Type C – Symplectic

The \(C_n\) Dynkin diagram (\(n \geq 3\)) is a directed multigraph: a path on n nodes with a (1,2) directed edge at one end.

\[0 - 1 - \cdots - (n{-}2) \overset{1}{\underset{2}{\rightleftarrows}} (n{-}1)\]

That is, there is 1 directed edge from node \(n{-}2\) to \(n{-}1\), and 2 edges back. This is the reverse of the B-type double edge.

The root system has \(n^2\) positive roots and \(2n^2\) total – the same count as \(B_n\). These correspond to the root system of the symplectic Lie algebra \(\mathfrak{sp}_{2n}\).

Note

The B/C notation here follows Wildberger’s convention (based on the directed multigraph structure), which is swapped relative to the Bourbaki convention.

C3

9 positive roots, 18 total. The root system of \(\mathfrak{sp}_6\).

>>> from mutation_game import MutationGame
>>> game = MutationGame.from_dynkin("C3")
>>> print(game.adj)
[[0 1 0]
 [1 0 1]
 [0 2 0]]

Here adj[2,1] = 2 (two edges from node 2 back to 1), while adj[1,2] = 1 – the opposite direction from B3.

C3 positive roots

C4

16 positive roots, 32 total. The root system of \(\mathfrak{sp}_8\).

C4 positive roots