An elementary construction of monomial bases of modified quantum affine ${\mathfrak{g}\mathfrak{l}}_n$

In 1990 Beilinson, Lusztig and MacPherson provided a geometric realization of modified quantum ${\mathfrak{g}\mathfrak{l}}_n$ and its canonical basis. A key step of their work is a construction of a monomial basis. Recently, Du and Fu provided an algebraic construction of the canonical basis for modified quantum affine ${\mathfrak{g}\mathfrak{l}}_n$, which among other results used an earlier construction of monomial bases using Ringel-Hall algebra of the cyclic quiver. In this paper, we give an elementary algebraic construction of a monomial basis for affine Schur algebras and modified quantum affine ${\mathfrak{g}\mathfrak{l}}_n$.