On homomorphisms between Specht modules in even characteristic

Over fields of characteristic 2, Specht modules may decompose and there is no upper bound for the dimension of their endomorphism algebra. A classification of the (in)decomposable Specht modules and a closed formula for the dimension of their endomorphism algebra remain two important open problems in the area. More generally, the space of homomorphisms between two Specht modules is of interest in its own right. In this thesis, we develop a novel description for the homomorphism space between two Specht modules, which we then utilise to deduce new results. Most notably, we provide infinite families of Specht modules with one-dimensional endomorphism algebra in characteristic 2. We conclude by providing a dimension formula for the space of homomorphisms between hook Specht modules in characteristic 2, thereby generalising a result of Murphy who provided an analogous formula covering the endomorphism case.