On the quiver and relations of the Borel Schur algebras

Let K be an infinite field of characteristic p ≥ 0 and let n, r be positive integers. Let S+(n, r) be the Borel Schur algebra over K, which is a sub-algebra of the Schur algebra S(n, r). We aim to give a description of the Borel Schur algebra S+ (n, r) by finding its quiver and relations. We give a complete description of the quiver and relations for S+(2, r). We also construct a family of embedding from S+(2, r) to S+(n, r + s) which induce embeddings of the corresponding quivers. This gives us some relations for S+(n, r) for n > 2.;We describe the quiver of S+( n, r) for both p = 0 and p > 0. We also describe some relations of special type for p > 0 and find all relations for p = 0.