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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.

MSC:
20C30 Representations of finite symmetric groups
20C20 Modular representations and characters
20C15 Ordinary representations and characters
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[1] Andrews, G.E.; Bessenrodt, C.; Olsson, J.B., Partition identities and labels for some modular characters, Trans. amer. math. soc., 344, 597-615, (1994) · Zbl 0806.05065
[2] Benson, D., Spin modules for symmetric groups, J. London math. soc. (2), 38, 250-262, (1988) · Zbl 0669.20005
[3] Bessenrodt, C.; Morris, A.O.; Olsson, J.B., Decomposition matrices for spin characters of symmetric groups at characteristic 3, J. algebra, 164, 146-172, (1994) · Zbl 0835.20018
[4] Brauer, R., On blocks and sections in finite groups, II, Amer. J. math., 90, 895-925, (1968)
[5] Cabanes, M., Local structure of thepsn, Math. Z., 198, 519-543, (1988) · Zbl 0646.20011
[6] Feit, W., The representation theory of finite groups, (1982), North-Holland Amsterdam · Zbl 0493.20007
[7] Fong, P.; Srinivasan, B., The blocks of the finite general linear and unitary groups, Invent. math., 69, 109-153, (1982) · Zbl 0507.20007
[8] Fong, P.; Srinivasan, B., The blocks of finite classical groups, J. reine angew. math., 396, 122-191, (1989) · Zbl 0656.20039
[9] Humphreys, J.F., Blocks of projective representations of the symmetric group, J. London math. soc. (2), 33, 441-452, (1986) · Zbl 0633.20007
[10] James, G.; Kerber, A., The representation theory of the symmetric group, (1981), Addison-Wesley Reading
[11] Morris, A.O., The spin representation of the symmetric group, Canad. J. math., 17, 543-549, (1965) · Zbl 0135.05602
[12] Morris, A.O.; Yaseen, A.K., Some combinatorial results involving shifted Young diagrams, Math. proc. Cambridge math. soc., 99, 23-31, (1986) · Zbl 0591.20010
[13] Nagao, H.; Tsushima, Y., Representations of finite groups, (1989), Academic Press New York
[14] Olsson, J.B., Lower defect groups in symmetric groups, J. algebra, 104, 37-56, (1986) · Zbl 0604.20016
[15] Olsson, J.B., Frobenius symbols for partitions and degrees of spin characters, Math. scand., 61, 223-247, (1987) · Zbl 0658.20012
[16] Olsson, J.B., The number of modular characters in certainp, Proc. London math. soc. (3), 65, 245-264, (1992) · Zbl 0784.20002
[17] L. Puig, The Nakayama conjecture and the Brauer pairs, Séminaire sur les groupes finis, Publ. Math. de l’U. Paris VII, 25, 171, 189
[18] Schur, I., Über die darstellung der symmetrischen und der alternierenden gruppe durch gebrochene lineare substitutionen, J. reine angew. math., 39, 155-250, (1911) · JFM 42.0154.02
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