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 an 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 automorphisms do not project to automorphisms of SO(8).

The word triality is constructed in analogy with the word duality, specifically with duality in projective geometry. Roughly speaking, symmetries of the Dynkin diagram lead to automorphisms of the Bruhat-Tits building associated with the group. For special linear groups, one obtains projective duality. For Spin(8), one finds a curious phenomenon involving 1, 2, and 4 dimensional subspaces of 8-dimensional space, historically known as "geometric triality".