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Decomposition matrices for spin characters of symmetric groups at characteristic $$3$$. (English) Zbl 0835.20018
Let $$\widehat{S}_n$$ denote either of the double covers of the symmetric group $$S_n$$ with $$\langle z\rangle$$ a central subgroup of order 2 in $$\widehat{S}_n$$. The irreducible characters of $$\widehat{S}_n$$ over an arbitrary field $$F$$ fall into two categories. Those with $$\langle z \rangle$$ in their kernels and those which do not have $$\langle z\rangle$$ in their kernels. The latter characters are called ordinary or modular spin characters of $$S_n$$ according to whether $$F$$ has characteristic 0 or non-zero respectively. In a series of papers G. D. James developed methods for finding the decomposition matrices of $$S_n$$. He also generalized the classical construction of Specht modules over a field of characteristic 0 to obtain irreducible modules for $$S_n$$ over a field of arbitrary characteristic.
In the present paper the authors attempt to obtain the shape of the decomposition matrix of a spin block of $$S_n$$ in the case $$p=3$$. In section 3 of the paper the authors describe the partitions of $$n$$ which label the modular spin characters. These partitions are called Schur regular partitions and Schur singular partitions. The section 4 of the paper contains a result called by the authors an “approximation matrix” whose consequence is the decomposition matrix of a spin block of $$S_n$$. Finally in section 5 they pose a number of questions and conjectures.

MSC:
 20C20 Modular representations and characters 05E10 Combinatorial aspects of representation theory
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