Quaternionic group representation
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In mathematics, especially in the field of representation theory, a quaternionic group representation is a homomorphism ρ from a group G to the group of endomorphisms of a module over the division algebra of quaternions. Being associative, quaternions admit modules. Being a division algebra, quaternionic modules have a fixed dimension.
A quaternionic representation assigns a quaternion-valued square matrix to each element of G such that
- ρ(gh) = ρ(g)ρ(h) for all g, h in G.
[edit] Example
An example would be the canonical quaternion representation of rotations in 3D. Each (proper) rotation is represented by a quaternion with unit norm. We can think of a one-dimensional module over the quaternions which is left-multiplied by the representative quaternion in question. This gives a one-dimensional quaternionic representation of the spinor group Spin(3), because a rotation by 2π is represented by -1 using quaternions.
This example also happens to be a unitary quaternion representation because
for all g in G.
Another unitary example is a two-dimensional quaternionic module forming an irreducible quaternion representation of Spin(5).
An example of a nonunitary representation would be the two dimensional irreducible representation of Spin(5,1).
