p-restriction of partitions and homomorphisms between Specht modules

Let λ be a partition of n. We show that the space of FSn-homomorphisms between the Specht modules Sλ and Sλr is one-dimensional, where F is a field of characteristic p and λr is the ‘p-restriction’ of λ. Equivalently, our result proves the corresponding theorem for the homomorphism space HomFSn(Sλreg,Sλ), where λreg is the ‘p-regularisation’ of λ, as defined by James.