On the composition factors of the Specht modules (Q799790)

An important result in the p-modular representation theory of finite symmetric groups says that the composition factors of the Specht module \(S^{\mu}\), \(\mu\) a partition of n, are of the form \(D^{\lambda}\) where \(\lambda\) dominates \(\mu\) (for short: \(\lambda {\underline\triangleright }\mu)\) and both \(\mu\) and \(\lambda\) have the same p-core (for short: \(\mu\sim_ p\lambda)\). The present author extends \(\sim_ p\) to tableaux as follows. \(t\sim_ pt^*\) means that each \(m\leq n\) occupies a node of the same p-residue in t and \(t^*\). Connecting this with G. D. James' concept of ladder he proves for example that in case of \(t_ 1^{\lambda}\) being a ladder tableau and \(t_ 1^{\lambda}\sim t_ k^{\mu}\), then \(\mu {\underline\triangleleft }{\tilde\mu }{\underline\triangleleft }{\tilde\lambda }\) (\({\tilde\mu }\),\({\tilde\lambda }\) the unique p-regular partition which have in each ladder the same number of nodes as \(\mu\), \(\lambda)\). It follows an easy proof of a result of G. D. James which gives the leading element of each row of the decomposition matrix.