A generic multiplication in quantised Schur algebras

TL;DR

This paper introduces a new algebra structure in quantised Schur algebras through a generic multiplication, linking it to Hall algebras and providing a geometric realization of 0-Schur algebras' positive part.

Contribution

It defines a generic multiplication in quantised Schur algebras and establishes connections to Hall algebras and geometric realizations, offering a new basis for 0-Schur algebras.

Findings

A new algebra structure in quantised Schur algebras is constructed.

A geometric realization of the positive part of 0-Schur algebras is achieved.

A multiplicative basis for the positive part of 0-Schur algebras is obtained.

Abstract

We define a generic multiplication in quantised Schur algebras and thus obtain a new algebra structure in the Schur algebras. We prove that via a modified version of the map from quantum groups to quantised Schur algebras, defined by A. A. Beilinson, G. Lusztig and R. MacPherson, a subalgebra of this new algebra is a quotient of the monoid algebra in Hall algebras studied by M. Reineke. We also prove that the subalgebra of the new algebra gives a geometric realisation of a positive part of 0-Schur algebras. Consequently, we obtain a multiplicative basis for the positive part of 0-Schur algebras.