Combinatorics of Minuscule Representations: 199 (Cambridge Tracts in Mathematics, Series Number 199)
Cambridge University Press, 2/21/2013
EAN 9781107026247, ISBN10: 1107026245
Hardcover, 330 pages, 23.1 x 15.5 x 2.3 cm
Language: English
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.
'This monograph could be read with profit not only by the specialist, but also by an interested graduate student with some background on Lie algebras.' Felipe Zaldivar, MAA Reviews (maa.org/publications/maa-reviews)