The 2-Blocks of the Covering Groups of the Symmetric Groups

LetŜnbe a double cover of the finite symmetric groupSnof degreen, i.e.,Ŝnhas a central involutionzsuch thatŜn/⦠z⦔≃Sn. An irreducible character ofŜnis calledordinaryorspinaccording to whether it haszin its kernel or not. The purpose of this paper is to determine the distribution of the spin characters ofŜninto 2-blocks. The methods applied here are essentially different from those applied to previous questions of this type. We also discuss some consequences of our main result for the decomposition numbers. An analogue of James' well-known result for the decomposition numbers of the symmetric groups is proved, providing also a generalization of a theorem of Benson [Ben, Theorem 1.2]. In Section 1 we present the background for our results and give some preliminaries. In Section 2 we give an explicit formula for the number of spin characters in a 2-block. We also prove a result about the weight of a block containing a given non-self-associate spin character which will be important for the proof of our theorem on the 2-block distribution of spin characters. Section 3 presents some fundamental combinatorial concepts used in Sections 4 and 5. The theorem concerning the spin characters in a given 2-block is proved in Section 4, and in Section 5 we present our results on the decomposition numbers.