You are previewing Combinatorics of Minuscule Representations.
O'Reilly logo
Combinatorics of Minuscule Representations

Book Description

Minuscule representations occur in a variety of contexts in mathematics and physics. They are typically much easier to understand than representations in general, which means they give rise to relatively easy constructions of algebraic objects such as Lie algebras and Weyl groups. This book describes a combinatorial approach to minuscule representations of Lie algebras using the theory of heaps, which for most practical purposes can be thought of as certain labelled partially ordered sets. This leads to uniform constructions of (most) simple Lie algebras over the complex numbers and their associated Weyl groups, and provides a common framework for various applications. The topics studied include Chevalley bases, permutation groups, weight polytopes and finite geometries. Ideal as a reference, this book is also suitable for students with a background in linear and abstract algebra and topology. Each chapter concludes with historical notes, references to the literature and suggestions for further reading.

Table of Contents

  1. Cover
  2. Cambridge Tracts in Mathematics
  3. Title Page
  4. Copyright
  5. Table of Contents
  6. Introduction
  7. Chapter 1: Classical Lie algebras and Weyl groups
    1. 1.1 Lie algebras
    2. 1.2 The classical Lie algebras
    3. 1.3 Classical Lie algebras and partially ordered sets
    4. 1.4 Classical Weyl groups and partially ordered sets
    5. 1.5 Notes and references
  8. Chapter 2: Heaps over graphs
    1. 2.1 Basic definitions
    2. 2.2 Full heaps over Dynkin diagrams
    3. 2.3 Local structure of full heaps
    4. 2.4 Quotient heaps
    5. 2.5 Notes and references
  9. Chapter 3: Weyl group actions
    1. 3.1 Linear operators and group actions
    2. 3.2 Proper ideals
    3. 3.3 Parabolic subheaps
    4. 3.4 Notes and references
  10. Chapter 4: Lie theory
    1. 4.1 Representations of Lie algebras from heaps
    2. 4.2 Review of Lie theory
    3. 4.3 Review of Weyl groups
    4. 4.4 Strongly orthogonal sets
    5. 4.5 Notes and references
  11. Chapter 5: Minuscule representations
    1. 5.1 Highest weight modules
    2. 5.2 Weights and heaps
    3. 5.3 Periodicity and trivialization
    4. 5.4 Reflections
    5. 5.5 Minuscule representations from heaps
    6. 5.6 Invariant bilinear forms
    7. 5.7 Notes and references
  12. Chapter 6: Full heaps over affine Dynkin diagrams
    1. 6.1 Full heaps in type Al(1)
    2. 6.2 Proper ideals in type Al(1)
    3. 6.3 Spin representations in type Dl
    4. 6.4 Types Bl(1) and Dl+1(2)
    5. 6.5 Full heaps in type E6(1) and E7(1)
    6. 6.6 The classification of full heaps over affine Dynkin diagrams
    7. 6.7 Notes and references
  13. Chapter 7: Chevalley bases
    1. 7.1 Kac's asymmetry function
    2. 7.2 Relations in simply laced simple Lie algebras
    3. 7.3 Folding
    4. 7.4 Long and short roots
    5. 7.5 Relations in non-simply laced simple Lie algebras
    6. 7.6 Notes and references
  14. Chapter 8: Combinatorics of Weyl groups
    1. 8.1 Minuscule systems
    2. 8.2 Weyl groups as permutation groups
    3. 8.3 Ideals of roots
    4. 8.4 Weight polytopes
    5. 8.5 Faces of weight polytopes
    6. 8.6 Graphs from minuscule representations
    7. 8.7 Notes and references
  15. Chapter 9: The 28 bitangents
    1. 9.1 The Gosset graph
    2. 9.2 Del Pezzo surfaces
    3. 9.3 Bitangents
    4. 9.4 Hesse–Cayley notation
    5. 9.5 Steiner complexes
    6. 9.6 Symplectic structure
    7. 9.7 Notes and references
  16. Chapter 10: Exceptional structures
    1. 10.1 The 27 lines on a cubic surface
    2. 10.2 Combinatorics of double sixes
    3. 10.3 2-graphs
    4. 10.4 Generalized quadrangles
    5. 10.5 Higher invariant forms
    6. 10.6 Notes and references
  17. Chapter 11: Further topics
    1. 11.1 Minuscule elements of Weyl groups
    2. 11.2 Principal subheaps as abstract posets
    3. 11.3 Gaussian posets
    4. 11.4 Jeu de taquin
    5. 11.5 Notes and references
  18. Appendix A Posets, graphs and categories
  19. Appendix B Lie theoretic data
  20. References
  21. Index