Triality

From Academic Kids

Dynkin diagram of spin(8)

In mathematics, triality is a peculiar property of the group Spin(8), the double cover of 8-dimensional rotation group SO(8). Of all simple Lie groups, Spin(8) has the most symmetrical Dynkin diagram. The diagram has four nodes with one node located at the center, and the other three attached symmetrically. The symmetry group of the diagram is the symmetric group S3 which acts by permuting the three legs. This gives rise to a S3 group of outer automorphisms of Spin(8). This automorphism group permutes the three 8-dimensional irreducible representations of Spin(8); these being the vector representation and two chiral spinor representations. As such, these automorphism do not project to automorphisms of SO(8).

The word triality is constructed in analogy with the word duality.

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