Circle Group - Representations

Representations

The representations of the circle group are easy to describe. It follows from Schur's lemma that the irreducible complex representations of an abelian group are all 1-dimensional. Since the circle group is compact, any representation ρ : T → GL(1, C) ≅ C×, must take values in U(1) ≅ T. Therefore, the irreducible representations of the circle group are just the homomorphisms from the circle group to itself. Every such homomorphism is of the form

These representations are all inequivalent. The representation φ−n is conjugate to φn,

These representations are just the characters of the circle group. The character group of T is clearly an infinite cyclic group generated by φ1:

The irreducible real representations of the circle group are the trivial representation (which is 1-dimensional) and the representations

\rho_n(e^{i\theta}) = \begin{bmatrix} \cos n\theta & -\sin n\theta \\ \sin n\theta & \cos n\theta \end{bmatrix},\quad n\in\mathbb Z^{+},

taking values in SO(2). Here we only have positive integers n since the representation is equivalent to .

Read more about this topic:  Circle Group