Lie Methods and Related Combinatorics in Sage

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 groups and algebras
  • Combinatorics
• 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
  • Weyl Character Rings
  • Methods of the ambient space
  • Methods of the Weyl Character Ring
  • Coroot Notation
  • Tensor products of representations
  • Weights
  • Frobenius-Schur indicator
  • Symmetric and Exterior Square
  • Weyl Dimension Formula
  • SL versus GL
  • Caching
• 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 and affine Weyl Groups
  • Affine Weyl Groups
  • Bruhat order
  • The Bruhat Graph
• 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
  • Intertwining Operators
  • Intertwining Operator Program
• Kazhdan-Lusztig Polynomials