On the decomposition matrices of the symmetric groups. II

We have long suspected the existence of two theorems about the decomposition matrices of the symmetric groups which involve the removal of the first row or the first column from a diagram. The purpose of this paper is to state and prove these theorems. It is interesting to note that we have been unable to construct proofs which lie entirely within the framework of the representation theory of symmetric groups, and the symmetric groups results are here deduced from two stronger theorems concerning the representation theory of the general linear group.