Decomposition matrices for spin characters of symmetric groups at characteristic \(3\).

*(English)*Zbl 0835.20018Let \(\widehat{S}_n\) 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.

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.

Reviewer: M.R.Darafsheh (Tehran)

##### MSC:

20C20 | Modular representations and characters |

05E10 | Combinatorial aspects of representation theory |