Decomposition numbers of \(2\)-parts spin representations of symmetric groups in characteristic \(2\) (Q6588189)

An irreducible representation of a double cover of some symmetric group \(S_n\) is said to be a \textit{spin representation} if it cannot be viewed as a representation of \(S_n\). The decomposition numbers of representations for \(S_n\) itself, never mind its double cover, are in general not known outside of a few specific cases or families. One such family in characteristic \(2\) are the Specht modules which are indexed by partitions with at most two parts.\N\NIn this paper, the author extends this result to include spin representations with at most two parts by providing the exact decomposition numbers in most cases (Theorems 1.4, 1.5) and upper bounds in most others (Theorem 1.6). In particular, only one column of the corresponding part of the decomposition matrix remains unknown without an upper bound. The composition factors of the modular reduction of these representations are either other irreducibles corresponding to partitions with at most two parts, or an irreducible corresponding to the \textit{double} of such a partition.