Type A – Path Graphs
The \(A_n\) Dynkin diagram is a path graph on n nodes:
The root system has \(\frac{n(n+1)}{2}\) positive roots and \(n(n+1)\) roots in total. These correspond to the root system of the special linear Lie algebra \(\mathfrak{sl}_{n+1}\).
The positive roots are exactly the vectors \(e_i + e_{i+1} + \cdots + e_j\) for all \(0 \leq i \leq j \leq n{-}1\), i.e. contiguous segments of 1s.
A1
The simplest case: a single node with no edges.
>>> from mutation_game import MutationGame
>>> game = MutationGame.from_dynkin("A1")
>>> print(game.adj)
[[0]]
>>> roots = game.calculate_roots()
>>> for r in roots:
... print(list(map(int, r)))
[-1]
[1]
There is 1 positive root (the simple root itself) and its negative. The Weyl group is \(\mathbb{Z}/2\mathbb{Z}\) – a single reflection.
A2
Two nodes connected by a single edge. The root system of \(\mathfrak{sl}_3\).
>>> game = MutationGame.from_dynkin("A2")
>>> for r in game.calculate_roots():
... print(list(map(int, r)))
[-1, -1]
[-1, 0]
[0, -1]
[0, 1]
[1, 0]
[1, 1]
3 positive roots, 6 total. The highest root is \((1, 1)\).
A3
The root system of \(\mathfrak{sl}_4\), with 6 positive roots and 12 total.
The mutation path table for A3:
>>> game = MutationGame.from_dynkin("A3")
>>> game.print_mutation_path_table()
Source Target Mutations Len
-------------------------------------------
(0, 0, 1) (0, 1, 0) 1 -> 2 2
(0, 0, 1) (0, 1, 1) 1 1
(0, 0, 1) (1, 0, 0) 1 -> 2 -> 0 -> 1 4
(0, 0, 1) (1, 1, 0) 1 -> 2 -> 0 3
(0, 0, 1) (1, 1, 1) 1 -> 0 2
(0, 1, 0) (0, 1, 1) 2 1
(0, 1, 0) (1, 0, 0) 0 -> 1 2
(0, 1, 0) (1, 1, 0) 0 1
(0, 1, 0) (1, 1, 1) 0 -> 2 2
(0, 1, 1) (1, 0, 0) 2 -> 0 -> 1 3
(0, 1, 1) (1, 1, 0) 2 -> 0 2
(0, 1, 1) (1, 1, 1) 0 1
(1, 0, 0) (1, 1, 0) 1 1
(1, 0, 0) (1, 1, 1) 1 -> 2 2
(1, 1, 0) (1, 1, 1) 2 1
The Cartan matrix and its spectrum:
>>> print(game.cartan_matrix())
[[ 2 -1 0]
[-1 2 -1]
[ 0 -1 2]]
>>> print(game.cartan_eigenvalues())
[0.58578644 2. 3.41421356]
A4
The root system of \(\mathfrak{sl}_5\), with 10 positive roots and 20 total.
A5
The root system of \(\mathfrak{sl}_6\), with 15 positive roots and 30 total. This is the largest A-type where the mutation graph remains visually manageable.
