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I am trying to solve the problem:

Decompose the standard representation of the cyclic group $C_{n}$ in $\mathbb{R}^{2}$ by rotations into a direct sum of irreducible representations.

What I have tried:

Taking $x$ to be a generator, I define $\rho:C_{n}\to\mathbb{R}^{2}$, $x^{k}\mapsto\left[\begin{array}{cc}cos(2k\pi/n) & -sin(2k\pi/n)\\ sin(2k\pi/n) & cos(2k\pi/n)\\\end{array}\right]$

But I can't think of any $C_{n}$-invariant subspace of $\mathbb{R}^{2}$. The $x$-axis, $\mathbb{R}\times \{0\}$ might be $H$-invariant for a subgroup $H\leq C_{n}$ which might give me an induction/restriction relationship. Anyhow that doesn't seem to go anywhere.

How can I find a $C_{n}$-invariant subspace $W\subset V$.

Or does this mean that $\rho$ is an irreducible representation?

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1 Answer 1

up vote 2 down vote accepted

When you say that you can't think of any $C_n$-invariant subspace of $\mathbb R^2$, you were probably assuming $n\geq3$. Treat $n=1$ and $n=2$ separately. Coming back to $n\geq3$, a good thing to do when you can't think of any [whatever] is to try to prove that there aren't any [whatever]. If you succeed, you've solved the problem (in this case, you'd have proved that the representation is irreducible); if you fail, the details of the failure may help focus your search for a [whatever].

By the way, in this particular problem, the answer changes if you think of the same matrices as defining a representation in a 2-dimensional vector space over the complex numbers (instead of the real numbers).

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Thank you. I was assuming $n\geq 3$ without realizing it. –  Kyle Schlitt Dec 8 '12 at 22:05
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