Mutation Game

A Python library for exploring root systems through the combinatorial mutation game on graphs and directed multigraphs. Given a Dynkin diagram name (A3, D5, E8, B4, C3, F4, G2, …) or an arbitrary adjacency matrix, the library computes the full root system via iterated mutations and visualizes the mutation graph.

The Code

• Getting Started
  • Installation
  • Basic Usage
  • Plotting
• Explorations
  • Simply-Laced (Simple Graphs)
  • Non-Simply-Laced (Directed Multigraphs)
• API Reference
  • MutationGame
  • Singularity Analysis

The Mathematics

• Mathematical Background
  • The mutation game
  • Root systems
  • Finite and infinite types
  • Where does the Cartan matrix come from?
  • The Cartan matrix and finite-type classification
  • Connection to Lie theory
  • Why not embed first?
  • The mutation graph
  • References
• Weyl Groups
  • 1. General properties
  • 2. Summary table
  • 3. Worked examples
  • 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
  • 7. The du Val Singularities
  • References
• From Singularities to Dynkin Diagrams
  • 1. The Setup: Critical Points of Smooth Functions
  • 2. Equivalence of Singularities
  • 3. Simple Singularities
  • 4. Why ADE? The Milnor Fiber and the Intersection Form
  • 5. Worked Example: A₃ in Detail
  • 6. Worked Example: D₄ in Detail
  • 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 A_n
  • Step 5: The Diophantine inequality
  • Step 6: Enumerating solutions
  • The complete classification
  • Connections