Quiver Schur algebras for linear quivers

TL;DR

This paper introduces graded quasi-hereditary quiver Schur algebras for linear quivers, establishing their cellular structure, compatibility with known gradings, and providing algorithms for decomposition number calculations.

Contribution

It constructs new graded quasi-hereditary algebras for type A quivers and links them to existing structures, including category O and Koszul properties.

Findings

Algebras are quasi-hereditary graded cellular with explicit bases.

Compatibility of KLR grading with category O gradings established.

Provides an LLT-like algorithm for graded decomposition numbers.

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 $\mathcal{O}$ 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 $e=0$.