Triality
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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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