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

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