The Unapologetic Mathematician

Mathematics for the interested outsider

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}.

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December 16, 2010 - Posted by | Algebra, Representation Theory, Representations of Symmetric Groups

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