Triality is a relationship among three vector spaces. It describes those special features of the Dynkin diagram D4 and the associated Lie group Spin(8), the double cover of 8-dimensional rotation group SO(8).

SO(8) is unique among the simple Lie groups in that its Dynkin diagram (below) (D4 under the Dynkin classification) possesses a three-fold symmetry. This gives rise to a surprising feature of Spin(8) known as triality. Related to this is the fact that the two spinor representations, as well as the fundamental vector representation, of Spin(8) are all eight-dimensional (for all other spin groups the spinor representation is either smaller or larger than the vector representation). The triality automorphism of Spin(8) lives in the outer automorphism group of Spin(8) which is isomorphic to the symmetric group $S_3$ that permutes these three representations.

What are physics applications of $SO(8)$ and Spin(8) triality?

For example, one of physics applications of $SO(8)$ and Spin(8) triality is that, in the classifications of interacting fermionic topological phases protected by global symmetries, the 1+1D BDI Time-Reversal invariant Topological Superconductor and 2+1D $Z_2$-Ising-symmetric Topological Superconductor have $\mathbb{Z}_8$ classifications (see a related post here), that can be deduced from adding non-trivial four-fermion interaction terms respect the $SO(8)$ and Spin(8) triality, see for example the Appendix A of this web version (free access).

Are there other examples, other applications in physics?

This post imported from StackExchange Physics at 2017-09-30 21:54 (UTC), posted by SE-user wonderich