In this thesis we study some subalgebras of the Schur algebra for the general linear group GLn(k) particularly the Schur algebra S(B+) for the Borel subgroup B+ of GLn(k). In many ways it is easier to work in S(B+) than in the more complicated algebra S(GLn(k). Using the properties of S(B+) we give a new treatment of the Weyl modules for GLn(k). We then construct 2-step minimal projective resolutions of the irreducible S(B+)-modules and from these we obtain very easily 2-step projective resolutions of the Weyl modules for GLn(k). We study the Cartan invariants of S(B+) and show that under certain conditions they satisfy an interesting identity. For particular cases of the field k and of the integer n we prove several results on minimal projective resolutions of the irreducible S(B+)-modules. The methods we use are combinatorial and do not involve algebraic group theory.