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

✍ Scribed by Christine Bessenrodt; Jørn B. Olsson

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 573 KB
• Volume: 129
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1997.1654

✦ Synopsis

Let S n be a double cover of the finite symmetric group S n of degree n, i.e., S n has a central involution z such that S n Â(z) & S n . An irreducible character of S n is called ordinary or spin according to whether it has z in its kernel or not.

The purpose of this paper is to determine the distribution of the spin characters of S n into 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.

1997 Academic Press

1. BACKGROUND AND PRELIMINARY RESULTS

For facts concerning the general representation theory of finite groups the reader is referred to [F, NT].