Schur Subalgebras and an Application to the Symmetric Group

Let K be an infinite field of prime characteristic p and let d ≤ r be positive integers of the same parity satisfying a certain congruence condition. We prove that the Schur algebra S(2, d) is isomorphic to a subalgebra of the form eS(2, r)e, where e is a certain idempotent of S(2, r). Translating this result via Ringel duality to the symmetric groups Σd and Σr, we obtain lattice isomorphisms between Specht modules, between Young modules, and between permutation modules. Here modules labelled by the partitions (r − k, k) correspond to modules labelled by (d − k, k). This provides a representation theoretical interpretation for part of the fractal structures observed for the decomposition numbers of the symmetric groups corresponding to two-part partitions.