Quiver Schur algebras for linear quivers

Understand

We define a graded quasi-hereditary covering for the cyclotomic quiver Hecke algebras $\mathcal{R}^\Lambda_n$ of type $A$ when $e=0$ (the linear quiver) or $e\ge n$.

• We show that these algebras are quasi-hereditary graded cellular algebras by giving explicit homogeneous bases for them.
• When $e=0$ we show that the KLR grading on the quiver Hecke algebras is compatible with the gradings on parabolic category $\mathcal{O}$ previously introduced in the works of Beilinson, Ginzburg and Soergel and Backelin.
• As a consequence, we show that when $e=0$ our graded Schur algebras are Koszul over field of characteristic zero.