| dc.description.abstract | This thesis consists of two research projects on the spin representation theory of the symmetric group. In Chapters 2 and 3, we determine the modular decomposition of the spin representation of Sn indexed by the partition (n − 2, 2). Whilst James provided a characteristic-free construction of the linear representations of the symmetric group Sn, there is no analogous construction for the spin (or projective) representations of Sn, i.e. the linear representations of a double cover Sn+ of Sn. The most crucial open problem in the spin representation theory of Sn is determining the number of times each prime characteristic irreducible appears in the decomposition of the modular reduction of a characteristic 0 irreducible. Inspired by James’ description of the linear representations of Sn in terms of submodules and induced modules, recovering the Specht modules, Wales showed that inducing the basic representation from Sn-1+ to Sn+ provides an irreducible 2-modular representation other than the basic representation, leading to a description of the modular decomposition of the spin representations denoted by the partitions (n) and (n − 1, 1). We extend this method to determine the decomposition of the spin representation corresponding to (n − 2, 2). In Chapter 4, we establish combinatorial results about bar-core partitions. When p and q are coprime odd integers no less than 3, Olsson proved that if λ is a p-bar-core partition, then the q-bar-core of λ is again a p-bar-core. We establish a generalisation of this theorem: that the p-bar-weight of the q-bar-core of any bar partition λ is at most the p-bar-weight of λ. We go on to study the set of bar partitions for which equality holds and show that it is a union of orbits for an action of a Coxeter group of type C ̃(p−1)/2 × C ̃(q−1)/2. We also provide an algorithm for constructing a bar partition in this set with a given p-bar-core and q-bar-core. | en_US |
