Suppose an n-dimensional irreducible complex representation is not faithful. Then a non-identity element gets mapped to the identity matrix in $GL_n(\mathbb{C})$ so that the value of its associated character on the conjugacy class of this element is $n$. Thus, $n$ appears at least twice in the corresponding row of the group's character table.
I suspect the converse is true: if the row corresponding to an irreducible $n$-dimensional complex representation contains the dimension of the representation in more than one column, then the representation is not faithful. I have looked in a few of the standard algebra references and have been unable to find a proof. Can anyone point me in the right direction? We proved this for $n=2$, but it seems that it would be difficult and messy to generalize. I wonder if there is a simpler proof.