On the combinatorics of core partitions and applications to the representation theory of the symmetric group.

We will start by attempting to generalise theorems on core-partitions for bar-core partitions. For example, it has been proved for two coprime positive integers s and t that the s-weight of the t-core of a partition is at most the s-weight of the partition itself, and it is also known that if s and t are also both odd then the s-bar-core of a t-bar-core is again a s-bar-core; we will try to show that the 's-bar-weight' of a t-bar-core of a partition is at most the 's-bar-weight' of the original partition.
Later, we will use these generalised theorems to establish results on the representation theory of double-covers of the symmetric group, as there is an intimate correspondence here with the combinatorics of bar-core partitions.