Mutation Game

The Code

  • Getting Started
    • Installation
    • Basic Usage
      • Create a game from a Dynkin diagram
      • Create a game from an adjacency matrix
      • Performing mutations
      • Computing the root system
      • Finding mutation paths
      • Mutation path table
    • Plotting
      • Positive roots (default)
      • All roots
      • Larger examples
  • Explorations
    • Simply-Laced (Simple Graphs)
      • Type A – Path Graphs
      • Type D – Forked Graphs
      • Type E – Exceptional Diagrams
    • Non-Simply-Laced (Directed Multigraphs)
      • Type B – Odd Orthogonal
      • Type C – Symplectic
      • Types F and G – Exceptional Multigraphs
  • API Reference
    • MutationGame
      • MutationGame
    • Singularity Analysis
      • milnor_number()
      • corank()
      • classify()
      • splitting_lemma()
      • jet_reduce()
      • homotopy_equivalence()

The Mathematics

  • Mathematical Background
    • The mutation game
    • Root systems
      • Roots vs. bases
      • The Weyl group as a group of symmetries
    • Finite and infinite types
    • Where does the Cartan matrix come from?
    • The Cartan matrix and finite-type classification
      • Eigenvectors of the Cartan matrix
    • Connection to Lie theory
    • Why not embed first?
    • The mutation graph
    • References
  • Weyl Groups
    • 1. General properties
    • 2. Summary table
    • 3. Worked examples
      • A2: the symmetric group S3
      • G2: the dihedral group D6
      • F4: generators, relations, and Coxeter element
      • D4: triality
    • 4. Recovering the Dynkin diagram from the Weyl group
  • From Platonic Solids to Dynkin Diagrams
    • 1. Euler’s Formula and Regular Polyhedra
    • 2. Enumerating Solutions
    • 3. The Rotation Groups
    • 4. The McKay Correspondence: From SO(3) to SU(2) to Dynkin
    • 5. Spherical Triangle Groups
    • 6. The Unified Picture
      • Root systems as geometric shapes
    • 7. The du Val Singularities
    • References
  • From Singularities to Dynkin Diagrams
    • 1. The Setup: Critical Points of Smooth Functions
    • 2. Equivalence of Singularities
      • How do we prove inequivalence?
      • How do we construct the coordinate change explicitly?
    • 3. Simple Singularities
    • 4. Why ADE? The Milnor Fiber and the Intersection Form
      • The Milnor Fiber
      • Vanishing Cycles
      • The Intersection Form
      • The Result
    • 5. Worked Example: A3 in Detail
      • The general case
    • 6. Worked Example: D4 in Detail
      • The general case
    • 7. The E-type Singularities
    • 8. Thom’s Seven Catastrophes
    • 9. Summary: The Chain of Connections
    • References
  • Proof of the ADE Classification
    • The spectral criterion
    • Step 1: The graph must be a tree
    • Step 2: Maximum degree at most 3
    • Step 3: At most one branch point
    • Step 4: No branch point gives An
    • Step 5: The Diophantine inequality
    • Step 6: Enumerating solutions
    • The complete classification
    • Connections
      • References
Mutation Game
  • Index

Index

C | F | H | I | J | M | P | S

C

  • calculate_roots() (mutation_game.MutationGame method)
  • cartan_eigenvalues() (mutation_game.MutationGame method)
  • cartan_eigenvectors() (mutation_game.MutationGame method)
  • cartan_matrix() (mutation_game.MutationGame method)
  • classify() (in module mutation_game)
  • corank() (in module mutation_game)

F

  • find_mutation_path() (mutation_game.MutationGame method)
  • from_dynkin() (mutation_game.MutationGame class method)

H

  • homotopy_equivalence() (in module mutation_game)

I

  • is_finite_type() (mutation_game.MutationGame method)

J

  • jet_reduce() (in module mutation_game)

M

  • milnor_number() (in module mutation_game)
  • mutate() (mutation_game.MutationGame method)
  • mutation_matrix() (mutation_game.MutationGame method)
  • mutation_path_table() (mutation_game.MutationGame method)
  • MutationGame (class in mutation_game)

P

  • plot_root_orbits() (mutation_game.MutationGame method)
  • print_mutation_path_table() (mutation_game.MutationGame method)

S

  • set_starting_population() (mutation_game.MutationGame method)
  • splitting_lemma() (in module mutation_game)

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