Now that we’ve shown the intertwinors that come from semistandard tableaux are independent, we want to show that they span the space . This is a bit fidgety, but should somewhat resemble the way we showed that standard polytabloids span Specht modules.
So, let be any intertwinor, and write out the image
Here we’re implicitly using the fact that .
First of all, I say that if and , then the coefficients of and differ by a factor of . Indeed, we calculate
This tells us that
Comparing coefficients on the left and right gives us our assertion.
As an immediate corollary to this lemma, we conclude that if has a repetition in some column, then . Indeed, we can let be the permutation that swaps the places of these two identical entries. Then , while the previous result tells us that , and so .