Generalised row and column removal results for decomposition numbers of spin representations of symmetric groups in characteristic \(2\) (Q6633080)

Let \(\mathsf{S}_{n}\) be the symmetric group on \(n\) letters. For a partition \(\lambda\) of \(n\) let \(S^{\lambda}\) be the corresponding Specht module, that is an irreducible representation of \(\mathbb{C}\mathsf{S}_{n}\) labeled by \(\lambda\), while for \(\mu\) a \(p\)-regular partition of \(n\) let \(D^{\mu}\) be the irreducible representation of \(\overline{\mathbb{F}}_{p}\mathsf{S}_{n}\) labeled by \(\mu\). Row and column removal results of \textit{G. D. James} [J. Algebra 71, 115-122 (1981; Zbl 0465.20010)] state that the multiplicity of \(D^{\mu}\) in the reduction modulo \(p\) of \(S^{\lambda}\) is equal to the multiplicity of \(D^{\overline{\mu}}\) in the reduction modulo \(p\) of \(S^{\overline{\lambda}}\), where \(\overline{\lambda}\) and \(\overline{\mu}\) are obtained from \(\lambda\) and \(\mu\) by removing their first rows or columns, provided the first rows or columns of \(\lambda\) and \(\mu\) have the same length.\N\NThe author proves generalised row and column removal results for decomposition numbers of spin representations of symmetric groups in characteristic \(2\), similar to \textit{S. Donkin}'s results for decomposition numbers of representations of symmetric groups [Math. Proc. Camb. Philos. Soc. 97, 57--62 (1985; Zbl 0557.20005)].