247
JO - Jahresbericht der Deutschen Mathematiker-Vereinigung
JF - Jahresbericht der Deutschen Mathematiker-Vereinigung
IS - 4
ER -