Representations of symmetric groups by bad shapes (Q909787)

A diagram of n nodes is a subset D of \({\mathbb{N}}\times {\mathbb{N}}\) containing n elements. For an arbitrary such diagram it is possible to define a \(KS_ n\)-module \(S^ D\), where K is a ring and \(S_ n\) the symmetric group of degree n. If D is a diagram of a partition, then \(S^ D\) is a Specht module. The authors discuss methods of finding the composition factors or Specht series of modules corresponding to arbitrary diagrams. The results presented are true for an arbitrary ring K, but the discussion is primarily concerned with scalars from a field of characteristic 0. The results are illustrated by some examples.