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The 2-blocks of the covering groups of the symmetric groups. (English) Zbl 0979.20014
Summary: Let \(\widehat S_n\) be a double cover of the finite symmetric group \(S_n\) of degree \(n\), i.e., \(\widehat S_n\) has a central involution \(z\) such that \(\widehat S_n/\langle z\rangle\simeq S_n\). An irreducible character of \(\widehat S_n\) 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 \(\widehat S_n\) 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 group is proved, providing also a generalization of a theorem of D. Benson [J. Lond. Math. Soc., II. Ser. 38, No. 2, 250-262 (1988; Zbl 0669.20005), 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.

20C30 Representations of finite symmetric groups
20C20 Modular representations and characters
20C15 Ordinary representations and characters
Full Text: DOI
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