TITLE: Faithful representations and character tables
QUESTION [8 upvotes]: 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.

REPLY [7 votes]: What you are saying is true and is in fact contained in all the standard references (e.g. Isaacs). The idea is that any character of an $n$-dimensional representation, evaluated on an element of order $d$ is a sum of $n$ $d$-th roots of unity (why?). Your statement now follows easily by the triangle inequality:
$|\chi(g)|\leq \chi(1)$ for all $g$ and $\chi(g)=\chi(1)$ iff $g$ is sent to the identity matrix.