Permutation matrix

Definition

The permutation matrix of a permutation $\sigma \in S_{n}$ is the $n\times n$ matrix $A$ such that $A_{ij}=1$ for $j=\sigma (i)$, $A_{ij}=0$ elsewhere. (Here, $S_{n}$ is the symmetric group on $n$ letters.)

Example

The permuation $(1\,2)\in S_{3}$ has permutation matrix ${\begin{pmatrix}0&1&0\\1&0&0\\0&0&1\end{pmatrix}}$.

The standard representation of the symmetric group

Further information: standard representation of the symmetric group

The standard representation of the symmetric group is the linear representation sending each element of the symmetric group to its corresponding permutation matrix.