## Consequences of the Submodule Theorem

We have a number of immediate consequences of the submodule theorem. First, and most important, the Specht modules form a complete list of irreducible modules for the symmetric group . We know that they’re irreducible, and that there’s one of them for each partition , which is the number of modules we’re looking for. But we need to show that the Specht modules corresponding to distinct partitions are themselves distinct. For this, we’ll use a lemma.

If is a nonzero intertwinor, then . Further, if , then must be multiplication by a scalar. Indeed, since there must be some polytabloid with . We decompose , and extent to all of by sending every vector in to . That is:

where the are -tableaux. Now, the can’t all be zero, so we must have at least one -tableau so that . But then our corollary of the sign lemma tells us that , as we asserted!

Further, if , then our other corollary shows us that for some scalar . We can thus calculate

and so multiplies every vector by .

As a consequence, the must be distinct for distinct permutations, since if then there is a nonzero homomorphism , and thus . But the same argument shows that , and thus .

More particularly, we have a decomposition

where the diagonal multiplicities are . The rest of these multiplicities will eventually have a nice interpretation.

[...] we’ve described the Specht modules, and we’ve shown that they give us a complete set of irreducible representations for the symmetric groups. But we [...]

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[...] tableaux of shape and content . This number we call the “Kostka number” . We’ve seen that there is a [...]

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