Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 19I / c

Let $e_0,\dots ,e_{n-1}$ be the standard basis and let the generator $g$ send $e_j$ to $e_{j+1}$ modulo $n$. The constant vectorspans a trivial module. Since $n$ is even, the alternating vectorspans a sign module on which $g$ acts by $-1$.

For each $1\le k<n/2$, define the real discrete Fourier modesThe plane $W_k=\mathrm{span}\{c_k,s_k\}$ is invariant, and $g$ acts on it by rotation through $2\pi k/n$. Since this angle is neither zero nor $\pi$, $W_k$ is irreducible over $\mathbb{R}$. The traces $2\cos (2\pi k/n)$ are distinct for $1\le k<n/2$, so these modules are pairwise nonisomorphic. Orthogonality of the discrete Fourier transform givesThe dimensions sum to $1+1+2(n-2)/2=n$, so this is the whole regular module: two nonisomorphic one-dimensional summands and $(n-2)/2$ nonisomorphic two-dimensional summands.

When $n$ is odd there is no alternating sign mode. The same cosine-sine construction gives invariant irreducible planes $W_k$ foragain pairwise nonisomorphic because their generator traces are distinct. The real Fourier decomposition isIts dimensions sum to $1+2(n-1)/2=n$. Thus the real regular representation of a finite cyclic group has one one-dimensional irreducible summand and $(n-1)/2$ nonisomorphic two-dimensional irreducible summands.