I'm trying to prove that if 2 complex representations have the same character, then they are the same. In my proof, I found that is sufficient that the following statement is true:
Let $\rho, \tau$ be representations such that: $$\begin{cases}\rho = \rho_1 \oplus \rho_2 \\ \tau = \tau_1 \oplus \tau_2 \\\rho_1 \cong \tau_1 \\\rho_2 \cong \tau_2\end{cases}$$ Then $\rho \cong \tau$
I tried using the definition of isomorphic complex representations to obtain this result, but I was unsuccesful. Please keep the proofs elementary. Thank you.
Let $V, V_i, W, W_i$ be vector spaces so that $$\rho\colon G \to \mathrm{GL}(V)$$ $$\rho_i\colon G \to \mathrm{GL}(V_i)$$ $$\tau\colon G \to \mathrm{GL}(W)$$ $$\tau_i\colon G \to \mathrm{GL}(W_i)$$ are the representations that you're looking at. Now to say that $\rho \simeq \rho_1 \oplus \rho_2$ means there is an isomorphism $\phi\colon V \to V_1 \oplus V_2$ such that if $\phi(v) = (v_1, v_2)$ then $\phi(\rho(g)(v)) = (\rho_1(g)(v_1), \rho_2(g)(v_2))$. Similarly $\tau \simeq \tau_1 \oplus \tau_2$ means there's an isomorphism $\psi\colon W \to W_1 \oplus W_2$ with a similar property. That $\rho_i \simeq \tau_i$ means there's an isomorphism $\sigma_i\colon V_i \to W_i$ such that $\sigma_i(\rho_i(g)(v)) = \tau_i(g)(\sigma_i(v))$.
All I've done so far is expand the definition of an isomorphism of representations so that everything has a name. Now the map you want is $$V \overset{\phi}{\to} V_1 \oplus V_2 \overset{\sigma_1 \oplus\sigma_2}{\to} W_1 \oplus W_2 \overset{\psi^{-1}}{\to} W.$$ You'll need to check that this is an isomorphism of vector spaces and that it has the required properties to be a map between the representations $\rho$ and $\tau$.