Homomorphisms between Specht modules

Abstract

In positive characteristic, the Specht modules Sλ corresponding to partitions λ are not necessarily irreducible, and understanding their structure is vital to understanding the representation theory of the symmetric group. In this paper, we address the related problem of finding the spaces of homomorphisms between Specht modules. Indeed in [2], Carter and Payne showed that the space of homomorphisms from Sλ to Sμ is non-zero for certain pairs of partitions λ and μ where the Young diagram for μ is obtained from that for λ by moving several nodes from one row to another. We also consider partitions of this type, and, by explicitly examining certain combinations of semi-standard homomorphisms, we are able to give a constructive proof of the Carter–Payne theorem and to generalise it.