The Dynkin diagram/Dynkin quiver D4 has an S3-symmetry group
(graphics grabbed from Wikipedia here)
This manifests itself in “triality” symmetries enjoyed by the objects that are labled by D4 in an ADE-classification.
Specifically for the simple Lie group SO(8) corresponding to D4, triality is the statement that the two spin representations of the corresponding spin group $Spin(8)$ and the defining vector representations are all isomorphic, and in a nice way.
See also
Wikipedia, Triality
Last revised on October 2, 2018 at 10:55:38. See the history of this page for a list of all contributions to it.