Standard representation of the symmetric group

Definition

Let $S_{n}$ be the symmetric group on $n$ letters.

Then the standard representation of $S_{n}$ is the linear representation sending each permutation $\sigma \in S_{n}$ to its corresponding permutation matrix, that is, the matrix $A$ such that $A_{ij}=1$ for $j=\sigma (i)$, $A_{ij}=0$ elsewhere.

Character

The standard representation has character $\chi (g)$ equal to the number of fixed points of the permutation $g\in S_{n}$, since $\chi (g)$ is equal to the trace of the corresponding permutation matrix - which has a $1$ on the diagonal for fixed points, $0$ for non-fixed points by definition.