§ Musing about Specht modules

If we generalize, to each point $x$, we are creating a group of orthogonal matrices $O_x$ (like $C_t$), such that

• All points in the orbit have (the same/similar, unsure?) $C_t$
• 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 $O_x$. These behave like nth roots of unity on averaging, so we get $1 + \omega + \omega^2 + \dots + \omega^{n-1} = 0$, 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 $O_p(q) = q - p = - (p-q)$.
• 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?

• What is a Coxeter diagram good for
• What is a Coxeter group