We want to “categorify” the relation we came up with last time:
That is, we want to replace these numbers with objects of a category, replace the sum with a direct sum, and replace the equation with a natural isomorphism.
It should be clear that an obvious choice for the objects is to replace with the Specht module , since we’ve seen that . But what category are they in? On the left side, is an -module, but on the right side all the are -modules. Our solution is to restrict , suggesting the isomorphism
This tells us what happens to any of the Specht modules as we restrict it to a smaller symmetric group. As a side note, it doesn’t really matter which we use, since they’re all conjugate to each other inside . So we’ll just use the one that permutes all the numbers but .
Anyway, say the inner corners of occur in the rows , and of course they must occur at the ends of these rows. For each one, we’ll write for the partition that comes from removing that inner corner. Similarly, if is a standard tableau with in the th row, we write for the (standard) tableau with removed. And the same goes for the standard tabloids and .
Our method will be to find a tower of subspaces
so that at each step we have as -modules. Then we can see that
And similarly we find , and step by step we go until we find the proposed isomorphism. The construction itself will be presented next time.