§ Musing about Specht modules

If we generalize, to each point xx, we are creating a group of orthogonal matrices OxO_x (like CtC_t), such that
  • All points in the orbit have (the same/similar, unsure?) CtC_t
  • For points outside the orbit, we evaluate to zero.
At least in dim=2, we can't take rotations (of finite order) as elements of OxO_x. These behave like nth roots of unity on averaging, so we get 1+ω+ω2+⋯+ωn−1=01 + \omega + \omega^2 + \dots + \omega^{n-1} = 0, leading to not giving any new points in OxO_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 Op={I,Y}O_p = \{I, Y\}, we get Op(p)=I−YO_p(p) = I - Y.
  • We need Op(q)=sgn(Y)Op(p)O_p(q) = sgn(Y)O_p(p), which does indeed happen, as Op(p)=p−qO_p(p) = p - q, with Op(q)=q−p=−(p−q)O_p(q) = q - p = - (p-q).
  • Let's add more points: 
  y
  |
p | q
c | d
  • The problem with the new points c,dc, d is that they are not in the orbit Op(p)O_p(p), but they also don't evaluate to zero!
  • This tells us that after we pick the points p,qp, 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,dc, d have as group Oc=Od={I,X}O_c = O_d = \{I, X\}, reflection about the XX axis. Check that all the axioms are satisfied: elements in the orbits Oc,Od,Op,OqO_c, O_d, O_p, O_q evaluate to ±Acc,±App\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 SnS_n. Is this why it is related to Coxeter theory?