Schur subalgebras II

Abstract

Let p be a prime, and k an infinite field of characteristic p. In [2], the author and Martin re-proved a result of Henke [4] in which the Schur algebra S(2 ,d) over k is shown to embed in the Schur algebra S(2 ,r) for certain values of d and r, corresponding to certain self-similarity properties of the decomposition matrices for S(2 ,r) . We also constructed embeddings of S(2 ,r) in S(2 ,r p)for all r, reflecting further the structure of the decomposition matrices. Here, an embedding is not necessarily an injective homomorphism of algebras, but simply a linear injection preserving the multiplication rule. In this paper we continue to study such embeddings, and examine their consequences for decomposition numbers. Essential results concerning Schur algebras can be found in the books of Green [3] and Martin [6]; further results and notation are taken from [2]. In Section 2 we construct embeddings