## Quiver Schur algebras for the linear quiver I

### Jun Hu and Andrew Mathas

#### Abstract

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 $$\mathcal O^\Lambda_n$$ 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. Finally, we give an LLT-like algorithm for computing the graded decomposition numbers of the quiver Schur algebras in characteristic zero when $$e=0$$.

Keywords: Cyclotomic Hecke algebras, Schur algebras, quasi-hereditary and graded cellular algebras, Khovanov-Lauda-Rouquier algebras.

: Primary 20C08; secondary 20C30, 05E10.

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 Monday, October 17, 2011