These notes explain how to use the mathematical software Sage for Lie group computations. Sage also contains many combinatorial algorithms. We will cover only some of these.

- The Scope of this Document
- Lie Group Basics
- Goals of this Section
- Semisimple and Reductive Groups
- Fundamental Group and Center
- Parabolic subgroups and Levi subgroups
- Cartan Types
- Dual Cartan Types
- Reducible Cartan Types
- Low dimensional Cartan types
- Affine Cartan types
- Relabeled Cartan Types
- Standard realizations of the ambient spaces
- Weights and the ambient space
- The Root System
- The Weyl Group
- The dual root system
- The Dynkin Diagram
- The Affine Root and the Extended Dynkin Diagram
- Fundamental Weights and the Weyl Vector
- Representations and Characters
- Representations: an example
- Partitions and Schur polynomials

- The Weyl Character Ring
- Maximal Subgroups and Branching Rules
- Branching Rules
- Levi Subgroups
- Subgroups Classified by the Extended Dynkin Diagram
- Orthogonal and symplectic subgroups of orthogonal and symplectic groups
- Symmetric Subgroups
- Tensor Products
- Symmetric Powers
- Plethysms
- Miscellaneous other subgroups
- Nuts and Bolts of Branching Rules
- Automorphisms and Triality

- Weight Rings
- Weyl Groups, Coxeter Groups and the Bruhat Order
- Classical Crystals
- Tableaux and Representations of
- Tableaux and representations of
- The Robinson-Schensted-Knuth correspondence
- Analogies between representation theory and combinatorics
- Interpolating between representation theory and combinatorics
- Kashiwara Crystals
- Installing Dot2tex
- Crystals of Tableaux in Sage
- Crystals of Letters
- Tensor Products of Crystals
- Crystals of Tableaux as tensor products of crystals
- Spin Crystals
- Levi Branching Rules for Crystals
- Affine Crystals

- Iwahori Hecke Algebras
- Kazhdan-Lusztig Polynomials

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© Copyright 2010, Daniel Bump.
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