§ Mnemonic for Specht module actions
Consider the two extreme cases, of wide v/s narrow:
x = [* * *]
y = [#]
x = [* * *]. It's very wide/fat, so it doesn't like much exercise, which is why it's columns stabilizer is trivial. Thus, the action .
y = [*][*][*]. It's very slim, and exercises quite a bit. So it's column stabilizer is , and its action has a lot of exercise.
- Anyone can participate in 's exercise regime. In particular, since doesn't tire out from the exercise regime of .
- On the other side, it's hard to take part in 's exercise regime and not get TODOed out. If we consider , we're going to get zero because by tableaux, there are swaps in that leave invariant, which causes sign cancellations. But intuitively, is asking to participate in 's exercise regmine, which it's not strong enough to do, and so it dies.
- In general, if , then is wider/fatter than . Thus we will have since is a harder exercise regime that has more permutations.
- Extend this to arrive at specht module morphism: If we have a non-zero morphism then [Check this?? Unsure ]