On projective resolutions of simple modules over the Borel subalgebra S^+(n, r) of the Schur algebra S(n, r) for n ≤3

Abstract

We construct projective resolutions for simple modules over the Borel subalgebras S^+(2,r) and S^+(3,r) of the Schur algebras S(2,r) and S(3,r) over a field of positive characteristic. In the case S^+(2,r) we get a minimal projective resolution. We also find projective resolutions of minimal possible length for Weyl modules over the Schur algebra S(2,r), corresponding to the regular weights.