I'm quite new in representations and I'm trying to do next problem: (It's supposed that I don't know anything about characters theory)
We want to study $S_3=(\tau=(123),\sigma=(1,2)\,|\, \sigma\tau=\tau^2\sigma)$ We have also $\omega=e^{{2i\pi/3}}$. I have to prove that we can find a representation $V$ with a base $(v,w)$ satifying $$\rho(\tau)v=\omega v\quad \rho(\sigma)v=w \quad \rho(\tau)w=\omega^2w\quad \rho(\sigma)w=v $$ and prove that $V$ is irreductible.
I have a solution of a classmate but I don't understand, it talks about eigenvalues of $\tau$. I will be very thankfull if anybody can explain this exercise.
I'll give some strategy how one could prove this. There are two problems here, first proving that you can define such a representation, second proving it is irreducible.
For the first part: The easiest two-dimensional vector space I can think of is $\mathbb{C}^2$ with its standard basis, let's denote this basis as $(v,w)$.
By definition giving $\mathbb{C}^2$ the structure of a representation is the same as defining a group homomorphism $\rho:S_3\to \operatorname{GL}(\mathbb{C}^2)$. Now, $\rho(\tau)$ should be a linear map sending $v$ to $\omega v$ and $w$ to $\omega^2 w$. This linear map is represented by the matrix $\rho(\tau)=\begin{pmatrix}\omega&0\\0&\omega^2\end{pmatrix}$. Finding a similar matrix for $\sigma$ should not be that difficult. Since $\sigma$ and $\tau$ generate $S_3$, you can now define a map $S_3\to \operatorname{GL}(\mathbb{C}^2)$ by extending this map multiplicatively, e.g. $\rho(\sigma\tau)=\rho(\sigma)\rho(\tau)$.
You have to check this is well-defined, it could be that by this definition $\tau^3=1$, but $\rho(\tau)^3\neq 1$ if we had done a mistake. An easy way to check this is to have a presentation of your group by generators and relations, e.g. for $S_3$ you have $S_3=\langle \sigma,\tau | \sigma^2=1, \tau^3=1, \sigma \tau=\tau^2\sigma\rangle$. Then for checking this map is well-defined, you just have to prove $\rho(\sigma)^2=1$, $\rho(\tau)^3=1$ and $\rho(\sigma\tau)=\rho(\tau^2\sigma)$. This is defining the representation.
For checking it is irreducible, by definition you have to prove it has no proper non-zero subrepresentation. Equivalently, you can prove that every vector in the representation generates the whole space. So, take a non-zero vector $u=av+bw$ with $a,b\in \mathbb{C}$. You have to prove that the one-dimensional subspace spanned by $u$ cannot be invariant under the action of $S_3$. Applying $\rho(\sigma)$ to this $u$ gives $\rho(u)=bv+aw$. Thus, if you assume that $u$ spans a subrepresentation you get $\rho(u)=\lambda u$, hence $bv+aw=\lambda av+\lambda bw$. Using it is a basis, you get $b=\lambda a$ and $a=\lambda b$. As a next step you can try to get some more conditions applying $\rho(\tau)$ and combining them should give you the only possibility $a=b=0$, a contradiction.