- #1

kexue

- 196

- 1

I would like to learn a bit more about triality of SO (8) as discussed in this http://en.wikipedia.org/wiki/SO%288%29" [Broken].

Especially the article says:

What do they mean by outer automorphism? How does this automorphism connect the three representations? What is meant by vector representation for the Spin(8), has a Spin group not only spinor representations?

thank you

Especially the article says:

SO(8) is unique among the simple Lie groups in that its Dynkin diagram (shown right) (D4 under the Dynkin classification) possesses a three-fold symmetry. This gives rise to peculiar 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 S3 that permutes these three representations.

What do they mean by outer automorphism? How does this automorphism connect the three representations? What is meant by vector representation for the Spin(8), has a Spin group not only spinor representations?

thank you

Last edited by a moderator: