# The Unapologetic Mathematician

## The Branching Rule, Part 1

We want to “categorify” the relation we came up with last time:

$\displaystyle f^\lambda=\sum\limits_{\lambda^-}f^{\lambda^-}$

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 $f^\lambda$ with the Specht module $S^\lambda$, since we’ve seen that $f^\lambda=\dim(S^\lambda)$. But what category are they in? On the left side, $S^\lambda$ is an $S_n$-module, but on the right side all the $S^{\lambda^-}$ are $S_{n-1}$-modules. Our solution is to restrict $S^\lambda$, suggesting the isomorphism

$\displaystyle S^\lambda\downarrow^{S_n}_{S_{n-1}}\cong\bigoplus\limits_{\lambda^-}S^{\lambda^-}$

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 $S_{n-1}\subseteq S_n$ we use, since they’re all conjugate to each other inside $S_n$. So we’ll just use the one that permutes all the numbers but $n$.

Anyway, say the inner corners of $\lambda$ occur in the rows $r_1<\dots, and of course they must occur at the ends of these rows. For each one, we’ll write $\lambda^i$ for the partition that comes from removing that inner corner. Similarly, if $t$ is a standard tableau with $n$ in the $i$th row, we write $t^i$ for the (standard) tableau with $n$ removed. And the same goes for the standard tabloids $\{t\}$ and $\{t^i\}$.

Our method will be to find a tower of subspaces

$\displaystyle0=V^{(0)}\subseteq V^{(1)}\subseteq\dots\subseteq V^{(k)}=S^\lambda$

so that at each step we have $V^{(i)}/V^{(i-1)}\cong S^{\lambda^i}$ as $S_{n-1}$-modules. Then we can see that

$\displaystyle S^\lambda\downarrow^{S_n}_{S_{n-1}}=V^{(k)}\cong V^{(k-1)}\oplus(V^{(k)}/V^{(k-1)}\cong V^{(k-1)}\oplus S^{\lambda^k}$

And similarly we find $V^{(k-1)}\cong V^{(k-2)}\oplus S^{\lambda^{k-1}}$, and step by step we go until we find the proposed isomorphism. The construction itself will be presented next time.