# Solution Verification: Irreducible representations of $C_n$ present in $V \otimes V$.

Let $$C_n$$ be the cyclic group order n. Let V be the faithful two dimensional representation (over complex field) denoted by:
$$\begin{equation} \rho(g^j)= \begin{pmatrix} \omega^j & 0 \\ 0 & \omega ^{-j} \end{pmatrix} \end{equation}$$ Where $$\omega^j =$$exp$$(\frac{2\pi i}{n}j)$$ and $$j=1,\dots ,n$$.

What are the irreducibles of $$C_n$$ in the direct sum decomposition of $$V\otimes V$$?

My attempt:
$$C_n$$ is cyclic (hence abelian) thus irreducible representations are one dimensional. There are n irreducible representations $$V_h$$ corresponding to the character $$\chi_h(g) = \omega ^h$$.
If $$V_h$$ apprears in the decomposition of $$V \otimes V$$ then the inner product $$<\chi, \chi_h>$$ is nonzero.
Now $$\begin{equation}\chi(g^j)=\chi_{V\otimes V}(g)= \omega^{2j}+\omega^{-2j} +2\end{equation}$$ Thus $$\begin{equation} <\chi,\chi_h>= \frac{1}{n}\Sigma_{j=1}^{n}(\omega^{j(2-h)}+\omega^{-j(2+h)}+2\omega^{-jh}) \end{equation}$$

Now this sum will equal 0 (by geometric series formula) unless $$h=2,0$$.

So $$V_2,V_0$$ are the only irreducibles appearing in $$V \otimes V$$.

Is this the correct way of approaching the problem? If not what is?

• This appears to be correct, but you might also argue that with respect to natural bases you have $(\rho \otimes \rho)(g^{j}) = \begin{pmatrix}\omega^{2j}\\&\omega^{-2j}\\&&1\\&&&1\end{pmatrix}$, with zeroes off the main diagonal. – Andreas Caranti Mar 7 at 15:54

## 1 Answer

Label the irreducible representations of $$C_n$$ by $$\{V_k \mid k \in \mathbb{Z} / n \mathbb{Z}\}$$, where $$V_k$$ is a one-dimensional representation where your chosen generator of $$C_n$$ acts by $$\exp(2 \pi i k / n)$$. Then it is clear that you have $$V \cong V_1 \oplus V_{-1}$$ as representations. Since tensor product distributes over direct sum, we must have \begin{aligned} V \otimes V & \cong (V_1 \oplus V_{-1}) \otimes (V_1 \oplus V_{-1}) \\ & \cong (V_1 \otimes V_1) \oplus (V_1 \otimes V_{-1}) \oplus (V_{-1} \otimes V_{1}) \oplus (V_{-1} \otimes V_{-1}) \\ & \cong V_2 \oplus V_0 \oplus V_0 \oplus V_{-2} \end{aligned} where the very last line follows from a direct computation that $$V_k \otimes V_r \cong V_{k + r}$$.

• Is $V_{-2}$ one of the irreducibles or is it isomorphic to another h greater than zero – Matthew Mar 8 at 12:05