I have the following map for a representations of $S_3$:
$$e \mapsto \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \quad (1\; 2) \mapsto \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \quad (1\; 3)\mapsto \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix}$$
$$(2\; 3)\mapsto \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}, \quad (1\; 2\; 3) \mapsto \begin{pmatrix} 0& 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}, \quad (1\; 3\; 2)\mapsto \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}$$
We can check that any $\sigma \in S_3$ and its image under the map represents the same permutation. For example, consider multiplying the matrix associated with $(2\; 3)$ with the column vector $[a\; b\; c]$:
$$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} \begin{pmatrix} a \\ b \\ c \end{pmatrix} = \begin{pmatrix} a\\c\\b \end{pmatrix}$$
The second and third element in the column vector are interchanged, with the first element remaining fixed. This is the kind of behavior I expected any representation of $S_3$ will be exhibit. However, when I consider the "standard representation" of $S_3$, given as: $$e \mapsto \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \quad (1\; 2) \mapsto \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 1 \\ 0 & 0 & 1 \end{pmatrix}, \quad (1\; 3)\mapsto \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & -1 & 0 \end{pmatrix}$$
$$(2\; 3)\mapsto \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & \\ 0 & 1 & -1 \end{pmatrix}, \quad (1\; 2\; 3) \mapsto \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & -1 \end{pmatrix}, \quad (1\; 3\; 2)\mapsto \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 1 \\ 0& -1 & 0 \end{pmatrix}$$
I don't know how to interpret the result I get from looking at it the way I was looking above, using matrix multiplication. For instance,
$$ \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 1 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} a\\b\\c\end{pmatrix}= \begin{pmatrix} a\\-b+c\\c \end{pmatrix},$$ from which I have no idea what to infer.
I realize I'm not explaining myself very well, but I hope someone can maybe take a stab at answering my question anyway. Thank you very much for your help.