Modular Representations of Symmetric Groups

Over the last few weeks, we have seen lots of things in representation theory—we began by looking at an onslaught of examples to see explicitly what representations look like, we discussed character theory and looked at character tables, we talked about representations from a module-theoretic perspective, and then we saw how we can model the representation theory of the symmetric group via the combinatorics of Young tableaux. But most of our discussion has been about the representation theory of finite groups over the complex numbers. With Maschke’s theorem in mind, it seems natural to ask what happens when the hypotheses of of this theorem fail. That is, what happens to the representations of a finite group G if we wish to work over a field of characteristic dividing the order of the group? (As a side comment: It seems like a shame that when we have charK ∣∣ |G|, KG is not a semisimple algebra. But the perspective we should actually take is that it is fantastically miraculous that KG is semisimple when charK |G|. For instance, if we were to think about some analogous statement for groups, it’s ridiculously false! I mean, it’s not even true that abelian groups are direct sums of simple groups! So we shouldn’t be depressed about the times when Maschke’s theorem fails. We should just be ecstatic that we have Maschke’s theorem!) In the last couple of lectures, we have looked at the specific case of when G = Sn, the symmetric group. Jeremy talked about how (miraculously!) we can parameterize representations of Sn over C (or an algebraically closed field of characteristic not dividing n! = |Sn|) by the partitions of n, which form a transversal for the conjugacy classes of Sn. We saw how the Specht modules Sλ, for partitions λ of n, form a transversal for the isomorphism classes of irreducible representations of Sn. In this lecture, we will try to obtain an analogous transversal for the case when the characteristic of the ground field of our representation space divides n!, the order of Sn.