On Schur algebras, Ringel duality and symmetric groups

Abstract

In Erdmann and Henke (Math. Proc. Cambridge Philos. Soc., to appear) we determine precisely the degrees r for which the Schur algebra S(2, r) is its own Ringel dual. Here we study some applications: We classify uniserial Weyl modules and tilting modules. Based on Doty (J. Algebra 95 (1985) 373), we describe the submodule lattice of Specht modules labelled by two-part partitions and we classify uniserial Specht modules and Young modules labelled by two-part partitions. Moreover, we determine extensions for simple modules for the Ringel duals of arbitrary S(2, r). As a consequence we obtain corresponding results on symmetric groups.