A generic multiplication in quantised Schur algebras

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.