Type E – Exceptional Diagrams
The exceptional Dynkin diagrams \(E_6\), \(E_7\), and \(E_8\) are the three “sporadic” simply-laced diagrams. Each is a path with a single branch off the third node:
These correspond to the exceptional simple Lie algebras \(\mathfrak{e}_6\), \(\mathfrak{e}_7\), and \(\mathfrak{e}_8\). No other E-type diagrams produce finite root systems.
E6
36 positive roots, 72 total. The Lie algebra \(\mathfrak{e}_6\) has dimension 78 (72 root vectors + 6 Cartan generators).
>>> from mutation_game import MutationGame
>>> game = MutationGame.from_dynkin("E6")
>>> print(game.adj)
[[0 1 0 0 0 0]
[1 0 1 0 0 0]
[0 1 0 1 0 1]
[0 0 1 0 1 0]
[0 0 0 1 0 0]
[0 0 1 0 0 0]]
Node 2 is the branch point, connected to nodes 1, 3, and 5.
The Cartan spectrum:
>>> print(game.cartan_eigenvalues())
[0.26794919 1. 1.26794919 2.73205081 3. 3.73205081]
E6 has a \(\mathbb{Z}/2\mathbb{Z}\) diagram automorphism (the diagram is symmetric under reflection), giving it a non-trivial outer automorphism.
E7
63 positive roots, 126 total. The Lie algebra \(\mathfrak{e}_7\) has dimension 133 (126 root vectors + 7 Cartan generators).
>>> game = MutationGame.from_dynkin("E7")
>>> print(game.cartan_eigenvalues())
[0.19806226 0.75487767 1.55495813 2. 2.80193774 3.24512233
3.44504187]
The highest root in E7 has height 17 (the sum of its coordinates), making the mutation graph 17 levels deep.
E8
120 positive roots, 240 total. \(E_8\) is the largest exceptional root system and one of the most remarkable objects in mathematics. The Lie algebra \(\mathfrak{e}_8\) has dimension 248 (240 root vectors + 8 Cartan generators).
>>> game = MutationGame.from_dynkin("E8")
>>> print(game.cartan_eigenvalues())
[0.12061476 0.47213595 1. 1.57357644 2.34729636 2.87938524
3.52786405 3.87938524]
>>> roots = game.calculate_roots()
>>> pos = [r for r in roots if all(x >= 0 for x in r)]
>>> highest = max(pos, key=lambda r: sum(r))
>>> print(list(map(int, highest)))
[2, 4, 6, 5, 4, 3, 2, 3]
The highest root \((2, 4, 6, 5, 4, 3, 2, 3)\) has height 29. The Weyl group of \(E_8\) has order 696,729,600.