irreducible representations of O(2) - reference?

Question

I am looking for a list of the irreducible representations of O(2). Could someone please provide a reference?

EDIT: I am particularly interested in the representations on IR^2 (irreducible or not)

Answer 1

I just see now, that the issue is appearently real representations. I consider complex representations. I not experienced with real representations and whether my strategy works there as well.

You can induce from $SO(2)$. Define on $SO(2)$ the rep $\epsilon_n: \theta \mapsto e^{i \theta n}$. Let $\rho_n$ be the induced one, then $\rho_n$ is irreducible if $n \neq 0$. You have $\rho_n \cong \rho_{-n}$ and $\rho_{0} = 1 \oplus det$. These are up to isomorphism all irreducible representations.

Reference: Traces of Hecke operator by Knightly and Li.

A proof also is in my thesis: http://ediss.uni-goettingen.de/bitstream/handle/11858/00-1735-0000-000D-F074-7/palm.pdf?sequence=1 on pg 101.

Answer 2

You can look into this section of the book Group theory in physics by Wu-Ki Tung to begin with.

EDIT: Theorem 11.3 from this book works for complex irreps only (see Jose's comments to the answer and to the question).