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

我们将首先尝试推广关于棒-核划分的核-划分的定理。例如，对于两个互质正整数S和t，证明了一个划分的t-核的S权至多就是这个划分本身的S-权，还证明了如果S和t都是奇数，则t-杆核的S-杆-芯又是S-杆-芯；我们将试图证明一个划分的一个t-棒核的‘S棒权’至多等于原划分的‘S-棒权’.然后，我们将利用这些推广的定理建立关于对称群的双覆盖表示理论的结果，因为这里与棒-核划分的组合学有密切的对应关系.

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.