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.