The study of the representation theory of the symmetric group can be carried out from a combinatorial point of view, avoiding the machinery of the representation theory of algebraic groups. This approach has the benefit of providing more insight into the subject as the study remains in the setting of the symmetric group. A number of combinatorial tools are explored throughout this dissertation, which as well as making advances in the representation theory of the symmetric group, also contributes to topics of a purely combinatorial interest.

The most important objects to be understood in the representation theory of the symmetric group are the Specht modules, which over fields of characteristic 0 are a complete set of simple modules. These can be defined combinatorially and thus allow an explicit combinatorial approach to be taken to their study. This dissertation, as with the modern study of the representation theory of the symmetric group, is concerned with modular representation theory - the theory over fields of positive characteristic. In this setting the Specht modules are not necessarily simple, however a complete set of simple modules can be found amongst their cosocles. A complete understanding of the Specht modules would, therefore, reveal the details of the simple modules.

To understand the structure of the Specht modules it is first necessary to understand the decomposition numbers, which count the multiplicity of each simple module as a factor of a Specht module. We shall examine the filtration introduced by Schaper, which remains the most powerful tool for determining decomposition numbers, and begin to classify Specht modules by the number of empty layers in this filtration. This knowledge improved the utility of Schaper’s sum formula and we shall demonstrate how our classification leads to new decomposition numbers.

We also study extensions of Specht modules by the trivial module, giving a number of classes of modules for which no such extensions exist and giving an upper bound on the dimension of the Ext-group for all Specht modules. In the case of two part partitions we see that this bound is obtained and we explicitly construct these extensions. Our methods here are entirely combinatorial and remain in the setting of the symmetric group. This work motivates the study of modular combinatorial designs.

Combinatorial designs over the integers are well studied and have numerous applications while designs over fields of positive characteristic, modular, or p-ary designs, have received less attention. In this dissertation universal p-ary designs are defined and classified. Like their integral counterparts we anticipate a broad range of applications for the theory of modular designs, amongst which are the applications to extensions of Specht modules seen in this dissertation.