Faithful representations and character tables 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.

If $\chi$ is the character, and $\chi(g)=\chi(1)=n$ for some group element $g$, then $\rho(g)$ is an $n\times n$ matrix $A$ whose eigenvalues are all complex numbers of modulus 1 and whose trace is $n$ (here $\rho$ is the representation whose character is $\chi$). Also, some power of $A$ is the identity. Can you see how this forces $A$ to be the identity matrix?
