Combinatorics of Minuscule Representations

Uses the combinatorics and representation theory to construct and study important families of Lie algebras and Weyl groups.

R. M. Green (Author)

9781107026247, Cambridge University Press

Hardback, published 21 February 2013

330 pages, 70 b/w illus. 7 tables 275 exercises
23.1 x 15.5 x 2.3 cm, 0.61 kg

'… useful as a book for students, the book under review is particularly useful as a reference for researchers of relevant fields.' Qendrim R. Gashi, Zentralblatt MATH

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.

Introduction
1. Classical Lie algebras and Weyl groups
2. Heaps over graphs
3. Weyl group actions
4. Lie theory
5. Minuscule representations
6. Full heaps over affine Dynkin diagrams
7. Chevalley bases
8. Combinatorics of Weyl groups
9. The 28 bitangents
10. Exceptional structures
11. Further topics
Appendix A. Posets, graphs and categories
Appendix B. Lie theoretic data
References
Index.

Subject Areas: Mathematical physics [PHU], Combinatorics & graph theory [PBV], Algebraic topology [PBPD], Algebra [PBF]