Three blocks and the representation of $S_3$

I’ve been studying chapter 1 of the famous group "Lie Algebras in Particles Physics" by Georgi.
I am rather confused by section 1.16.
The claim is the following. Consider a system of three bodies, say "A", "B" and "C", connected by springs and living on a two-dimensional plane.

Of course the system is invariant under the permutation group, $S_3$. Now, consider the 6-dimensional tensor product space, composed from the $(x,y)$ positions of each of the three bodies. This can be written as $r_{imu}$, where $i=1,2,3$ runs over the bodies and $mu=1,2$ over spatial coordinates.

Now, I understand that the index $i$ transforms under the 3-dimesional representation of $S_3$. This is simple. We can just represent the combination of the bodies with a 3-dimensional vector, $v_i = (A ; B ; C)$, which means that $A$ is in position 1, $B$ in position 2 and $C$ in position 3.

What I don’t understand is what it means for the index $mu$ to transform under the 2-dimensional representation of $S_3$. How should I think about this?

I thought of representing the configuration where the body at position $i$ has coordinates $(x_i,y_i)$ with the following $2times 3$ matrix with $(x_i,y_i)$ on the row and the column representation to which position those coordinates correspond, i.e.
begin{equation}
v =
begin{pmatrix}
x_1 & x_2 & x_3
y_1 & y_2 & y_3
end{pmatrix}
end{equation}
but this does not seem to work. Consider, in fact, for example, the following 2-dimensional representation
begin{equation}
D_2(a_1) =
begin{pmatrix}
-frac{1}{2} & – frac{sqrt{3}}{2}
frac{sqrt{3}}{2} & – frac{1}{2}
end{pmatrix},.
end{equation}
This is supposed to implement the cyclic permutations of the three positions, but it clearly does not do that when acting on $v$.

Could anyone help me understand this?