Presenting Schur algebras

Motivated by the work of R. M. Green (1995), we obtain a presentation of Schur algebras (both the classical and quantized versions) in terms of generators and relations. The presentation is compatible with the usual presentation of the (quantized or classical) enveloping algebra of $g\ell_n$ . As a result, we obtain a new integral basis for Schur algebras which is a subset of Kostant's basis of the integral form of the enveloping algebra (or its q-analogue). Projection onto an appropriate component gives a new integral basis and a presentation for the Hecke algebra, compatible with the basis and presentation for the Schur algebra. Finally, we find a second presentation of Schur algebras which is similar to Luzstig's modified form of the quantized enveloping algebra.