Real regular representation of cyclic group I am looking for help to answer the following questions:
What are the irreducible real representions $ρ: C_n → GL(V ) $ of a cyclic group of order n?  How does the real regular representation $RC_n$ of $C_n$ break up as a direct sum
of irreducible representations?
Thank you.
 A: There is an isomorphism $\Bbb R[C_n]\cong\Bbb R[X]/(X^n-1)$. We can factor this polynomial over $\Bbb R$ by pairing off complex conjugate roots of unity, and then invoking Sun-Ze (better known as the Chinese Remainder Theorem) to decompose (as an $\Bbb R$-algebra or as an $\Bbb R[C_n]$-module).
To write down what the summands look like, let's invent the following notation:


*

*$\Bbb R[X]/(X-1)\cong V_1$ is the trivial representation, since $X\equiv1$.

*$\Bbb R[X]/(X+1)\cong V_{-1}$ is a "sign" representation, which appears if $n$ is even (as $-1$ is always an even root of unity but never an odd). Since $X\equiv-1$, the generator of $C_n$ simply negates a vector in this one-dimensional representation.

*$\Bbb R[X]/((X-\zeta)(X-\bar{\zeta}))\cong V_\zeta$ has as its underlying space $\Bbb C$, and $X$ acts as $\xi$ on it (or equivalently as $\bar{\xi}$ on it; either way it's the same real representation of dimension two). This occurs for every complex conjugate pairs of $n$th roots of unity $\{\zeta,\bar{\zeta}\}$.


And so with the above definitions we have
$$\Bbb R[C_n]=\bigoplus_{\substack{\zeta^n=1 \\ {\rm Im}(\zeta)\ge0}} V_\zeta.$$
As an $\Bbb R$-algebra we can say $\Bbb R[C_n]\cong\Bbb R^{\oplus r}\oplus\Bbb C^{\oplus s}$, where $r=1$ if $n$ is odd and $r=2$ if $n$ is even, and we can determine $s$ via the relation $r+2s=n$.
