ions have labelling partitions of the form $(a,3,1^b)$ (and their conjugates) with $a,b$ even positive integers. So in this paper we make a case study of this family of partitions. Our technique is different from that of Murphy [@gm], and is weaker in the sense that we cannot always when tell for certain whether one of our Specht modules is decomposable. However, in the cases where we can show decomposability, we have the advantage of being able to describe one summand explicitly.

More specifically, our main result is a determination of exactly which irreducible Specht modules occur as summands of the Specht modules $S^{(a,3,1^b)}$. The technique we use is to consider homomorphisms between Specht modules, and the set-up for computing such homomorphisms is described in Section \[homsec\].

We use homomorphisms between Specht modules in the following way. Suppose $\la,\mu$ are partitions of $n$. Then it is a straightforward result that $S^\mu$ occurs as a summand of $S^\la$ if and only if there are homomorphisms $\gamma:S^\mu\to S^\la$ and $\delta:S^\la\to S^\mu$ such that $\delta\circ\gamma$ is the identity on $S^\mu$. If we assume in addition that $S^\mu$ is irreducible, then by Schur’s Lemma we just need to show that $\delta\circ\gamma$ is non-zero.

Some effort has been devoted to computing the space of homomorphisms between two Specht modules, beginning with the paper of the second author and Martin [@fm]. In fact, there is now an explicit algorithm which computes the homomorphism space ${\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\la,S^\mu)$ except when ${\operatorname{char}}(\bbf)=2$ and $\la$ is $2$-singular. Even in this exceptional case, this technique can be used to construct some homomorphisms between $S^\la$ and $S^\mu$, though only if $\la$ dominates $\mu$.

In our situation, the partitions $\mu$ we shall consider are always $2$-regular, because (as long as $n\neq4$) every irreducible Specht module in characteristic $2$ has the form $S^\mu$ for $\mu$ $2$-regular. So we are able to compute the spa