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Kleinian singularities: some geometry, combinatorics and representation theory

Lukas Bertsch, Ádám Gyenge, Balázs Szendrői

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.

Kleinian singularities: some geometry, combinatorics and representation theory

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.

Paper Structure

This paper contains 17 sections, 26 theorems, 70 equations, 11 figures.

Table of Contents

  1. The basic characters of our story
  2. Some groups of rotations
  3. Invariant theory and algebraic geometry
  4. Equivariant geometry and the McKay Quiver
  5. Hilbert and Quot schemes
  6. The abelian case: partition combinatorics
  7. Combinatorics around labelled partitions
  8. Equivariant Hilbert schemes and labelled partitions
  9. Some abelian groups in $\mathop{\mathrm{GL}}\nolimits(2,\mathop{\mathrm{\mathbb{C}}}\nolimits)$
  10. Quot schemes and specialised generating functions
  11. Representation theory around coloured partitions
  12. The non-abelian cases
  13. Type D: the challenge
  14. Type D: Young walls
  15. Stratifying the equivariant Hilbert scheme in Type D
  16. ...and 2 more sections

Key Result

Theorem 1.1

The following is a complete list of finite subgroups of $\mathop{\mathrm{SL}}\nolimits(2,\mathop{\mathrm{\mathbb{C}}}\nolimits)$ up to conjugation.

Figures (11)

  • Figure 1: The real locus of the singular surface $u^2 - w^2 - v^3=0$
  • Figure 2: Young diagram of the partition $\lambda=(4,2,2,1)$ of $n=9$
  • Figure 3: The pattern of type $A_r$: periodic labelling of $\mathop{\mathrm{\mathbb{N}}}\nolimits\times \mathop{\mathrm{\mathbb{N}}}\nolimits$ with $(r+1)$ labels
  • Figure 4: Young diagram of the partition $\lambda=(4,2,2,1)$ labelled by $I=\mathop{\mathrm{\mathbb{Z}}}\nolimits/3$, of multiweight (4,2,3). The gray boxes form the only removable border strip of length $3$
  • Figure 5: Periodic labelling of $\mathop{\mathrm{\mathbb{N}}}\nolimits\times \mathop{\mathrm{\mathbb{N}}}\nolimits$ with a pair of labels mod $2$
  • ...and 6 more figures

Theorems & Definitions (38)

  • Theorem 1.1: Klein Klein
  • Theorem 1.2: Klein Klein
  • Example 1.3
  • Theorem 1.4
  • Example 1.5
  • Definition 1.6
  • Definition 1.7
  • Proposition 1.8
  • Proposition 1.9
  • Theorem 1.10
  • ...and 28 more