The Unapologetic Mathematician

Dimensions of Young Tabloid Modules

Last time our efforts to calculate the characters of the modules $M^\lambda$ were stymied. But at least we can calculate their dimensions. The dimension of $M^\lambda$ is the number of Young tabloids of shape $\lambda$.

Again, we pick some canonical Young tableau $Y$ of shape $\lambda$ so that every other tableau $t$ can be written uniquely as $t=\tau Y$ for some $\tau\in S_n$. That is, the set of all Young tabloids $\{t\}$ is the orbit $S_n\{Y\}$ of the canonical one. By general properties of group actions we know that there is a bijection between the orbit and the index of the stabilizer of $\{Y\}$ in $S_n$. That is, we must count the number of permutations $\tau\in S_n$ with $\tau Y$ row-equivalent to $Y$.

It doesn’t really matter which $Y$ we pick; any two tableaux in the same orbit — and they’re all in the same single orbit — have isomorphic stabilizers. But like we mentioned last time the usual choice lists the numbers from $1$ to $\lambda_1$ on the first row, from $\lambda_1+1$ to $\lambda_1+\lambda_2$ on the second row, and so on. We write $S_\lambda$ for the stabilizer of this choice, and this is the subgroup of $S_n$ we will use. Notice that this is exactly the same subgroup we described earlier.

Anyway, now we know that Young tabloids $\{\tau Y\}$ correspond to cosets of $S_\lambda$; if $\tau'=\tau\pi$ for some $\pi\in S_\lambda$, then

$\displaystyle\{\tau' Y\}=\{\tau\pi Y\}=\tau\{\pi Y\}=\tau\{Y\}=\{\tau Y\}$

So we can count these cosets in the usual way:

$\displaystyle[S_n:S_\lambda]=\lvert S_n\rvert/\lvert S_\lambda\rvert=n!/\lvert S_\lambda\rvert$

How big is $S_\lambda$? Well, we know that

$\displaystyle S_\lambda\cong S_{\lambda_1}\times\dots\times S_{\lambda_k}$

and so

$\displaystyle\lvert S_\lambda\rvert=\lvert S_{\lambda_1}\rvert\dots\lvert S_{\lambda_k}\rvert=\lambda_1!\dots\lambda_k!$

Since it will come up so often, we will write this product of factorials as $\lambda!$ for short. We can then write $S_\lambda=\lambda!$ and thus we calculate $n!/\lambda!$ for the number of cosets of $S_\lambda$ in $S_n$. And so this is also the number of Young tabloids of shape $\lambda$, and also the dimension of $M^\lambda$.

Now, along the way we saw that the Young tabloid $\{\tau Y\}$ corresponds to the coset $\tau S_\lambda$. It should be clear that the action of $S_n$ on the Young tabloids is exactly the same as the coset action corresponding to $S_\lambda$. And thus the permutation module $M^\lambda$ must be isomorphic to the induced representation $1\!\!\uparrow_{S_\lambda}^{S_n}$.