Spin(8) triality - Knowlpedia

Statement

The compact simply connected Lie group $\operatorname{Spin}(8)$ has outer automorphism group

\operatorname{Out}(\operatorname{Spin}(8))\cong S_3.

Under twisting by these automorphisms, $S_3$ permutes the three inequivalent eight-dimensional irreducible real representations:

8_v,\qquad 8_s=\Delta^+,\qquad 8_c=\Delta^-.

Here $8_v$ is the vector representation factoring through $SO(8)$ , while $8_s$ and $8_c$ are the two half-spin representations. This symmetry is called triality.

Dynkin-diagram origin

The Dynkin diagram of type $D_4$ has one central node joined to three outer nodes. Every permutation of the outer nodes preserves the diagram, giving its automorphism group $S_3$ . The three outer nodes label the highest weights of $8_v,8_s,8_c$ , so the diagram symmetry permutes these representations.

Triality is exceptional to $D_4$ . For $D_m$ with $m\geq5$ , the two spin nodes can still be exchanged, but the vector node is distinguished and the diagram automorphism group has only two elements.

Invariant trilinear form

Clifford multiplication gives a nonzero $\operatorname{Spin}(8)$ -equivariant map

8_v\otimes8_s\longrightarrow8_c.

After choosing invariant inner products, this is equivalent to an invariant trilinear form

t:8_v\otimes8_s\otimes8_c\longrightarrow\mathbb R.

The triality symmetry can be realized so that it permutes the three factors together with the corresponding group automorphisms. This is a representation-theoretic shadow of the close relation between $\operatorname{Spin}(8)$ and the octonions.

What triality does not say

The three representations are not isomorphic as representations for a fixed, unchanged action of $\operatorname{Spin}(8)$ . Rather, an outer automorphism of the group changes which representation is being used: for a suitable $\varphi$ , one has $8_v\circ\varphi\cong8_s$ , and similarly for the other permutations.

Likewise, the labels $8_s$ and $8_c$ , or $+$ and $-$ , depend on a choice of the two spin nodes and on chirality conventions. Triality preserves the unordered triple.

References

John F. Adams, Lectures on Exceptional Lie Groups, University of Chicago Press, 1996, Chapter 3. Publisher record.
John C. Baez, “The Octonions,” Bulletin of the American Mathematical Society 39 (2002), 145–205, §§2.4 and 4.1. DOI record.
William Fulton and Joe Harris, Representation Theory: A First Course, Springer, 1991, §20. Publisher record.