Contents Menu Expand Light mode Dark mode Auto light/dark, in light mode Auto light/dark, in dark mode Skip to content
Thematic Tutorials
Light Logo Dark Logo
Version 10.6 Documentation
  • Home - Thematic Tutorials
  • Thematic tutorial document tree
    • Algebraic Combinatorics in Sage
      • Walks in graphs
      • \(n\)-Cube
      • The Tsetlin library
      • Young’s lattice and the RSK algorithm
    • Abelian Sandpile Model
    • Group Theory and Sage
    • Lie Methods and Related Combinatorics in Sage
      • The Scope of this Document
      • Lie Group Basics
      • The Weyl Character Ring
      • Maximal Subgroups and Branching Rules
      • Weyl Groups, Coxeter Groups and the Bruhat Order
      • Classical Crystals
      • Affine Root System Basics
      • Integrable Highest Weight Representations of Affine Lie algebras
      • Affine Finite Crystals
      • Affine Highest Weight Crystals
      • Elementary crystals
      • Infinity Crystals
      • Iwahori Hecke Algebras
      • Kazhdan-Lusztig Polynomials
      • Bibliography
    • Tutorial: Vector Calculus in Euclidean Spaces
      • How to compute a gradient, a divergence or a curl
      • How to perform vector calculus in curvilinear coordinates
      • How to change coordinates
      • Advanced aspects: the Euclidean space as a Riemannian manifold
      • Vector calculus in the Euclidean plane
    • Linear Programming (Mixed Integer)
    • Number Theory and the RSA Public Key Cryptosystem
    • Coding Theory in Sage
    • How to write your own classes for coding theory
    • Polyhedra
      • A Brief Introduction to Polytopes in Sage
      • A Longer Introduction to Polyhedral Computations in Sage
      • Quick reference for polyhedra in Sage
      • Polyhedra tips and tricks
      • Visualization of polyhedron objects in Sage
      • Draw polytopes in LaTeX using TikZ
    • Steenrod Algebra Modules
    • Tutorial: Programming in Python and Sage
    • Tutorial: Comprehensions, Iterators, and Iterables
    • Tutorial: Objects and Classes in Python and Sage
    • Functional Programming for Mathematicians
    • How to implement new algebraic structures in Sage
    • Tutorial: Implementing Algebraic Structures
    • How to call a C code (or a compiled library) from Sage ?
    • Numerical Computing with Sage
      • Numerical Tools
        • NumPy
        • SciPy
        • Cvxopt
      • Using Compiled Code Interactively
        • f2py
        • More Interesting Examples with f2py
        • Ctypes
        • More complicated ctypes example
        • Comparison to Cython/Pyrex
      • Parallel Computation
        • mpi4py
        • Parallel Laplace Solver
    • Three Lectures about Explicit Methods in Number Theory Using Sage
      • Introduction
      • Number Fields
        • Introduction to Number Fields
        • Number Fields: Galois Groups and Class Groups
        • Orders and Relative Extensions
      • A Bird’s Eye View
        • Integer Factorization
        • Elliptic Curves
        • The Matrix of Frobenius on Hyperelliptic Curves
        • Modular Symbols
        • Enumerating Totally Real Number Fields
        • Bernoulli Numbers
        • Polynomial Arithmetic
      • Modular Forms
        • Modular Forms and Hecke Operators
        • Modular Symbols
        • Method of Graphs
        • Level One Modular Forms
        • Half Integral Weight Forms
        • Generators for Rings of Modular Forms
        • \(L\)-series
        • Modular Abelian Varieties
    • Profiling in Sage
    • Creating a Tutorial from an old Sage Worksheet (.sws)
Back to top
View this page
Edit this page

Lie Methods and Related Combinatorics in Sage¶

Author: Daniel Bump (Stanford University), Ben Salisbury (Central Michigan University), and Anne Schilling (UC Davis)

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
    • 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 Cartan matrix
    • Fundamental weights and the Weyl vector
    • Representations and characters
    • Representations: an example
    • Partitions and Schur polynomials
    • Affine Cartan types
    • The affine root and the extended Dynkin diagram
    • Twisted affine root systems
    • Further Generalizations
  • The Weyl Character Ring
    • Weyl character rings
    • Methods of the ambient space
    • Methods of the Weyl character ring
    • Coroot notation
    • Tensor products of representations
    • Weight multiplicities
    • Example
    • Frobenius-Schur indicator
    • Symmetric and exterior powers
    • Weyl dimension formula
    • SL versus GL
    • Integration
    • Invariants and multiplicities
    • Weight Rings
  • Maximal Subgroups and Branching Rules
    • Branching rules
    • What’s in a branching rule?
    • Maximal subgroups
    • Levi subgroups
    • Subgroups classified by the extended Dynkin diagram
    • Levi subgroups of \(G_2\)
    • Orthogonal and symplectic subgroups of orthogonal and symplectic groups
    • Non-maximal Levi subgroups and Projection from Reducible Types
    • Symmetric subgroups
    • Tensor products
    • Symmetric powers
    • Plethysms
    • Miscellaneous other subgroups
    • Maximal A1 subgroups of Exceptional Groups
    • Writing your own branching rules
    • Automorphisms and triality
  • 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 \(GL(n)\)
    • Frobenius-Schur Duality
    • Counting pairs of tableaux
    • 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
    • Lusztig involution
    • Levi branching rules for crystals
    • Subcrystals
  • Affine Root System Basics
    • Cartan Matrix
    • Untwisted Affine Kac-Moody Lie Algebras
    • Twisted Types
    • Roots and Weights
    • Affine Root System and Weyl Group
    • Labels and Coxeter Number
    • The Weyl Group
    • The extended affine Weyl Group
  • Integrable Highest Weight Representations of Affine Lie algebras
    • The affine case
    • The support of an integrable highest weight representation
    • Modular Forms
    • Sage methods for integrable representations
  • Affine Finite Crystals
    • Type \(A_n^{(1)}\)
    • Types \(D_n^{(1)}\), \(B_n^{(1)}\), \(A_{2n-1}^{(2)}\)
    • Type \(C_n^{(1)}\)
    • Types \(D_{n+1}^{(2)}\), \(A_{2n}^{(2)}\)
    • Exceptional nodes
    • Type \(E_6^{(1)}\)
    • Single column KR crystals
    • Applications
    • Energy function and one-dimensional configuration sum
  • Affine Highest Weight Crystals
    • Littelmann path model
    • Modified Nakajima monomials
  • Elementary crystals
    • T-crystal
    • C-crystal
    • R-crystal
    • \(i\)-th elementary crystal
  • Infinity Crystals
    • Marginally large tableaux
    • Generalized Young walls
    • Modified Nakajima monomials
    • Rigged configurations
  • Iwahori Hecke Algebras
  • Kazhdan-Lusztig Polynomials
  • Bibliography

Preparation of this document was supported in part by NSF grants DMS-0652817, DMS-1001079, OCI-1147463, DMS–0652641, DMS–0652652, DMS–1001256, OCI–1147247 and DMS-1601026.

Next
The Scope of this Document
Previous
Group Theory and Sage
Copyright © 2005--2025, The Sage Development Team
Made with Sphinx and @pradyunsg's Furo