leading to not giving any new points in $O_x$.
 The only way to get new points in dim=2 is by taking reflections. So, for example:
y

p  q

 reflection of
p
about the y
axis gives us q
. So if we set $O_p = \{I, Y\}$, we get $O_p(p) = I  Y$.  We need $O_p(q) = sgn(Y)O_p(p)$, which does indeed happen, as $O_p(p) = p  q$, with $O_p(q) = q  p =  (pq)$.
 Let's add more points:
y

p  q
c  d
 The problem with the new points $c, d$ is that they are not in the orbit $O_p(p)$, but they also don't evaluate to zero!
 This tells us that after we pick the points $p, q$, any new points we pick must lie on the axis of reflection to be annhilated.
 Thus, one valid way of adding new points is:
y

c
p  q
+x

d

 Here, $c, d$ have as group $O_c = O_d = \{I, X\}$, reflection about the $X$ axis. Check that all the axioms are satisfied: elements in the orbits $O_c, O_d, O_p, O_q$ evaluate to $\pm A_c c, \pm A_p p$. While elements not the orbit become zero.
 Thus, it seems like the Specht module attempts to construct "reflections" that somehow represent $S_n$. Is this why it is related to Coxeter theory?