If R:U(1)→GL(n,𝐂) is a smooth representation then there exists a basis of 𝐂n with respect to which R(eiθ)=(eim1θ0⋱0eimnθ) where m1,…,mn are integers called the weights of the representation. A fancier way of saying this is that 𝐂n=⊕ni=1Vi where each Vi is a 1-dimensional subrepresentation and R=R1⊕⋯⊕Rn with Ri=R|Vi .
Representations of U(1), part 1
Representations of U(1), part 1
We now state the classification theorem for representations of U(1) and illustrate it with an example. We will prove the theorem next time.
This means that the basis with respect to which R has this form is a basis of eigenvectors v1,…,vn . Moreover, vk is simultaneously an eigenvector of all the matrices R(eiθ) with eigenvalue eimkθ .
Take R(eiθ)=(cosθ-sinθsinθcosθ)∈GL(2,𝐂) . The characteristic polynomial of this matrix is det(cosθ-λ-sinθsinθcosθ-λ)=λ2-2λcosθ+1, so the eigenvalues are λ=2cosθ±√4cos2θ-42=cosθ±isinθ=e±iθ . Therefore the weights of this representation are ±1 .
The eigenvectors are (i,1) and (-i,1) . These are therefore our vectors v1∈V1 and v2∈V2 . With respect to this basis of eigenvectors, R(eiθ)=(eiθ00e-iθ) .
Pre-class exercise
Check that (±i,1) is an eigenvector of (cosθ-sinθsinθcosθ) with eigenvalue e±iθ .