How to tell whether a representation of a group is faithful or unfaithful? From just the character table and the basis functions of the irreducible representations, how do I know whether a representation is faithful or unfaithful?
For the 1-D representation it is trivial to know the answer, of course, so I am only talking about the 2-dimensional ones.
For example, the hexagon group, $D_6$:

How do I know whether $\Gamma_5$ and $\Gamma_6$ are faithful or unfaithful?

The numbers in the right hand section of the table are called the character values. A representation is faithful if and only if the number in the $E$ column of that row only appears once in that row. So $\Gamma_1$ is not faithful, since all the columns have the same value as $E$ (namely, $1$). For nearly the same reason, $\Gamma_2$ is not faithful (1 appears four times, rather than once).
The only faithful representations listed are $\Gamma_6$ and $\chi^{\text{atom sites}} = \Gamma_6 + \Gamma_5 + \Gamma_3 + \Gamma_1$.
If a representation $\Gamma_i$ is faithful, so is $\Gamma_i + \Gamma_j$ for any representation $\Gamma_j$.
By the way, $\Gamma_1$ and $\Gamma_2$ are both 1-dimensional (at least to mathematicians). The dimension is that number in the $E$ column, also called the degree.
