Given the character table for a certain group $G$, The first row of the table is filled with ones.
I understand that it's because the first row corresponds to the trivial representation, let us denote it by $ \Gamma^{(1)}$, which maps every group element $g$ to the identity map $I_d$, i.e $\Gamma^{(1)}(g)=I_{dV}$. The identity is here defined over the vector field $V$.
But, I don't get why the dimension of $V$ is one. Is that some sort of convention ? Then, why would $\Gamma^{(1)}$ necessarily appear in the character table ? Is it necessarily present in any irreducible representations decomposition ? I'd say no, but I'm a bit lost.
By the way, what's the point in representing a group via a representation, and in that case, why would one decompose the latter in irreducible representations ?
Thanks in advance
Regards