§ Mnemonic for Specht module actions

Consider the two extreme cases, of wide v/s narrow:
x = [* * *]
y = [#]
    [#]
    [#]
  • Consider x = [* * *]. It's very wide/fat, so it doesn't like much exercise, which is why it's columns stabilizer Cx={e}C_x =\{ e\} is trivial. Thus, the action AxidA_x \equiv id.
  • Consider y = [*][*][*]. It's very slim, and exercises quite a bit. So it's column stabilizer is S3S_3, and its action AyA_y \equiv \dots has a lot of exercise.
  • Anyone can participate in xx's exercise regime. In particular, Ax(y)=id(y)=yA_x(y) = id(y) = y since yy doesn't tire out from the exercise regime of xx.
  • On the other side, it's hard to take part in yy's exercise regime and not get TODOed out. If we consider Ay(x)A_y(x), we're going to get zero because by tableaux, there are swaps in AyA_y that leave xx invariant, which causes sign cancellations. But intuitively, Ay(x)A_y(x) is asking xxto participate in yy's exercise regmine, which it's not strong enough to do, and so it dies.
  • In general, if λμ\lambda \triangleright \mu, then λ\lambda is wider/fatter than μ\mu. Thus we will have Aμ(λ)=0A_\mu(\lambda) = 0 since AμA_\mu is a harder exercise regime that has more permutations.
  • Extend this to arrive at specht module morphism: If we have a non-zero morphism ϕ:SλSμ\phi: S^\lambda \rightarrow S^\mu then λμ\lambda \rightarrow \mu [Check this?? Unsure ]