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In mathematics, triality is a relationship among three vector spaces, analogous to the duality relation between dual vector spaces. Most commonly, it describes those special features of the Dynkin diagram D 4 and the associated Lie group Spin(8) , the double cover of 8-dimensional rotation group SO(8) , arising because the group has an outer ...
The triality automorphism of Spin(8) lives in the outer automorphism group of Spin(8) which is isomorphic to the symmetric group S 3 that permutes these three representations. The automorphism group acts on the center Z 2 x Z 2 (which also has automorphism group isomorphic to S 3 which may also be considered as the general linear group over the ...
The symmetries of the Dynkin diagram, D 4, correspond to the outer automorphisms of Spin(8) in triality. Let G now be a connected reductive group over an algebraically closed field. Then any two Borel subgroups are conjugate by an inner automorphism, so to study outer automorphisms it suffices to consider automorphisms that fix a given Borel ...
The simply connected split algebraic group of type D 4 has a triality automorphism σ of order 3 coming from an order 3 automorphism of its Dynkin diagram. If L is a field with an automorphism τ of order 3, then this induced an order 3 automorphism τ of the group D 4 (L).
Elementary particle states assigned to E 8 roots corresponding to their spin, electroweak, and strong charges according to E 8 Theory, with particles related by triality. This eight-dimensional root diagram is shown projected onto a Coxeter plane.
In mathematics, specifically in group theory, the phrase group of Lie type usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field.
The Platonic solids, seen here in an illustration from Johannes Kepler's Mysterium Cosmographicum (1596), are an early example of exceptional objects. The symmetries of three-dimensional space can be classified into two infinite families—the cyclic and dihedral symmetries of n-sided polygons—and five exceptional types of symmetry, namely the symmetry groups of the Platonic solids.
The shaded region is the fundamental Weyl chamber for the base {,}. If is a root system, we may consider the hyperplane perpendicular to each root .Recall that denotes the reflection about the hyperplane and that the Weyl group is the group of transformations of generated by all the 's.