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.