Quiver Schur algebras for linear quivers

We define a graded quasi-hereditary covering of the cyclotomic quiver Hecke algebras R^Λ_n of type A when e = 0 (the linear quiver) or e > n.We prove that these algebras are quasi-hereditary graded cellular algebras by giving explicit homogeneous bases for them. When e = 0, we show that the Khovanov-Lauda-Rouquier grading on the quiver Hecke algebras is compatible with the Koszul grading on the blocks of parabolic category O^Λ given by Backelin, building on the work of Beilinson, Ginzburg and Soergel. As a consequence, e = 0 our cyclotomic quiver Schur algebras are Koszul over fields of characteristic zero. Finally, we give an Lascoux-Leclerc-Thibonlike algorithm for computing the graded decomposition numbers of the cyclotomic quiver Schur algebras in characteristic zero.