Linear quivers and the geometric setting of quantum GL n

Abstract

This paper presents a connection between the defining basis presented by Beilinson-Lusztig-MacPherson [1] in their geometric setting for quantum GL n and the isomorphism classes of linear quiver representations. More precisely, the positive part of the basis in [1] identifies with the defining basis for the relevant Ringel-Hall algebra; hence, it is a PBW basis in the sense of quantum groups. This approach extends to q-Schur algebras, yielding a monomial basis property with respect to the Drinfeld-Jimbo type presentation for the positive (or negative) part of the q-Schur algebra. Finally, the paper establishes an explicit connection between the canonical basis for the positive part of quantum GL n and the Kazhdan-Lusztig basis for q-Schur algebras.