Characters and permutation matrices Suppose I represent the group $S_{\,n}$ using $n \times n$ permutation matrices. This is a valid group representation. Let $\chi$ be its character. Since $\chi(g)$ is complex and since $1=\chi(gg^{-1})=\chi(g)\chi(g^{-1})=\chi(g)\chi(g)^*=|\chi(g)|^2$, the value of $\chi(g)$ lies on the unit circle. However, for certain permutations $g$, the permutation matrix has 0's down the diagonal and the value of $\chi(g)$, being the trace of this matrix, is 0. Am I missing something here? 

Call the representation $\rho$, then $\chi$ is the trace of $\rho$. The trace is a homomorphism of the additive group, not the multiplicative; your $\chi(gg^{-1})=\chi(g)\chi(g^{-1})$ is false. Only characters of degree 1 are multiplicative. 
